Abstract
Matroid models are used to approximate complex systems that can be used to solve problems in the real world. The main goal of this paper is to show how matrices and rough sets on simplicial complexes can be used to create new types of matroids called simplicial matroids. We will look at some of their material properties. Because of these results, we are interested in learning about circuit and base axioms, rank functions, and closure operators. We also give more equivalent relations that can be used to make other equivalent simplicial matroids, such as 2circuit simplicial matroids.
1 Introduction
The simplicial complex is a collection of some simplices or simplexes. A zerodimensional simplex is a vertex \((s^0)\). A onedimensional simplex is a line segment \((s^1)\). A two dimensional simplex is a field triangle \((s^2)\). A threedimensional simplex is a tetrahedron \((s^3)\) (Stolz 2014), there is two types of simplicial complexes the first one is the ordinary (undirected) simplicial complex which can be induced from ordinary (undirected) graph, if the graph has ncliques which contain nvertices and every vertex is connected to any other vertex. The second type is called the directed simplicial complex; it is generated from directed graph. That has a directed clique in which a collection of nvertices are connected all to all like the case in ordinary clique in an undirected graph, but where in every subclique there is a unique vertex that is a source, i.e., all directed edges in the subclique are pointing out of it, and unique vertex that is a sink, that is, all directed edges in the subclique are pointing into it. In other words, a set of n all to all connected vertices forming a subgraph in a graph G is a directed nclique if its vertices can be ordered linearly \(v_1,v_2\),\(\ldots \), \(v_n\) such that for every \(i<j\) there is a directed edge in the subgraph from \(v_i\) to \(v_j\). Restricted intersecting families on simplicial complex were studied by Wang in Wang (2021).
Rough sets first introduced by Pawlak (1982, 1991) has attracted increasing research interests in recent years. It has been applied in different fields, such as economics, engineering, medicine (El Atik and Wahba 2020, 2022), biology (Nawar et al. 2020), chemistry, banking, market research, speech recognition, information analysis (Hu et al. 2010; Qian et al. 2009), data analysis (Herawan et al. 2010; Zhan et al. 2020), material science (Pal and Mitra 2004), data mining (Huang et al. 2011), networking (Qian et al. 2009), linguistics and other fields. Relations can be used to illustrate graphs as finite topological structures (El Atik and Nasef 2020; El Atik and Hassan 2020; Nada et al. 2018). In theory, it has been strained to reflexive (Kondo 2005; Qin et al. 2008), similarity (Cattaneo and Ciucci 2002; El Atik 2020; Slowinski 2000), tolerance (Bartol et al. 2004; Ouyang et al. 2010; Skowron and Stepaniuk 1996; Skowron et al. 2012) and arbitrary relation based on rough sets (Diker 2010; Liu and Zhu 2008; Zhu 2007a), coveringbased on rough sets (Zhu 2007b; Zhu and Wang 2007), probabilistic rough sets (Yao 2010, 2011), fuzzy rough sets (Deng et al. 2007; Gong et al. 2008; Jensen and Shen 2004) and topology on rough sets (Lashin et al. 2005).
Matroids (Edmonds (1971); Mao (2006); Oxley (1993)) have been proposed as abstract extensions to linear independent vectors in vector spaces. They have wellfounded theories and a wide range of applications. Tsumoto et al. in Tsumoto and Tanaka (1994, 1996) have characterized relevance and irrelevance in empirical learning based on the relations between rough set and matroids theories. Furthermore, Tusmoto in Tsumoto (2002) has developed an induction formula algorithm induced by matroids and rough sets. In addition, Li and Liu (2012) have created an axiomatic arrangement of rough sets using matroidal advances. Furthermore, Wang et al. (2012, 2014) have demonstrated equivalence between 2circuit matroids and rough sets and provided an equivalent depiction for reducing uncertainty attributes via matroids. Some interesting results concerning the combination of matroids and rough sets can be found in the literature (Wang and Zhu 2011; Wang et al. 2013; Yang and Li 2006; Zhu and Wang 2011).
In this paper, matroidal structures of simplicial complexes are constructed by two methods. First, we establish special types of matrices that can be used to generate matroids called simplicial matroids. Circuits and base on simplicial matroids are defined and their properties are studied. Simplicial matroids satisfy circuit and base axioms. A closure operator, rank function and restriction matroids are studied with some examples. Second, we use rough sets to generate simplicial matroids through an equivalence relation on a universe set, that is the collection of all vertices, edges, triangles and tetrahedrons of simplicial complexes. The relationship between upper and lower approximations with respect to an equivalence relation, closure and interior operators for corresponding simplicial matroids is discussed. On the other side, we relate more equivalence relations to simplicial matroids. The relationship between two inductions with some examples is studied. This paper consists of five sections as follows: Sections 1 and 2 are introduction and preliminaries. Section 3 aimed to construct some new types of simplicial matroids in terms of matrices and some their properties are studied. In Section 4, simplicial matroids which are generated by rough sets are established. Also, equivalence relations are represented by \(01\) matrices which are efficient for fast parallel computing. Finally, Section 5 is devoted to defining new kinds of rough sets through the proposed simplicial matroids.
2 Preliminaries
Several fundamental terminologies of simplicial complexes, rough sets and matroids will be stated and studied throughout this section.
2.1 Simplicial complexes
Definition 1
(ksimplex Cavaliere et al. (2017)). A ksimplex \(s^k= (s_1^0\), \(s_2^0\), \(s_3^0\), \(\ldots \), \(s_{k+1}^0)\), is a set of independent abstract vertices \(s_1^0\) \(s_2^0\), \(s_3^0\), \(\ldots \), \(s_{k+1}^0\) that constitutes a convex hull of \(k+1\) points, where 0 is the dimension for vertices and k is the dimension of the simplex.
Definition 2
(hface Cavaliere et al. (2017)). Let \(s^k=\) \((s_1^0\), \(s_2^0\), \(s_3^0\), \(\ldots \), \(s_{k+1}^0)\) be a ksimplex. An hface is an hsimplex whose vertices are a subset of \(s_1^0\), \(s_2^0\), \(s_3^0\), \(\ldots \), \(s_{k+1}^0\) with cardinality \(h+1\).
Definition 3
(simplicial complex Cavaliere et al. (2017)). The simplicial complex \(\sigma \) is a finite set of simplices that satisfies conditions:

(i)
Any face of a simplex from \(\sigma \) is in \(\sigma \).

(ii)
The intersection of any two simplices is a face of each of them.
The dimension of \(\sigma \) corresponds to the highest dimension of its ksimplices.
Definition 4
(boundary Estrada and Ross (2018)). The boundary of a ksimplex consists of \((k+1)\)simplices with dimension \(k1\); for instance, the boundary of a 1simplex is two vertices and the boundary of a 2simplex is three edges.
Definition 5
(directly connected Cavaliere et al. (2017)). If the intersection of two simplices generates a nonempty hface with \(h\le k\), then they are directly linked.
Definition 6
(hconnected Cavaliere et al. (2017)). Let \(A=s_1^k, s_2^h,\ldots , s_{m1}^h, s_m^k= B\) be nonempty ksimplices. Then, A and B are hconnected if each pair of successive \(s_i^h\) and \(s_{i+1}^h\) shares an hface with \(i=0, 1, 2,\) \(\ldots ,\) \(m1\), where \(h\le k\).
From Definitions 5 and 6, it is noted that each directly connected is hconnected.
2.2 Rough sets
Some basic ideas and characteristics of rough sets and approximation operators are discussed.
Definition 7
Pawlak (1982). Let U be a finite universal set, \(X\subseteq U\), and R be an equivalence relation on U. Then, a lower (resp., upper) approximation of X with respect to R is defined by

\({\underline{R}}(X)=\) \(\{x\in U:[x]_R \subseteq X \}\),

\({\overline{R}}(X)=\) \(\{x\in U:[ x ]_R\cap X\ne \phi \}\),
where \([x]_R=\) \(\{y\in U: xRy\}\) is the x equivalence class with regard to R.
According to Pawlak’s definition, X is a rough set if \({\underline{R}} (X)\ne {\overline{R}} (X)\). All equivalence classes with regard to R shall be indicated by U/R.
Proposition 1
Yu et al. (2013). Let \(X^c\) be the complement of X in U. Pawlak’s rough sets have the following properties:
(L1) \({\underline{R}} (X)\subseteq X\).  (U1) \(X \subseteq {\overline{R}} (X)\). 
(L2) \({\underline{R}} (\phi )= \phi \).  (U2) \({\overline{R}} (\phi )= \phi \). 
(L3) \({\underline{R}} (U)= U\).  (U3) \({\overline{R}} (U)= U\). 
(L4) \({\underline{R}} (X\cap Y)=\) \({\underline{R}}(X)\cap \) \({\underline{R}}(Y)\).  (U4) \({\overline{R}} (X\cup Y)=\) \({\overline{R}}(X)\cup \) \({\overline{R}}(Y)\). 
(L5) If \(X\subseteq Y\), then \({\underline{R}}(X)\subseteq \) \({\underline{R}}(Y)\).  (U5) If \(X\subseteq Y\), then \({\overline{R}}(X)\subseteq \) \({\overline{R}}(Y)\). 
(L6) \({\underline{R}}(X)\cup \) \({\underline{R}}(Y)\) \(\subseteq \) \({\underline{R}} (X\cup Y)\).  (U6) \({\overline{R}}(X)\cap \) \({\overline{R}}(Y)\) \(\supseteq \) \({\overline{R}} (X\cap Y)\). 
(L7) \({\underline{R}}(X^c )= ({\overline{R}}(X))^c\).  (U7) \({\overline{R}}(X^c )= ({\underline{R}}(X))^c\). 
(L8) \({\underline{R}}({\underline{R}}(X))=\) \({\underline{R}}(X)\).  (U8) \({\overline{R}}({\overline{R}}(X))=\) \({\overline{R}}(X)\). 
(L9) \({\underline{R}}(({\underline{R}}(X))^c)=\) \(({\underline{R}}(X))^c\).  (U9) \({\overline{R}}(({\overline{R}}(X))^c)=\) \(({\overline{R}}(X))^c\). 
Definition 8
Pawlak (1982); Yao (2011). The boundary region for \(X\subseteq U\) is represented by \(BN_g (X)\) and is given by \(BN_g (X)=\) \({\overline{R}}(X) {\underline{R}}(X)\). In other words, \(BN_g (X)=\) \(\bigcup \{[x]_R \in U/R: [x]_R\cap X\ne \phi \wedge [x]_R \nsubseteq X\}\).
2.3 Matroids
Definition 9
(matroids Oxley (1993)). Let \(P({\mathcal {U}})\) be the power set of a ground set \({\mathcal {U}}\) and \({\mathcal {I}}\) be a collection of subsets of \({\mathcal {U}}\) satisfying the following conditions:

(I1)
\(\phi \) is an element of \({\mathcal {I}}\).

(I2)
If \(A\in {\mathcal {I}}\) and \(B\subseteq A\), then \(B\in {\mathcal {I}}\).

(I3)
If \(A, B\in {\mathcal {I}}\) and \(A<B\), then \(\exists \) \(b\in BA\) such that \(A\cup \{b\}\in {\mathcal {I}}\), where A is the cardinality of A. Each element in \({\mathcal {I}}\) is said to be an independent set. Each element in \(P({\mathcal {U}}) {\mathcal {I}}\) is dependent. The pair \({\mathcal {M}}=\) \(({\mathcal {U}}, {\mathcal {I}})\) is called a matroid.
Definition 10
(circuits Oxley (1993)). A minimal dependent set of \({\mathcal {M}}\) is said to be a circuit. The family of all circuits is denoted by \({\mathcal {C}}({\mathcal {M}})\).
Proposition 2
(circuit axioms Oxley (1993)). Let \({\mathcal {C}}\) be a collection of subsets of \({\mathcal {U}}\). Then, \(\exists \) \({\mathcal {M}}=({\mathcal {U}},{\mathcal {I}})\) such that \({\mathcal {C}}=\) \({\mathcal {C}}({\mathcal {M}})\) iff \({\mathcal {C}}\) satisfies conditions:

(C1)
\(\phi \) is not in \({\mathcal {C}}\).

(C2)
If \(C_1\subseteq C_2\), then \(C_1= C_2\), \(\forall \) \(C_1, C_2\in {\mathcal {C}}\).

(C3)
If \(C_1\ne C_2\), and \(x\in C_1\cap C_2\), then \(\exists \) \(C_3\in {\mathcal {C}}\) such that \(C_3\subseteq (C_1\cup C_2)\{x\}\), \(\forall \) \(C_1, C_2\in {\mathcal {C}}\).
Definition 11
(base Oxley (1993)). The base of a matroid \({\mathcal {M}}\) is the maximal independent sets of \({\mathcal {M}}\). The collection of all bases of M is denoted by \({\mathcal {B}}({\mathcal {M}})\).
Proposition 3
(base axioms Oxley (1993)). Let \({\mathcal {B}}\) be a collection of subsets of U. Then, \(\exists \) \({\mathcal {M}}=({\mathcal {U}},{\mathcal {I}})\) such that \({\mathcal {B}}= B({\mathcal {M}})\) iff it satisfies conditions:

(B1)
\(\phi \) is not in \({\mathcal {B}}\).

(B2)
If \(A, B \in {\mathcal {B}}\) and \(x\in AB\), then \(\exists \) \(y\in BA\) such that \((A \{x\})\) \(\cup \{y\}\in {\mathcal {B}}\).
Definition 12
(rank function Lai (2002)). A rank function \(r_{{\mathcal {M}}}\) of \({\mathcal {M}}\) for \(X\subseteq {\mathcal {U}}\) is defined by \(r_{{\mathcal {M}}}(X)=\) \(Max~\{l: l\in {\mathcal {I}}~~\text{ and }~~l\subseteq X\}\).
Definition 13
(closure operator Lai (2002)). A closure operator \(Cl_{{\mathcal {M}}}\) of \({\mathcal {M}}\) is defined by \(Cl_{{\mathcal {M}}}(X)=\) \(\{u\in {\mathcal {U}}:\) \(r_{{\mathcal {M}}}(X)=\) \(r_{{\mathcal {M}}}(X\cup \{u\})\}\). \(Cl_{{\mathcal {M}}}(X)\) is said to be the closure of X in \({\mathcal {M}}\).
In terms of the circuits of a matroid \({\mathcal {M}}\), there is an analogous definition for closure operator is given in Oxley (1993).
Definition 14
\(Cl_{{\mathcal {M}}}(X)=\) X \(\cup \) \(\{u\in {\mathcal {U}}:\) \(\exists \) \(C\in {\mathcal {C}}({\mathcal {M}})\) such that \(u\in C\) \(\subseteq \) \(X\cup \{u\}\}\), \(\forall \) \(X\subseteq {\mathcal {U}}\).
Proposition 4
(2circuit matroids Wang et al. (2014)). \({\mathcal {M}}\) is a 2circuit matroid if and only if the following are held:

(i)
\(Cl_{{\mathcal {M}}}(\phi )=\) \(\phi \).

(ii)
\(Cl_{{\mathcal {M}}}(X\cup Y)=\) \(Cl_{{\mathcal {M}}}(X)\cup Cl_{{\mathcal {M}}}(Y)\), \(\forall \) \(X,Y\in {\mathcal {U}}\).
Corollary 1
Wang et al. (2014). \(Cl_{{\mathcal {M}}}(X\cup Y)= Cl_{{\mathcal {M}}}(X)\cup Cl_{{\mathcal {M}}}(Y)\), \(\forall \) \(X,Y\in {\mathcal {U}}\) if and only if \(C\le 2\), \(\forall \) \(C\in {\mathcal {C}}({\mathcal {M}})\).
3 Matroids on simplicial complex via matrices
In this section, we will build several matrices on a simplicial complex \(\sigma \). These matrices will be used to generate new kinds of matroids known as simplicial matroids, and their characteristics such as circuits, bases, closure and rank function will be investigated.
Definition 15
Let \(\sigma \) be a simplicial complex and represented as a union of their simplices. If simplices approach to \(S^0\), \(S^1\), \(S^2\), \(S^3\), \(\ldots \), where

\(S^0=\) \(\{s_i^0: i\in I_1\}\), in which each point is called a 0face;

\(S^1=\) \(\{s_j^1: j\in I_2\}\), in which each point is called a 1face;

\(S^2=\) \(\{s_k^2: k\in I_3\}\), in which each point is called a 2face;

\(S^3=\) \(\{s_m^3: m\in I_4\}\), in which each point is called a 3face, respectively, and so on. The universal set of \(\sigma \) is \({\mathcal {U}}_\sigma =\) \(S^0 \cup S^1 \cup S^2 \cup S^3 \cup \ldots \), where \(I_1, I_2, I_3, I_4, \ldots \) are indices.
Definition 16
Let \(s_j^n\) and \(s_i^{n1}\) be distinct simplices in \(\sigma \) with dimensions n and \(n1\), respectively. A matrix \({\mathcal {D}}\) is defined by
Remark 1
The dimension of the simplicial complex is equal to the number of matrices that it induces. This is shown in Examples 1 and 2.
Example 1
Figure 1 depicts a 2simplicial complex \(\sigma _1\) with just one 2face \(s_1^2\), three 1faces \(s_1^1, s_2^1, s_3^1\) and three 0faces \(s_1^0, s_2^0, s_3^0\). The universal set has the following form: \({\mathcal {U}}_\sigma =\) \(\{s_1^0, s_2^0, s_3^0, s_1^1, s_2^1, s_3^1, s_1^2\}\). According to Remark 1, the dimension of \(\sigma _1\) is 2. Thus, there are two matrices.
Example 2
Figure 2 shows a 2simplicial complex \(\sigma _2\) with two 2faces \(s_1^2, s_2^2\), five 1faces \(s_1^1, s_2^1, s_3^1, s_4^1, s_5^1\), and four 0faces \(s_1^0, s_2^0, s_3^0, s_4^0\). Because the dimension of \(\sigma _2\) is 2, there are two matrices.
The following describes the fundamental idea of simplicial matroids, which is based on linear independence for columns of matrices.
Definition 17
Let \({\mathcal {V}}_\sigma \subseteq {\mathcal {U}}_\sigma \), also known as a ground set be the set of column labels of matrix \({\mathcal {D}}_\ell \), where \(\ell =\) \(1,2,\ldots \), based on the number of matrices for \(\sigma \), and \({\mathcal {W}}\) be the set of linearly independent columns of \({\mathcal {V}}_\sigma \). Then, \({\mathcal {M}}_\sigma ({\mathcal {W}})=\) \(({\mathcal {V}}_\sigma , {\mathcal {W}})\) is referred to as a simplicial matroid on \({\mathcal {V}}_\sigma \) (\({\mathcal {M}}_\sigma \), for short).
Proposition 5
A simplicial matroid \({\mathcal {M}}_\sigma ({\mathcal {W}})\) on \({\mathcal {W}}\) is a matroid.
Proof
According to Definition 9, the conditions I1 and I2 are obvious. It is sufficient to prove I3. Consider \(I, J\in {\mathcal {W}}\), where \(I=\) \(\{s_1^{n}, s_2^{n}, \ldots , s_{t}^{n}\}\) and \(I\subseteq J\). This means that \(I<J\). Suppose that \(I\cup \{j\}\) is dependent for any \(j \in JI\). Then, \(c_{1} s_{1}^{n}\) \(+\) \(c_{2} s_{2}^{n}\) \(+\) \(\ldots \) \(+ c_{t} s_{t}^{n}\) \(+ c j =0\), where \(c_i, c\) for \(i=1,2 \ldots , t\) are scalars. So, \(j=\) \(\frac{c_1}{c} s_1^{n}  \frac{c_2}{c} s_2^{n} \ldots \frac{c_t}{c} s_t^{n}\). This means that j \(\in \) Span \((s_1^{n}, s_2^{n}, \ldots , s_t^{n})\). Since \(j\in JI\), \(JI\) \(\subseteq \) Span I and \(I\subseteq \) Span I. Hence, \((JI)\cup I\) \(\subseteq \) Span I and so \(J\subseteq \) Span I. Then, Span J \(\subseteq \) Span I and so \(J=\) \(\{s_1^{n}, s_2^{n}, \ldots , s_{t^{'}}^{n}\}\); \(t^{'} \le t\) means that \(J\le I\), which gives a contradiction. Therefore, \(I\cup \{j\}\) \(\in I\). \(\square \)
Remark 2
Any nonindependent subset in \({\mathcal {V}}_\sigma \) in \({\mathcal {M}}_\sigma ({\mathcal {W}})\) is called dependent. The circuits of \({\mathcal {M}}_\sigma \) are obviously a subset of linearly dependent columns of \({\mathcal {V}}_\sigma \).
Example 3
(continued for Example 2).
The simplicial complex \(\sigma _2\) yields two matrices, \({\mathcal {D}}_1\) and \({\mathcal {D}}_2\). A simplicial matroid \({\mathcal {M}}_\sigma \) may be introduced from each matrix. For matrix \({\mathcal {D}}_1\), \({\mathcal {V}}_{\sigma _2}=\) \(\{s_1^1, s_2^1,\) \(s_3^1, s_4^1, s_5^1\}\), and \({\mathcal {W}}_{1}=\) \(\{\phi ,\) \(\{s_1^1\},\) \(\{s_2^1\},\) \(\{s_3^1\},\) \(\{s_4^1\},\) \(\{s_5^1\},\) \(\{s_1^1, s_2^1\},\) \(\{s_1^1, s_3^1\},\) \(\{s_1^1, s_4^1\},\) \(\{s_1^1, s_5^1\},\) \(\{ s_2^1,\) \( s_3^1\},\) \(\{s_2^1, s_4^1\},\) \(\{s_2^1, s_5^1\},\) \(\{s_3^1, s_4^1\},\) \(\{s_3^1, s_5^1\},\) \(\{s_4^1, s_5^1\},\) \(\{s_1^1, s_2^1, s_3^1\},\) \(\{s_1^1,\) \(s_2^1, s_4^1\},\) \(\{s_1^1, s_2^1, s_5^1\},\) \(\{s_1^1, s_3^1, s_4^1\},\) \(\{s_1^1,\) \( s_3^1, s_5^1\},\) \(\{s_3^1, s_4^1, s_5^1\},\) \(\{s_2^1, s_3^1, s_4^1\},\) \(\{s_2^1, s_3^1, s_5^1\},\) \(\{s_1^1, s_4^1, s_5^1\},\) \(\{s_1^1, s_2^1, s_3^1, s_4^1\},\) \(\{s_2^1, s_4^1, s_5^1\},\) \(\{s_1^1, s_2^1,\) \(s_3^1,s_5^1\},\) \(\{s_2^1,\) \(s_3^1, s_4^1, s_5^1\},\) \(\{s_1^1, s_3^1,\) \(s_4^1, s_5^1\}\}\). By similarity, for matrix \({\mathcal {D}}_2\), we have \({\mathcal {V}}^{'}_{\sigma _2}=\) \(\{s_1^2, s_2^2\}\) and \({\mathcal {W}}_{2}=\) \(\{\phi ,\) \(\{s_1^2\},\) \(\{s_2^2\},\) \(\{s_1^2, s_2^2\}\}\).
Now, some fundamental ideas about a simplicial matroid \({\mathcal {M}}_\sigma ({\mathcal {W}})\) will be discussed.
Definition 18
Let \({\mathcal {V}}_\sigma \) be a ground set and \({\mathcal {A}}\subseteq P({\mathcal {V}}_\sigma )\). The following families induced by \({\mathcal {A}}\) are defined by the following:

(i)
\(Opp({\mathcal {A}})=\) \(\{X \subseteq {\mathcal {V}}_\sigma :\) \(X\notin {\mathcal {A}}\}\).

(ii)
\(Upp({\mathcal {A}})=\) \(\{X \subseteq {\mathcal {V}}_\sigma :\) \(\exists \) \(A\in {\mathcal {A}}~\text{ such } \text{ that }~A\subseteq X\}\).

(iii)
\(Low({\mathcal {A}})=\) \(\{X \subseteq {\mathcal {V}}_\sigma :\) \(\exists \) \(A\in {\mathcal {A}}~\text{ such } \text{ that }~X\subseteq A\}\).

(iv)
\(Max({\mathcal {A}})=\) \(\{X \in {\mathcal {A}}:\) \(\forall \) \(Y \in {\mathcal {A}}, X \subseteq Y \Rightarrow X= Y\}\).

(v)
\(Min({\mathcal {A}})=\) \(\{X\in {\mathcal {A}}:\) \(\forall \) \(Y\in {\mathcal {A}}, Y\subseteq X \Rightarrow X= Y\}\).
Definition 19
A circuit of a simplicial matroid \({\mathcal {M}}_\sigma ({\mathcal {W}})\) is the minimal of dependent sets of \({\mathcal {M}}_\sigma ({\mathcal {W}})\). The class of all circuits of \({\mathcal {M}}_\sigma ({\mathcal {W}})\) is denoted by \({\mathcal {C}}({\mathcal {M}}_\sigma )\) such that \({\mathcal {C}}({\mathcal {M}}_\sigma )=\) \(Min(P({\mathcal {V}}_\sigma ) {\mathcal {W}})\), where \(P({\mathcal {V}}_\sigma ) {\mathcal {W}}\) is a family of dependent sets.
Example 4
(continued for Example 3). The circuit for \({\mathcal {M}}_{\sigma _2} ({\mathcal {W}}_1)\) is \({\mathcal {C}}=\) \(\{s_1^1, s_2^1, s_4^1, s_5^1\}\) and the circuit for \({\mathcal {M}}_{\sigma _2} ({\mathcal {W}}_2)\) is \({\mathcal {C}}=\) \(\phi \).
Circuits can be used to give a simplicial matroid \({\mathcal {M}}_\sigma ({\mathcal {W}})\) as shown in Proposition 6.
Proposition 6
Let \({\mathcal {C}}\) be a class of \({\mathcal {V}}_\sigma \). \({\mathcal {C}}\) satisfies the circuit axioms if and only if we build a simplicial matroid \({\mathcal {M}}_\sigma ({\mathcal {W}})\) with \({\mathcal {C}}=\) \({\mathcal {C}}({\mathcal {M}}_\sigma )\).
Proof
Let \({\mathcal {M}}_\sigma =\) \(({\mathcal {V}}_\sigma , {\mathcal {W}})\) be a simplicial matroid and \({\mathcal {C}}({\mathcal {M}}_\sigma )\) be a collection of circuits. The conditions of circuits will be proved

(C1)
Since \(\phi \in {\mathcal {W}}\), then \(\phi \notin {\mathcal {C}}\).

(C2)
Let \(C_1, C_2 \in {\mathcal {C}}\). Then, by Definition 19, \(C_1, C_2\in \) \(Min (P({\mathcal {V}}_\sigma ) {\mathcal {W}})\). If \(C_1 \subseteq C_2\), then \(C_1= C_2\).

(C3)
Let \(C, D \in {\mathcal {C}}\). Assume that there is no a circuit subset from \((C\cup D)\{e\}\) such that \(e\in C\cap D\). This means that \((C\cup D)\{e\}\) is an independent set. Take \(f\in C D\). Then, \(C\{f\}\) is not a circuit and so is an independent set. Hence, \(C\{f\}<(C\cup D)\{e\}\). Since \({\mathcal {M}}_\sigma \) satisfies I3 in Definition 9, then there is an independent set \((C\{f\}) \cup \{a\}\), where \(a\in D\) and so \((C\{f\})\cup \{a\}<(C\cup D)\{e\}\). In addition, there is an independent set \((C\{f\})\cup (\{a\}\cup \{b\})\), where \(b\in D\). Continue in the same manner for all elements of D; we have \((C\{f\})\cup D=\) \((C\cup D)\{e\}\). Therefore, \((C\{f\})\cup D\) is an independent set, which contradicts with the dependence of C and D. Conversely, consider that \({\mathcal {C}}\) satisfies C1, C2, and C3. We need to generate a simplicial matroid \({\mathcal {M}}^{'}_{\sigma }=\) \(({\mathcal {V}}_\sigma , {\mathcal {W}}^{'})\) with a circuit \({\mathcal {C}}\). (I’1) From C1, since \(\phi \notin {\mathcal {C}}\), then \(\phi \in {\mathcal {W}}^{'}\). (I’2) If \(S_1\in {\mathcal {W}}^{'}\), then, there is no a circuit subset of \(S_1\). In other words, if \(S_2\subseteq S_1\), then \(S_2\) has no such circuit and so \(S_2\in {\mathcal {W}}^{'}\). (I’3) Let \(C, D\in {\mathcal {C}}\) such that \(x\in C \cap D\). Using C3, \(C\{f\}\) and \(D\{g\}\) are independent sets, for \(f\in C D\) and \(g\in D C\). Now, let \(C\{f\}<D\{g\}\) and \(S=\) \((C\{f\})\cup (D\{g\})\) is an independent set. Assume that \((C\{f\})\cup \{j\} \in {\mathcal {C}}\), where \(j\in (D\{g\})(C\{f\})\). Then, there are two circuits \((C\{f\})\cup \{j\}\) and D such that \([(C\{f\})\cup \{j\}] \cap D=\) \(\{x,j\}\). By C3, there is \(C_1\in {\mathcal {C}}\) such that \(C_1 \subseteq \) \([(C\{f\})\cup \{j\} \cup D] \{j\}\subseteq \) \((C\{f\})\cup (D\{j\})=\) S, which contradict with the independence of S. Now, we prove that \({\mathcal {C}}=\) \({\mathcal {C}}({\mathcal {M}}_\sigma )\). Consider \({\mathcal {W}}^{'}=\) \((Upp ({\mathcal {C}}))^{c}\) such that \(Upp~({\mathcal {C}})=\) \(\{X\subseteq {\mathcal {V}}_\sigma :\) \(\exists \) \(C\in {\mathcal {C}}\) and \(C\subseteq X\}\). Since \({\mathcal {C}}\subseteq Upp({\mathcal {C}})\), then we need to prove that \({\mathcal {C}}\) is the minimum set for \(Upp({\mathcal {C}})\). Sine \(\forall \) \(C_{1}, C_{2}\in {\mathcal {C}}\) and \(C_{1}\subseteq C_{2}\). Using C2, we get \(C_{1}= C_{2}\) which means that \({\mathcal {C}}\) is the minimum set for \(Upp({\mathcal {C}})\). Also, we prove that if \(C_{1} \in {\mathcal {C}}\), then \(\forall \) \(x\in C_{1}\), then we get \(C_{1}\{x\}\in {\mathcal {W}}^{'}\). From C2, if \(C_{1} \in {\mathcal {C}}\) such that \(C_{2}\in {\mathcal {V}}_\sigma \), \(C_{2}\subset C_{1}\), and \(C_{2}\ne C_{1}\), then \(C_{2}\in {\mathcal {W}}^{'}\). Therefore, \({\mathcal {C}}\) contains all circuits. The proof is completed.
\(\square \)
Definition 20
Let \({\mathcal {M}}_\sigma ({\mathcal {W}})\) be a simplicial matroid. For all \(A\subseteq {\mathcal {V}}_\sigma \), the closure of A with respect to \({\mathcal {M}}_\sigma \) is defined by \(Cl_{{\mathcal {M}}_\sigma }(A)=\) A \(\cup \) \(\{s_i^n\in {\mathcal {V}}_\sigma :\) \(\exists \) \(C\in {\mathcal {C}}({\mathcal {M}}_\sigma )\) such that \(s_i^n\) \(\in C\subseteq \) \(A\cup \) \(\{s_i^n\}\}\). \(Cl_{{\mathcal {M}}_\sigma }\) is said to be a closure operator with respect to \({\mathcal {M}}_\sigma \).
A closure operator \(Cl_{{\mathcal {M}}_\sigma }\) can satisfy Kuratowski closure operator (Oxley 1993). In the following, it is easy to prove Proposition 7. So, the proof is omitted.
Proposition 7
Let \(Cl: P({\mathcal {V}}_\sigma ) \rightarrow P({\mathcal {V}}_\sigma )\) be an operator. Then, there is a simplicial matroid \({\mathcal {M}}_\sigma \) such that \(Cl=\) \(Cl_{{\mathcal {M}}_\sigma }\) if and only if Cl satisfies the following conditions:

(CL1)
\(\forall \) \(A\subseteq {\mathcal {V}}_\sigma \), \(A\subseteq Cl(A)\).

(CL2)
\(\forall \) \(A\subseteq B\), \(Cl(A) \subseteq Cl(B)\).

(CL3)
\(\forall \) \(A\subseteq {\mathcal {V}}_\sigma \), \(Cl(Cl(A))= Cl(A)\).

(CL4)
\(\forall \) \(s_i^n, s_j^n\in {\mathcal {V}}_\sigma \) and \(A\subseteq {\mathcal {V}}_\sigma \), if \(s_j^n\in Cl(A \cup \{s_i^n\}) Cl(A)\), then \(s_i^n\in Cl (A\cup \{s_j^n\})\).
Remark 3
Let \({\mathcal {M}}_\sigma \) be a simplicial matroid such that \(C\le 2\), \(\forall \) \(C\in {\mathcal {C}}({\mathcal {M}}_\sigma )\). Then, by Corollary 1, the condition CL4 in Proposition 7 has the form \(s_j^n\in Cl (\{s_i^n\})\), for all \(s_i^n, s_j^n\in {\mathcal {V}}_\sigma \).
Example 5
(continued for Example 3).
Consider the simplicial matroid \({\mathcal {M}}_{\sigma _2}({\mathcal {W}}_1)\) with \(A=\) \(\{s_1^1,\) \(s_3^1,\) \(s_4^1\}\). Then, \(Cl_{{\mathcal {M}}_{\sigma _2}}(A)\) \(=\{s_1^1,s_3^1,s_4^1\}\), while the closure of any set A with regard to \({\mathcal {M}}_{\sigma _2}\) \(({\mathcal {W}}_2)\) is A since the circuit equals \(\phi \).
There is another method to define the closure operator for any subset \(A\subseteq {\mathcal {V}}_\sigma \). That depends on rank function of simplicial matroids.
Definition 21
Let \({\mathcal {M}}_\sigma \) be a simplicial matroid. The rank function for any \(A\subseteq {\mathcal {V}}_\sigma \) is defined by \(r_{{\mathcal {M}}_\sigma }(A)=\) Max \(\{S:\) \(S\subseteq A\) and \(S\in {\mathcal {W}}\}\).
Definition 22
The closure operator of \({\mathcal {M}}_\sigma \) is given by \(Cl_{{\mathcal {M}}_\sigma }(A)=\) \(\{s_i^n\in {\mathcal {V}}_\sigma : r_{{\mathcal {M}}_\sigma }(A)=\) \(r_{{\mathcal {M}}_\sigma }(A\cup \{s_i^n\})\}\), for any \(A\subseteq {\mathcal {V}}_\sigma \).
According to Definition 22 and Example 5, the closure of \(A=\) \(\{s_1^1, s_3^1, s_4^1\}\) is \(Cl_{{\mathcal {M}}_{\sigma _2}}(A)=\) \(\{s_1^1, s_3^1, s_4^1\}\) in relation to \({\mathcal {W}}_{1}\).
Definition 23
Let \({\mathcal {M}}_\sigma =\) \(({\mathcal {V}}_{\sigma }, {\mathcal {W}})\) be a simplicial matroid. A maximal independent set of \({\mathcal {W}}\) is called a base of \({\mathcal {M}}_\sigma \). The class of all bases of \({\mathcal {M}}_\sigma \) is denoted by \({\mathcal {B}}({\mathcal {M}}_\sigma )=\) Max\(({\mathcal {W}})\).
Example 6
(continued for Example 3). The bases \({\mathcal {B}}({\mathcal {M}}_{\sigma _2}({\mathcal {W}}_1))=\) \(\{\{s_1^1,\) \(s_2^1,\) \(s_3^1, s_4^1 \}\), \(\{s_1^1,\) \(s_2^1,s_3^1,\) \(s_5^1\}\), \(\{ s_2^1, s_3^1,\) \(s_4^1, s_5^1\}\), \(\{s_1^1, s_3^1,\) \(s_4^1, s_5^1\}\}\). Moreover, the base for \({\mathcal {M}}_{\sigma _2}\) is \({\mathcal {B}}({\mathcal {M}}_{\sigma _2}({\mathcal {W}}_2))=\) \(\{s_1^2, s_2^2\}\).
Proposition 8
Let \({\mathcal {B}}\) be a class of subsets of \({\mathcal {V}}_\sigma \). Then, there is a simplicial matroid \({\mathcal {M}}_\sigma \) such that \({\mathcal {B}}=\) \({\mathcal {B}}({\mathcal {M}}_\sigma )\) if and only if \({\mathcal {B}}\) fulfills base axioms.
Proof
Let \({\mathcal {M}}_\sigma \) be a simplicial matroid with a base \({\mathcal {B}}\). By Proposition 3, if \({\mathcal {B}}\) is a maximal in \({\mathcal {W}}\), then \({\mathcal {B}}\ne \phi \). Since \(A, B\in {\mathcal {B}}\), \(a\in AB\), and \(b\in BA\), \(A\{a\}\in {\mathcal {W}}\), \(B\in {\mathcal {W}}\), and \(A\{a\}<B\). By Definition 9(I3), since \(b\in B\) and \(b\notin A\{a\}\), \((A\{a\})\cup \{b\}\in {\mathcal {W}}\) and \((A\{a\})\cup \{b\}=\) A. Therefore, \((A\{a\})\cup \{b\} \in {\mathcal {B}}\). Conversely, suppose that \({\mathcal {B}}\) satisfies (B1) and (B2) in Proposition 3. We prove that there exists a simplicial matroid \({\mathcal {M}}_\sigma \) with a base \({\mathcal {B}}\). Assume that \({\mathcal {W}}=\) \(\{S: S\subseteq B\) for some \(B\in {\mathcal {B}}({\mathcal {M}}_\sigma )\}\). First, we prove that all sets in \({\mathcal {B}}({\mathcal {M}}_\sigma )\) have the same size. If \(S_1,S_2 \in {\mathcal {B}}({\mathcal {M}}_\sigma )\) are minimal, then \(S_1=\) \(S_2=\) 1. In general, suppose that \(S_1,S_2 \in {\mathcal {B}}({\mathcal {M}}_\sigma )\); by Proposition 3, if \(a \in S_1S_2\), then there is \(b\in S_2S_1\) such that \((S_1\{a\})\cup \{b\}\) \(\in \) \({\mathcal {B}}({\mathcal {M}}_\sigma )\). Since \((S_1\{a\}) \cup \{b\}=\) \(S_1\), all sets in \({\mathcal {B}}({\mathcal {M}}_\sigma )\) have the same size. From \({\mathcal {W}}\), all independent sets are in \({\mathcal {B}}({\mathcal {M}}_\sigma )\). Hence, \({\mathcal {B}}({\mathcal {M}}_\sigma )\) is a base. Second, it is sufficient to prove that \({\mathcal {M}}_\sigma \) is a simplicial matroid. Since \(\phi \in {\mathcal {B}}\) for some \(B \in {\mathcal {B}}({\mathcal {M}}_\sigma )\), then \(\phi \in {\mathcal {W}}\). Let \(S_1\in {\mathcal {W}}\) such that \(S_2 \subseteq S_1\) for some \(S_2\in {\mathcal {V}}_\sigma \). Then, \(S_1, S_2 \subseteq B\) for some \(B \in {\mathcal {B}}({\mathcal {M}}_\sigma )\) and so \(S_2\in {\mathcal {W}}\). Finally, suppose that there are \(S_1\) and B in \({\mathcal {W}}\). Then, by definitions of \({\mathcal {W}}\) and \({\mathcal {B}}\), \(A\in {\mathcal {B}}({\mathcal {M}}_\sigma )\) such that \(S_1\subseteq A\) and \(S_1<B\). So, we have two cases
Case 1. \(a\in BS_1\) (see Fig. 3). Since \(A\in {\mathcal {B}}({\mathcal {M}}_\sigma )\), \((S_1\cup \{a\})\subseteq A\) and so \(S_1\cup \{a\}\in {\mathcal {W}}\).
Case 2. \(A, B\in {\mathcal {B}}({\mathcal {M}}_\sigma )\) and \(a\in AB\), \(b\in BA\). Then, by (B2) in Proposition 3, \((A\{a\})\cup \{b\}\in {\mathcal {B}}({\mathcal {M}}_\sigma )\). Since \(S_1\subseteq A\{a\}\) (see Fig. 4), \((S_1\cup \{b\})\subseteq ((A\{a\})\cup \{b\})\). Hence, \(S_1\cup \{b\}\in {\mathcal {W}}\). \(\square \)
Now, any simplicial matroid can be generated by a subset from \({\mathcal {V}}_\sigma \) and is called a restriction simplicial matroid.
Definition 24
Let \({\mathcal {M}}_\sigma ({\mathcal {W}})\) be a simplicial matroid and \(D\subseteq {\mathcal {V}}_\sigma \). Then, \({\mathcal {M}}_\sigma /D=\) \((D, {\mathcal {W}}_D)\) is said to be a restriction simplicial matroid of \({\mathcal {M}}_\sigma \) on D, where \({\mathcal {W}}_D=\) \(\{S\in {\mathcal {W}}: S \subseteq D\}\).
Lemma 1
A restriction simplicial matroid \({\mathcal {M}}_\sigma /D=\) \((D, {\mathcal {W}}_D)\) is a matroid.
Proof

(i)
Since \(\phi \in {\mathcal {W}}\) and \(\phi \subseteq D\), \(\phi \in {\mathcal {W}}_D\).

(ii)
Let \(I \in {\mathcal {W}}_D\) and \(J \subseteq I\). Then, \(I \in {\mathcal {W}}\) and \(I \subseteq D\). Hence, \(J \in {\mathcal {W}}\) and \(J \subseteq D\). Therefore, \(J \in {\mathcal {W}}_D\).

(iii)
Let \(A,B \in {\mathcal {W}}_D\) such that \(A\le B\). Then, \(A,B \in {\mathcal {W}}\) such that \(A,B \subseteq D\). Since \({\mathcal {W}}\) is a simplicial matroid, \(A\cup \{a\}\in {\mathcal {W}}\) and \(A\cup \{a\}\subseteq D\), for some \(a\in BA\). Therefore, \(A\cup \{a\}\in {\mathcal {W}}_D\).
\(\square \)
Example 7
(continued for Example 3). Let \(D=\) \(\{s_1^1,s_2^1,s_3^1\}\). Then, \({\mathcal {M}}_{\sigma _2}({\mathcal {W}}_1)/D=\) \(\{\phi ,\{s_1^1\}\), \(\{s_2^1\}\), \(\{s_3^1\}\), \(\{s_1^1, s_2^1\}\), \(\{s_1^1,s_3^1\},\) \(\{s_2^1, s_3^1\}, \{s_1^1, s_2^1,s_3^1\}\}\).
In the following, we generate the base of \({\mathcal {M}}_\sigma (M)/D\).
Proposition 9
Let \(D\subseteq {\mathcal {V}}_\sigma \). Then, \({\mathcal {B}}({\mathcal {M}}_\sigma /D)=\) \(\{B\cap D: B\in {\mathcal {B}}({\mathcal {M}}_\sigma ) \wedge B\cap D=\) \(r_{{\mathcal {M}}_\sigma }(D)\}\).
Proof
It is sufficient to prove that \({\mathcal {B}}({\mathcal {M}}_\sigma /D)=\) Max \(({\mathcal {W}}_{D})\). Let \(S\in {\mathcal {B}}({\mathcal {M}}_\sigma /D)\). Then, \(S= B\cap D\) and \(S=\) \(r_{{\mathcal {M}}_\sigma }(D)\). Moreover, \(S\subseteq B \in {\mathcal {W}}\) and \(S\subseteq D\); by Definition 24, we have \(S\in {\mathcal {W}}_D\). Now, it is needed to prove that \(S=\) Max \(({\mathcal {W}}_{D})\). By a contradiction, assume that \(S\notin \) Max \(({\mathcal {W}}_{D})\). Then, there is \(S^{'}\in {\mathcal {W}}_{D}\) such that \(S^{'}>S\). Therefore, \(S^{'}\in {\mathcal {W}}\) and \(S^{'}\subseteq D\) and then \(r_{{\mathcal {M}}_\sigma }(D)= S^{'}\) that contradicts with \(S= r_{{\mathcal {M}}_\sigma }(D)\) and hence \({\mathcal {B}}({\mathcal {M}}_\sigma /D)\) \(\subseteq \) Max \(({\mathcal {W}}_{D})\). On the other hand, let \(S\in \) Max \(({\mathcal {W}}_{D})\). Then, by Definition 12, we have \(S=\) \(r_{{\mathcal {M}}_\sigma }(D)\). Therefore, \(S\in {\mathcal {B}}({\mathcal {M}}_\sigma /D)\) and so Max \(({\mathcal {W}}_{D})\) \(\subseteq \) \({\mathcal {B}}({\mathcal {M}}_\sigma /D)\). \(\square \)
Example 8
(continued for Example 3). Let \(D=\) \(\{s_1^1,s_2^1,s_3^1\}\). Then, the base on D is \({\mathcal {B}}({\mathcal {M}}_{\sigma _2}({\mathcal {W}}_1)/D )=\) \(\{s_1^1,s_2^1,s_3^1\}\). According to Proposition 9, since \({\mathcal {B}}({\mathcal {M}}_{\sigma _2}({\mathcal {W}}_1))=\) \(\{\{s_1^1,s_2^1,s_3^1,s_4^1\}\), \(\{s_1^1,s_2^1\), \(s_3^1, s_5^1\}\), \(\{s_2^1,s_3^1,s_4^1,s_5^1\}\), \(\{s_1^1,s_3^1,s_4^1,s_5^1\}\}\) and \(r_{{\mathcal {M}}_{\sigma _2}({\mathcal {W}}_1)}(D)=3\), then \({\mathcal {M}}_{\sigma _2}({\mathcal {W}}_1)/D)=\) \(\{s_1^1, s_2^1,s_3^1\}\).
4 Simplicial matroids generated by a relation and its rough set
In this section, we define an equivalence relation on a simplicial complex \(\sigma \). We introduce independent sets of simplicial matroids \({\mathcal {M}}_{\sigma }\) by different methods. Some of their propositions will be studied.
Definition 25
Let \(\sigma \) be a simplicial complex. For all \(s_i^n, s_j^n\in \sigma \), a relation \(R_\sigma \) is defined by the following:
Moreover, either both \(s_i^n\) and \(s_j^n\) are contained in \((n+1)\)face called \(s_k^{n+1}\) or there are two different \((n+1)\)faces called \(s_\ell ^{n+1}\) and \(s_m^{n+1}\) such that \(s_\ell ^{n+1}\) contains \(s_i^n\) and \(s_m^{n+1}\) contains \(s_j^n\).
Remark 4
In Definition 25, \(R_\sigma \) may be represented using a matrix named \([a_{ij}]\) such that if \(s_i^n R_\sigma s_j^n\), then \(a_{ij}=1\). Otherwise, \(a_{ij}=0\). The circuits \({\mathcal {C}}\) in this instance are a subset of linearly dependent columns in \([a_{ij}]\).
Proposition 10
\(R_\sigma \) is an equivalence relation on \({\mathcal {U}}_\sigma \).
Proof
Since each \(s_i^n\) is hconnected (directly connected) with itself, \(R_\sigma \) is reflexive. Since \(s_i^n R_\sigma s_j^n\), then \(s_i^n\) is hconnected with \(s_j^n\), that is, \(s_i^n=s_1^n, s_2^h,\) \(\ldots ,\) \(s_{m1}^h, s_m^n =s_j^n\) and can also be reformulated to \(s_j^n=s_m^n, s_{m1}^h, \ldots , s_2^h, s_1^n=s_i^n\). So, \(s_j^n\) and \(s_i^n\) are hconnected, \(s_j^n R_\sigma s_i^n\), and then \(R_\sigma \) is symmetric. Finally, let \(s_i^n R_\sigma s_a^n\) and \(s_a^n R_\sigma s_j^n\). Then, \(s_i^n = s_1^n, s_2^h,\) \(\ldots \) \(, s_{m1}^h, s_m^n = s_a^n\), and \(s_a^n=s_m^n, s_{m+1}^h,\) \(\ldots \) \(, s_\ell ^n =s_j^n\). Hence, \(s_i^n = s_1^n, s_2^h\), \(\ldots \) \(, s_m^n, s_{m+1}^h, \) \(\ldots \) \(, s_\ell ^n =s_j^n\), meaning that \(s_i^n\) and \(s_j^n\) are hconnected. Therefore, \(R_\sigma \) is transitive. \(\square \)
Remark 5

(i)
The equivalence class for the simplex \(s_i^n\) will be defined by \([s_i^n]_{R_\sigma }=\) \(\{s_j^n\in {\mathcal {U}}_\sigma : s_i^n R_\sigma s_j^n\}\). The set of all equivalence classes of \({\mathcal {U}}_\sigma \) is \({\mathcal {U}}_\sigma /R_\sigma =\) \(\{[s_i^n]_{R_\sigma }: s_i^n\in {\mathcal {U}}_\sigma \}\).

(ii)
\((s_i^n, s_j^n)\notin R_\sigma \) if \(s_i^n\) and \(s_j^n\) are not boundaries for some \((n+1)\)faces.

(iii)
0faces are only related to itself.
Example 9
(continued for Example 1). The matrix \([a_{ij}]\) represents the equivalence relation \(R_{\sigma _1}\):
Because columns 4, 5 and 6 are linearly dependent, the circuits \({\mathcal {C}}=\) \(\{ \{s_1^1, s_2^1\},\) \(\{s_1^1, s_3^1\},\) \(\{s_2^1, s_3^1\}\}\). Therefore, the matroid is immediately derived. Furthermore, the equivalence classes are \({\mathcal {U}}_{\sigma _1}/R_{\sigma _1}=\) \(\{\{s_1^0\}\), \(\{s_2^0\}\), \(\{s_3^0\}\), \(\{s_1^1, s_2^1, s_3^1\}\), \(\{s_1^2\}\}\).
Example 10
(continued for Example 2). The family of equivalence classes of \({\mathcal {U}}_{\sigma _2}\) is \({\mathcal {U}}_{\sigma _2}/R_{\sigma _2}=\) \(\{\{s_1^0\},\) \(\{s_2^0\},\) \(\{s_3^0\},\) \(\{s_4^0\},\) \(\{s_1^1, s_2^1, s_3^1,s_4^1, s_5^1\},\) \(\{s_1^2\},\{s_2^2\}\}\). It is noted that \((s_1^2, s_2^2)\notin R_{\sigma _2}\) because there is no 3face containing both \(s_1^2\) and \(s_2^2\).
Definition 26
A class \(C(R_\sigma )\) on \({\mathcal {U}}_\sigma \) is defined by \(C(R_\sigma )=\) \(\{\{s_i^n, s_j^n\}:\) \(s_i^n R_\sigma s_j^n,\) \(s_i^n\) \(\ne \) \(s_j^n\}\), where \(R_\sigma \) is equivalence on \({\mathcal {U}}_\sigma \). In other words, C is the family of all two point sets that are not reflexive in \(R_\sigma \).
For instance, in Example 9, \(C(R_\sigma )=\) \(\{\{s_1^1, s_2^1\},\) \(\{s_3^1, s_1^1\},\) \(\{s_2^1, s_3^1\}\}\).
Proposition 11
\(C(R_\sigma )\) satisfies circuit axioms for \(\sigma \).
Proof
The conditions C1 and C2 are obvious; thus, we only prove condition C3 in Proposition 2. Let \(C, D\in C(R_\sigma )\), \(C\ne D\) and \(s_i^n\in C\cap D\). Suppose that \(C=\) \(\{s_i^n, s_j^n\}\), and \(D=\) \(\{s_i^n, s_k^n\}\), which means that \(s_j^n R_\sigma s_i^n\) and \(s_i^n R_\sigma s_k^n\) imply that \(s_j^n R_\sigma s_k^n\). Hence, \(\exists \) \(E=\) \(\{s_j^n, s_k^n\}\in C(R_\sigma )\). Therefore, \(E\subseteq (C\cup D) \{s_i^n\}\). \(\square \)
Remark 6
According to Proposition 2, there is a simplicial matroid on \({\mathcal {U}}_\sigma \) whose circuits are the family \(C(R_\sigma )\).
Definition 27
The simplicial matroid whose circuits are \(C(R_\sigma )\) is denoted by \({\mathcal {M}}_\sigma (R_\sigma )=\) \(({\mathcal {U}}_\sigma , {\mathcal {W}}(R_\sigma ))\), where \({\mathcal {W}}(R_\sigma )=\) \([Upp(C(R_\sigma ))]^c\).
Example 11
(continued for Example 9). The simplicial matroid \({\mathcal {M}}_{\sigma _1}(R_{\sigma _1})\) generated by \(R_{\sigma _1}\) consists of independent sets \({\mathcal {W}}(R_{\sigma _1})=\) [Upp \(\{\{s_1^1, s_2^1\},\) \(\{s_2^1, s_3^1\},\) \(\{s_3^1, s_1^1\}\}]^c=\) \(Low\{\{s_1^2, s_3^1,\) \(s_1^0, s_2^0, s_3^0\},\) \(\{s_1^2, s_2^1,\) \(s_1^0, s_2^0, s_3^0\},\) \(\{s_1^2, s_1^1,\) \(s_1^0, s_2^0, s_3^0\}\}\).
Rough sets may be used to express certain comparable formulations of independent sets for simplicial matroids.
Proposition 12
Let \({\mathcal {M}}_\sigma (R_\sigma )=\) \(({\mathcal {U}}_\sigma , {\mathcal {W}}(R_\sigma ))\) be a simplicial matroid induced by the equivalence relation \(R_\sigma \). Then,
Proof
Proving that \([Upp(C(R_\sigma ))]^c=\) \(\{S\subseteq {\mathcal {U}}_\sigma \): \(\forall \) \(s_i^n, s_j^n \in S,\) \(s_i^n \ne s_j^n\) implies that \((s_i^n, s_j^n)\notin R_\sigma \}\) is needed. For all \(S\notin \{S\subseteq {\mathcal {U}}_\sigma :\) \(\forall \) \(s_i^n, s_j^n\in S, s_i^n\ne s_j^n\) implying that \((s_i^n, s_j^n)\) \(\notin \) \(R_\sigma \}\), we have \(s_i^n, s_j^n\in S\), and \(s_i^n\ne s_j^n\) such that \((s_i^n, s_j^n)\in R_\sigma \), which means that \(\{s_i^n, s_j^n\}\in C(R_\sigma )\). Since \(\{s_i^n, s_j^n\}\subseteq S\), \(S\in Upp(C(R_\sigma ))\) and so \(S\notin [Upp(C(R_\sigma ))]^c\). Therefore, \(\{S\subseteq {\mathcal {U}}_\sigma :\) \(\forall \) \(s_i^n, s_j^n\in S, s_i^n\ne s_j^n\) implies that \((s_i^n, s_j^n)\notin R_\sigma \}\) \(\subseteq \) \([Upp(C(R_\sigma ))]^c\). Conversely, \(\forall \) \(S\notin [Upp(C(R_\sigma ))]^c\), which means that \(S\in (Upp(C(R_\sigma )))\); then, we get \(C_1\in C(R_\sigma )\) such that \(C_1\subseteq S\). Consider \(C_1=\) \(\{s_i^n, s_j^n\}\). Then, \((s_i^n, s_j^n)\in R_\sigma \) implies that \(S\notin \{S\subseteq {\mathcal {U}}_\sigma :\) \(\forall \) \(s_i^n, s_j^n\in S, s_i^n\ne s_j^n\) implies that \((s_i^n, s_j^n)\notin R_\sigma \}\). Therefore, \([Upp(C (R_\sigma ))]^c \subseteq \{S\subseteq {\mathcal {U}}_\sigma :\) \(\forall \) \(s_i^n, s_j^n\in S, s_i^n\ne s_j^n\) implies that \((s_i^n, s_j^n)\notin R_\sigma \}\). \(\square \)
Proposition 13
Let \({\mathcal {M}}_\sigma (R_\sigma )=\) \(({\mathcal {U}}_\sigma , {\mathcal {W}}(R_\sigma ))\) be a simplicial matroid induced by \(R_\sigma \). Then, \({\mathcal {W}}(R_\sigma )=\) \(\{S\subseteq {\mathcal {U}}_\sigma :\) \(\forall \) \(s_i^n\in \) \({\mathcal {U}}_\sigma ,\) \(S\cap [s_i^n]_{R_\sigma } \le 1\}\).
Proof
Proposition 12 is sufficient to prove that \(\{S\subseteq {\mathcal {U}}_\sigma :\) \(\forall \) \(s_i^n\in {\mathcal {U}}_\sigma ,\) \(S\cap [s_i^n]_{R_\sigma }\le 1\}=\) \(\{S\subseteq \) \({\mathcal {U}}_\sigma :\) \(\forall \) \(s_i^n, s_j^n\in S,\) \(s_i^n\ne s_j^n\) implying that \((s_i^n, s_j^n)\notin R_\sigma \}\). Suppose that \(\forall \) \(S\in \{S\subseteq {\mathcal {U}}_\sigma :\) \(\forall \) \(s_i^n\in {\mathcal {U}}_\sigma , S\cap [s_i^n]_{R_\sigma } \le 1\}\). Let \(s_i^n, s_j^n\in S\), \(s_i^n\ne s_j^n\), \((s_i^n, s_j^n)\) \(\in \) \(R_\sigma \). Since \((s_i^n, s_j^n)\) \(\in \) \(R_\sigma \), then \(\{s_i^n, s_j^n\} \subseteq [s_i^n]_{R_\sigma }\). Therefore, \(S \cap [s_i^n]_{R_\sigma }\ge 2\). This contradicts with \(S\cap [s_i^n]_{R_\sigma } \le 1\). Hence, \(\{S\subseteq {\mathcal {U}}_\sigma :\) \(\forall \) \(s_i^n\in {\mathcal {U}}_\sigma , S\cap [s_i^n]_{R_\sigma }\le 1\}\) \(\subseteq \) \(\{S\subseteq {\mathcal {U}}_\sigma :\) \(\forall \) \(s_i^n, s_j^n\in S, s_i^n\ne s_j^n\) implying that \((s_i^n, s_j^n)\notin R_\sigma \}\). Conversely, \(\forall \) \(S\in \{S\subseteq {\mathcal {U}}_\sigma :\) \(\forall \) \(s_i^n, s_j^n\in S, s_i^n \ne s_j^n\) implying that \((s_i^n, s_j^n)\notin R_\sigma \}\). This means that \(s_j^n\notin [s_i^n]_{R_\sigma }\) \(\forall \) \(s_j^n\in S\) and \(s_i^n\ne s_j^n\). Then, \(S\cap [s_i^n]_{R_\sigma }\le 1\), \(\forall \) \(s_i^n\in S\). \(\square \)
In the following, we investigate some basic properties of the simplicial matroid \({\mathcal {M}}_\sigma (R_\sigma )\).
Proposition 14
Let \({\mathcal {M}}_\sigma (R_\sigma )=\) \(({\mathcal {U}}_\sigma , {\mathcal {W}}(R_\sigma ))\) be a simplicial matroid. Then,

(i)
S is dependent in \({\mathcal {M}}_\sigma (R_\sigma )\) if and only if \(\exists \) \(s_i^n\in {\mathcal {U}}_\sigma \) such that \([s_i^n]_{R_\sigma }\cap S>1\).

(ii)
C is a circuit of \({\mathcal {M}}_\sigma (R_\sigma )\) if and only if \(\exists \) \(s_i^n\in {\mathcal {U}}_\sigma \) such that \(C\subseteq [s_i^n]_{R_\sigma }\) and \(C= 2\).

(iii)
\({\mathcal {B}}\) is a base of \({\mathcal {M}}_\sigma (R_\sigma )\) if and only if \({\mathcal {B}}\cap [s_i^n]_{R_\sigma }=\) 1, \(\forall \) \(s_i^n\in {\mathcal {U}}_\sigma \).
Proof

(i)
From Proposition 13, S is dependent if and only if \(S\notin {\mathcal {W}}(R_\sigma )\) and so \([s_i^n]_{R_\sigma }\cap S>1\).

(ii)
Let C be a circuit of \({\mathcal {M}}_\sigma (R_\sigma )\) and \(s_j^n \in C\). Then, by Definition 26, \(\exists \) \(s_i^n \in {\mathcal {U}}_\sigma \) such that \(s_j^n R_\sigma s_i^n\) and so \(s_j^n \in [s_i^n]_{R_\sigma }\). Hence, \(C\subseteq [s_i^n]_{R_\sigma }\) and obviously by Definition 26, \(C= 2\). Conversely, let \(C\subseteq [s_i^n]_{R_\sigma }\) and \(C=\) 2. Then, by (i), \([s_i^n]_{R_\sigma }\cap C=\) 2. This means that C is a minimal dependent set and so it is a circuit.

(iii)
By Proposition 13, B is a base element if and only if it is a maximal independent set and so \(B \cap [s_i^n]_{R_\sigma }=\) 1, \(\forall \) \(s_i^n\in {\mathcal {U}}_\sigma \).
\(\square \)
The relation between upper approximations related to \(R_\sigma \) and the closure operator for simplicial matroids will be discussed.
Proposition 15
Let \(R_\sigma \) be an equivalence relation on \({\mathcal {U}}_\sigma \). Then, \({\overline{R}}_\sigma (A)=\) \(Cl_{{\mathcal {M}}_\sigma (R_\sigma )}(A)\), \(\forall \) \(A\subseteq {\mathcal {U}}_\sigma \).
Proof
Since \(s_i^n R_\sigma s_j^n\), \(\{s_i^n, s_j^n\} \in C(R_\sigma )\). By Definition 20, we get the following:
\(\square \)
Corollary 2
Let \(R_\sigma \) be an equivalence relation on \({\mathcal {U}}_\sigma \). Then, \({\text {Int}}_{{\mathcal {M}}_\sigma (R_\sigma )}(A)=\) \(\underline{R_\sigma }(A)\), \(\forall \) \(A\subseteq {\mathcal {U}}_\sigma \).
Proof
Since \({\text {Int}}_{{\mathcal {M}}_\sigma (R_\sigma )}(A)=\) \([(Cl_{{\mathcal {M}}_\sigma (R_\sigma )}(A^c))]^c=\) \(({\overline{R}}_\sigma (A^c))^c=\) \(\underline{R_\sigma }(A)\), \(\forall \) \(A\subseteq \) \({\mathcal {U}}_\sigma \). \(\square \)
Proposition 16
\(R_\sigma \) is an equivalence relation on \({\mathcal {U}}_\sigma \) if and only if \({\overline{R}}_\sigma \) satisfies closure axioms.
Proof
Let \(R_\sigma \) be an equivalence relation on \({\mathcal {U}}_\sigma \). Then, by Proposition 15, \({\overline{R}}_\sigma (A)=\) \(Cl_{{\mathcal {M}}_\sigma (R_\sigma )}(A)\) is verified. By Proposition 7, the closure axioms are held. Conversely, let \({\overline{R}}_\sigma \) satisfy closure axioms. Take a relation \(R^{*}_\sigma \) by \(s_i^n R^{*}_\sigma s_j^n\) if and only if \(s_j^n\in [s_i^n]_{R^{*}_\sigma }\) and so \(s_i^n\in {\overline{R}}_\sigma (\{s_j^n\})\). Using Proposition 7 (CL1), \(s_i^n\in \{s_i^n\}\subseteq {\overline{R}}_\sigma (\{s_i^n\})\). Then, \(R^{*}_\sigma \) is reflexive. Now, let \(s_i^n R^{*}_\sigma s_j^n\) and \(s_j^n R^{*}_\sigma s_k^n\). Since \(s_j^n\in \{s_j^n\}\), \(s_j^n \in {\overline{R}}_\sigma (\{s_k^n\})\), \(\{s_j^n\}\subseteq {\overline{R}}_\sigma (\{s_k^n\})\). Using CL2, \({\overline{R}}_\sigma (\{s_j^n\})\subseteq {\overline{R}}_\sigma ({\overline{R}}_\sigma (\{s_k^n\}))\). Moreover, from CL3, we get \({\overline{R}}_\sigma (\{s_j^n\})\subseteq {\overline{R}}_\sigma (\{s_k^n\})\). Since \(s_i^n R^{*}_\sigma s_j^n\), then \(s_i^n \in {\overline{R}}_\sigma (\{s_j^n\})\). Hence, \(s_i^n \in {\overline{R}}_\sigma (\{s_j^n\})\) \(\subseteq {\overline{R}}_\sigma (\{s_k^n\})\). Therefore, \(s_i^n R^{*}_\sigma s_k^n\) and so \(R^{*}_\sigma \) is transitive. Let \(s_i^n R^{*}_\sigma s_j^n\). Then, \(s_i^n\in {\overline{R}}_\sigma \{s_j^n\}\); by CL4 in Proposition 7, we have \(s_i^n\in {\overline{R}}_\sigma (\phi \cup \{s_j^n\}) {\overline{R}}_\sigma (\phi )\). Therefore, \(s_j^n\in {\overline{R}}_\sigma (\phi \cup \{s_i^n\})\) and so \(s_j^n \in {\overline{R}}_\sigma (\{s_i^n\})\). This implies that \(s_j^n R^{*}_\sigma s_i^n\) and hence \(R^{*}_\sigma \) is symmetric. \(\square \)
In the following, the boundary region for the matroidal approach is determined.
Proposition 17
Let \(C(R_\sigma )\) be a collection of circuits for \({\mathcal {M}}_\sigma (R_\sigma )\) with an equivalence relation \(R_\sigma \) on \({\mathcal {U}}_\sigma \). Then, the boundary region of X with respect to \(R_\sigma \) is \(BN_{R_\sigma }(X)=\) \(\bigcup \{C\in C(R_\sigma ): C\cap X= 1\}\).
Proof
Proving that \(\bigcup \{[s_i^n]_{R_\sigma }\in {\mathcal {U}}_\sigma /R_\sigma :\) \([s_i^n]\cap X\ne \phi \wedge [s_i^n]_{R_\sigma }\nsubseteq X\}=\) \(\bigcup \{C\in C(R_\sigma ):\) \(C\cap X=1\}\) is needed. Let \([s_i^n]_{R_\sigma }\in \bigcup \{[s_i^n]_{R_\sigma }\in {\mathcal {U}}_\sigma / R_\sigma :\) \([s_i^n]_{R_\sigma } \cap X\ne \phi , [s_i^n]_{R_\sigma }\nsubseteq X\}\). Then, \(\exists \) \(s_i^n\in [s_i^n ]_{R_\sigma }\) and \(s_i^n \notin X\); we find \(C\in C(R_\sigma )\) such that \(C\subseteq [s_i^n]_{R_\sigma }\) implies that \(s_i^n\in C\) and \(s_i^n \notin X\) \(\Rightarrow \) \(C\cap X= 1\). Therefore, \(\bigcup \{[s_i^n]_{R_\sigma }\in {\mathcal {U}}_\sigma /R_\sigma :\) \([s_i^n]\cap X\ne \phi \wedge [s_i^n]_{R_\sigma }\) \(\nsubseteq X\}\subseteq \) \(\bigcup \{C\in C (R_\sigma ):\) \(C\cap X= 1\}\). On the other hand, assume that \(C\in \bigcup \{C\in C (R_\sigma ):\) \(C\cap X= 1\}\) such that \([s_i^n]_{R_\sigma } \subseteq X\). Then, we can find \([s_i^n]_{R_\sigma }\in {\mathcal {U}}_\sigma /R_\sigma \) such that \(C\subseteq [s_i^n]_{R_\sigma }\) implies that \(C\subseteq X\). So, \(C\cap X= 2\), which contradicts with \(C\cap X= 1\). Hence, \(\bigcup \{C\in C(R_\sigma ):\) \(C\cap X =1\}\subseteq \) \(\bigcup \{[s_i^n]_{R_\sigma }\in {\mathcal {U}}_\sigma /R_\sigma :\) \([s_i^n]_{R_\sigma } \cap X\ne \phi \) \(\wedge [s_i^n]_{R_\sigma }\nsubseteq X\}\). \(\square \)
Example 12
(continued for Example 9) .
Let \({\mathcal {U}}_\sigma =\) \(\{s_1^0,\) \(s_2^0,\) \(s_3^0,\) \(s_1^1,\) \(s_2^1,\) \(s_3^1,\) \(s_1^2\}\). Then, \(C(R_\sigma )=\) \(\{\{s_1^1, s_2^1\},\) \(\{s_2^1,s_3^1\}\), \(\{s_1^1, s_3^1\}\). Consider \(X=\) \(\{s_1^0, s_1^1, s_1^2\}\). By Proposition 17, we get \(B N_{R_\sigma } (X)=\) \(\{s_1^1, s_2^1, s_3^1\}\).
5 Rough sets induced by simplicial matroids
In this section, simplicial matroids are used to generate a special type of equivalence relations. By this relation, we can also generate other simplicial matroids in terms of rough sets, say, 2circuit simplicial matroids. The relation between these new kinds of simplicial matroids will be studied.
Definition 28
Let \({\mathcal {M}}_\sigma =\) \(({\mathcal {V}}_\sigma , {\mathcal {W}})\) (resp., \({\mathcal {M}}_\sigma (R_\sigma )=\) \(({\mathcal {U}}_\sigma , {\mathcal {W}}(R_\sigma )\)) be a simplicial matroid. A relation \(R_1\) on \({\mathcal {V}}_\sigma \) (resp., \(R_2\) on \({\mathcal {U}}_\sigma \)) is given by \(s_i^n\) \(R_1\) \(s_j^n\) (resp., \(s_i^n\) \(R_2\) \(s_j^n\)) if \(s_i^n= s_j^n\) or \(\exists \) \(C\in {\mathcal {C}}({\mathcal {M}}_\sigma )\) (resp., \(C\in {\mathcal {C}}(R_\sigma ))\) such that \(\{s_i^n, s_j^n\} \subseteq C\), \(\forall \) \(s_i^n, s_j^n\in {\mathcal {V}}_\sigma \) (resp., \(s_i^n, s_j^n\in {\mathcal {U}}_\sigma \)).
Proposition 18
Let \({\mathcal {M}}_\sigma \) (resp., \({\mathcal {M}}_\sigma (R_\sigma )\)) be a simplicial matroid. Then, \(R_1\) and \(R_2\) are equivalence relations.
Proof
Proving that \(R_1\) is an equivalence relation and \(R_2\) is so by similarity is needed. It is clear that \(R_1\) is reflexive. Let \(s_i^n\) \(R_1\) \(s_j^n\). Then, \(\exists \) \(C\in {\mathcal {C}}({\mathcal {M}}_\sigma )\) such that \(\{s_i^n, s_j^n\} \subseteq C\), which can be rewritten by \(\{s_j^n, s_i^n\} \subseteq C\), and so \(s_j^n\) \(R_1\) \(s_i^n\). Finally, let \(s_i^n\) \(R_1\) \(s_j^n\) and \(s_j^n\) \(R_1\) \(s_k^n\). Then, we have the following two cases:
Case 1. There is only one circuit \(C\in {\mathcal {C}}({\mathcal {M}}_\sigma )\) such that \(\{s_i^n, s_j^n\} \subseteq C\) and \(\{s_j^n, s_k^n\} \subseteq C\). Then, using Definition 28, \(\{s_i^n, s_k^n\} \subseteq C\) and so \(s_i^n\) \(R_1\) \(s_k^n\).
Case 2. There are more than one circuit in \({\mathcal {C}}({\mathcal {M}}_\sigma )\). Then, by Definition 28, we find \(C_1, C_2\) \(\in \) \({\mathcal {C}}({\mathcal {M}}_\sigma )\) such that \(\{s_i^n, s_j^n\} \subseteq C_1\) and \(\{s_j^n, s_k^n\} \subseteq C_2\). By Proposition 2 (C3), we can find \(C_3\in {\mathcal {C}}({\mathcal {M}}_\sigma )\) such that \(C_3\subseteq (C_1\cup C_2) \{s_j^n\}\). This means that \(C_3=\) \(\{s_i^n, s_k^n\}\). Therefore, \(s_i^n\) \(R_1\) \(s_k^n\). \(\square \)
Remark 7
According to Definition 28 and Proposition 18, there is an equivalence relation \(R_1\) on \({\mathcal {M}}_\sigma \). We can also construct a simplicial matroid \({\mathcal {M}}_\sigma (R_1)\) using \(R_1\). In addition, on \({\mathcal {M}}_\sigma (R_\sigma )\), another equivalence relation \(R_2\) may be given. Furthermore, another simplicial matroid \({\mathcal {M}}_\sigma (R_2)\) may be constructed through \(R_2\) so that \(R_\sigma = R_2\).
Definition 29
Let \({\mathcal {M}}_\sigma =\) \(({\mathcal {V}}_\sigma , {\mathcal {W}})\) be a simplicial matroid such that \(C=\) 2, \(\forall \) \(C\in {\mathcal {C}}({\mathcal {M}}_\sigma )\). Then, \({\mathcal {M}}_\sigma \) is said to be a 2circuit simplicial matroid.
Proposition 19
Let \(R_\sigma \) be an equivalence relation on \({\mathcal {U}}_\sigma \). Then, \({\mathcal {M}}_\sigma (R_\sigma )=\) \({\mathcal {M}}_\sigma ({\mathcal {U}}_\sigma , {\mathcal {W}}(R_\sigma ))\) is a 2circuit simplicial matroid.
Proof
By Definition 26 and Proposition 11, the circuits of \({\mathcal {M}}_\sigma (R_\sigma )\) have the form \(C(R_\sigma )=\) \(\{\{s_i^n, s_j^n\}: s_i^n R_\sigma s_j^n\}\). Therefore, \(C= 2\), \(\forall \) \(C\in {\mathcal {C}}(R_\sigma )\). \(\square \)
Proposition 20
Let \({\mathcal {M}}_\sigma =\) \(({\mathcal {V}}_\sigma , {\mathcal {W}})\) be a simplicial matroid. Then, \({\mathcal {M}}_\sigma =\) \({\mathcal {M}}_\sigma (R_1)\) if and only if \({\mathcal {M}}_\sigma \) is a 2circuit matroid.
Proof
Let \({\mathcal {M}}_\sigma =\) \({\mathcal {M}}_\sigma (R_1)\). Then, by Proposition 19, \({\mathcal {M}}_\sigma (R_1)\) is a 2circuit matroid and hence \({\mathcal {M}}_\sigma \) is so. Conversely, let \({\mathcal {M}}_\sigma \) be a 2circuit matroid and \(R_1\) be an equivalence relation on \({\mathcal {V}}_\sigma \). Then, using Definitions 26, 27, and 28, we get \({\mathcal {C}}({\mathcal {M}}_\sigma )=\) \({\mathcal {C}}(R_1)\). Therefore, \({\mathcal {M}}_\sigma =\) \({\mathcal {M}}_\sigma (R_1)\). \(\square \)
In Propositions 19 and 20, a 2circuit is both a necessary and sufficient condition, which is shown in Example 13.
Example 13
(continued for Examples 3 and 4).
The sets
\({\mathcal {C}}=\) \(\{s_1^1, s_2^1, s_4^1, s_5^1\}\).
\({\mathcal {C}}({\mathcal {M}}_{\sigma _2}(R_1))=\) \(\{\{s_1^1,s_2^1\},\! \{s_1^1,s_4^1\}, \!\{s_1^1,s_5^1\}, \{s_2^1,s_4^1\}, \{s_2^1,s_5^1\},\! \{s_4^1,s_5^1\}\}\).
\({\mathcal {M}}_{\sigma _2}(R_1)=\) \(\{\phi \), \(\{s_1^1\}\), \(\{s_2^1\}\), \(\{s_3^1\}\), \(\{s_4^1\}\), \(\{s_5^1\}\), \(\{s_1^1, s_3^1\}\), \(\{s_2^1, s_3^1\}\), \(\{s_3^1, s_4^1\}\), \(\{s_3^1, s_5^1\}\}\).
show the circuit for \({\mathcal {M}}_{\sigma _2}({\mathcal {W}}_1)\) and a simplicial matroid \({\mathcal {M}}_{\sigma _2}(R_1)\) and its matrix. It is obvious that \({\mathcal {M}}_{\sigma _2}({\mathcal {W}}_1)\ne {\mathcal {M}}_{\sigma _2}(R_1)\).
Proposition 21
Let \({\mathcal {M}}_\sigma =\) \(({\mathcal {V}}_\sigma , {\mathcal {W}})\) be a simplicial matroid. Then, \({\mathcal {M}}_\sigma \) is a 2circuit simplicial matroid if and only if \(\exists \) an equivalence relation \(R_\sigma \) on \({\mathcal {V}}_\sigma \) such that \({\mathcal {M}}_\sigma =\) \({\mathcal {M}}_\sigma (R_\sigma )\).
Proof
Let \({\mathcal {M}}_\sigma \) be a 2circuit simplicial matroid. By Proposition 20, \({\mathcal {M}}_\sigma =\) \({\mathcal {M}}_\sigma (R_1)\). Suppose that \(R_\sigma =\) \(R_1\). Then, \(R_\sigma \) is an equivalence relation and \({\mathcal {M}}_\sigma =\) \({\mathcal {M}}_\sigma (R_\sigma )\). Conversely, if \(R_\sigma \) is an equivalence relation, then, by Proposition 19, \({\mathcal {M}}_\sigma =\) \({\mathcal {M}}_\sigma (R_\sigma )\) is a 2circuit simplicial matroid. \(\square \)
Proposition 22
Let \({\mathcal {M}}_\sigma =\) \(({\mathcal {V}}_\sigma , {\mathcal {W}})\) be a simplicial matroid. Then, \({\mathcal {M}}_\sigma \) is a 2circuit simplicial matroid if and only if \(\exists \) an equivalence relation \(R_\sigma \) such that \({\mathcal {W}}=\) \(\{S\subseteq {\mathcal {V}}_\sigma :\) \(\forall \) \(s_i^n\in S, [s_i^n]_R\cap S\) \(\le 1\}\).
Proof
Let \({\mathcal {M}}_\sigma \) be a 2circuit simplicial matroid. Then, using Proposition 21, \(\exists \) an equivalence relation \(R_\sigma \) on \({\mathcal {V}}_\sigma \) such that \({\mathcal {M}}_\sigma =\) \({\mathcal {M}}_\sigma (R_\sigma )\), which implies that \({\mathcal {W}}=\) \(\{S\subseteq {\mathcal {V}}_\sigma :\) \(\forall \) \(s_i^n \in S, [s_i^n]_R \cap S\) \(\le 1\}\). Conversely, if \(\exists \) an equivalence relation \(R_\sigma \) such that \({\mathcal {W}}=\) \(\{S\subseteq {\mathcal {V}}_\sigma :\) \(\forall \) \(s_i^n\in S, [s_i^n]_R\cap S\) \(\le 1\}\), then \({\mathcal {C}}({\mathcal {M}}_\sigma )=\) \(\{\{s_i^n, s_j^n\}:\) \(s_i^n, s_j^n \in {\mathcal {V}}_\sigma , s_i^n \ne s_j^n,\) \((s_i^n, s_j^n)\in R_\sigma \}\). Therefore, \({\mathcal {M}}_\sigma \) is a 2circuit matroid. \(\square \)
Proposition 23
Let \({\mathcal {M}}_\sigma =\) \(({\mathcal {V}}_\sigma , {\mathcal {W}})\) be a 2circuit simplicial matroid. Then, \(\forall \) \(A\subseteq {\mathcal {V}}_\sigma \) \(\overline{R_1}(A)=\) \(Cl_{{\mathcal {M}}_\sigma }(A)\).
Proof
Since \({\mathcal {M}}_\sigma \) is a 2circuit simplicial matroid, by Proposition 21, then \(\exists \) an equivalence relation \(R_\sigma \) on \({\mathcal {V}}_\sigma \) such that \({\mathcal {M}}_\sigma =\) \({\mathcal {M}}_\sigma (R_\sigma )\). Using Remark 7, we have \(R_2=\) \(R_\sigma \). Therefore, \(\overline{R_1}(A)=\) \(\overline{R_2}(A)=\) \(\overline{R_\sigma }(A)=\) \(Cl_{{\mathcal {M}}_\sigma (R_\sigma )}(A)=\) \(Cl_{{\mathcal {M}}_\sigma }(A)\), \(\forall \) \(A\subseteq {\mathcal {V}}_\sigma \). \(\square \)
Proposition 24
Let \({\mathcal {M}}_\sigma =\) \(({\mathcal {V}}_\sigma , {\mathcal {W}})\) be a 2circuit simplicial matroid. Then, the following are held:

(i)
\(Cl_{{\mathcal {M}}_\sigma }(\phi )=\) \(\phi \).

(ii)
\(Cl_{{\mathcal {M}}_\sigma } (A\cup B)=\) \(Cl_{{\mathcal {M}}_\sigma }(A)\cup \) \(Cl_{{\mathcal {M}}_\sigma }(B)\), \(\forall \) \(A, B\in {\mathcal {V}}_\sigma \).
Proof
Since \({\mathcal {M}}_\sigma \) is a 2circuit simplicial matroid, by Proposition 23, \(\overline{R_1}(A)\) \(=Cl_{{\mathcal {M}}_\sigma }(A)\). Therefore, \(Cl_{{\mathcal {M}}_\sigma }(\phi ) =\) \(\phi \). Also, \(Cl_{{\mathcal {M}}_\sigma }(A \cup B)=\) \(Cl_{{\mathcal {M}}_\sigma }(A)\cup \) \(Cl_{{\mathcal {M}}_\sigma }(B)\) \(\forall \) \(A, B \subseteq {\mathcal {V}}_\sigma \). \(\square \)
Example 14
Consider \({\mathcal {V}}_{\sigma _3}\) is a ground set for \({\mathcal {D}}_1\) as shown in Fig. 5. Its circuit, a 2circuit simplicial matroid, an equivalence relation \(R_1\), and its matrix and a simplicial matroid by \(R_1\) are
\({\mathcal {U}}_{\sigma _3}=\) \(\{s_1^0, s_2^0, s_3^0, s_1^1, s_2^1, s_3^1\}\).
\({\mathcal {V}}_{\sigma _3}({\mathcal {D}}_1)=\) \(\{s_1^1, s_2^1, s_3^1, s_4^1\}\).
\({\mathcal {W}}=\) \(\{\phi , \{s_1^1\},\) \(\{s_2^1\},\) \(\{s_3^1\},\) \(\{s_4^1\},\) \(\{s_1^1, s_2^1\},\) \(\{s_1^1, s_3^1\},\) \(\{s_1^1, s_4^1\},\) \(\{s_2^1, s_3^1\},\)
\(\{s_2^1,s_4^1\},\) \(\{s_1^1, s_2^1, s_3^1\},\) \(\{s_1^1, s_2^1, s_4^1\}\}\).
\({\mathcal {C}}({\mathcal {M}}_{\sigma _3})=\) \(\{s_3^1, s_4^1\}\).
\({\mathcal {M}}_{\sigma _3}(R_1)=\) \((Upp \{s_3^1,s_4^1\})^c=\) \({\mathcal {W}}\).
\({\mathcal {M}}_{\sigma _3} (R_{\sigma _3})=\) \({\mathcal {M}}_{\sigma _3} (R_1)\).
There is also an equivalence relation \(R_{\sigma _3}\) and its matrix on \({\mathcal {U}}_{\sigma _3}\) in terms of \(R_{\sigma _3}\) is given. The simplicial matroid on \(\sigma _3\) clearly corresponds with the simplicial matroid produced by \(R_{\sigma _3}\).
Example 15
(continued for Example 11).
The ground set \({\mathcal {U}}_\sigma \) and its \({\mathcal {W}}(R_{\sigma _1})\) are
\({\mathcal {U}}_{\sigma _1}=\) \(\{s_1^0, s_2^0,\) \(s_3^0, s_1^1,\) \(s_2^1, s_3^1, s_1^2\}\).
\({\mathcal {C}}(R_{\sigma _1})=\) \(\{\{s_1^1, s_2^1\},\) \(\{s_2^1, s_3^1\}, \{s_3^1, s_1^1\}\}\).
\({\mathcal {W}}(R_{\sigma _1})=\) \(Low \{\{s_1^2, s_3^1,\) \(s_1^0,\) \(s_2^0, s_3^0\},\) \(\{s_1^2, s_2^1, s_1^0, s_2^0, s_3^0\}\), \(\{s_1^2, s_1^1, s_1^0, s_2^0\), \(s_3^0\}\}\).
It is obvious that the equivalence relation established on \({\mathcal {M}}_{\sigma _1} (R_{\sigma _1})\) is \(R_2=\) \(R_{\sigma _1}\) and so \({\mathcal {M}}_{\sigma }(R_2)\) \(={\mathcal {M}}_{\sigma _1}(R_{\sigma _1})\).
6 Conclusion and further works
Topological graphs (El Atik et al. 2021; El Atik and Hassan 2020; Nada et al. 2018) and fuzzy topological graphs (Atef et al. 2021) may represent many structures. This transformation has numerous applications, including selfsimilar fractals (El Atik and Nasef 2020) and smart cities (Atef et al. 2021). Moreover, matroids may be used to depict simplicial complexes. The primary goal of this study is to provide new types of simplicial matroids. The first type arises as a result of matrices; then, its circuits and underlying principles are described. Furthermore, rank functions and closure operators on simplicial matroids are investigated. The base of a restriction simplicial matroid is studied. Rough sets may generate additional simplicial matroids in terms of certain equivalence relations. Furthermore, closure and interior operators are proposed, and the border area is properly defined as a result. On the other hand, the resultant simplicial matroids are used to provide more equivalent relations. New equivalence relations are generated to create more equal simplicial matroids. Finally, our proposals are feasible because they pave the way for certain actual problems in more matroidal applications (Serrano et al. 2020; Wang et al. 2020).
Using the results of this article, we will do further research in the following areas: topological and big data analysis, rough membership function, fuzzy and soft set theory, some medical applications, and so on. Moreover, a topological model for simplicial complexes can be represented. In this case, the brain cab covered by a union of simplicial complexes which may be used to give a diagnosis for brain cancer.
Availability of data and material
Not applicable.
Abbreviations
 \(\sigma \) :

Simplicial complex
 \(s^k\) :

ksimplex
 U :

Finite universe set
 R :

Equivalence relation
 \([x]_R\) :

Equivalence classes
 \({\underline{R}}\) :

Lower approximation
 R :

Upper approximation
 \(\text {BN}_g\) :

Boundary region
 \({\mathcal {M}}\) :

Matroids
 \({\mathcal {C}}({\mathcal {M}})\) :

Circuits of matroids
 \(B({\mathcal {M}})\) :

Base of matroids
 \(r_{{\mathcal {M}}}\) :

Rank function
 \(\text {Cl}_{{\mathcal {M}}}\) :

Closure operator
 \({\mathcal {D}}\) :

Matrix
 \({\mathcal {V}}_\sigma \) :

Column labels set (or, a ground set)
 \({\mathcal {W}}\) :

Linearly independent columns set
 \({\mathcal {M}}_\sigma \) :

Simplicial matroids
 \(\text {Max}({\mathcal {W}})\) :

Maximal independent set
 \({\mathcal {B}}({\mathcal {M}}_\sigma )\) :

Base matroid set
 \({\mathcal {M}}_\sigma /D\) :

Restriction simplicial matroids
References
Atef M, El Atik AA, Nawar A (2021) Fuzzy topological structures via fuzzy graphs and their applications. Soft Comput 25:6013–6027. https://doi.org/10.1007/s00500021055948
Bartol W, Miro J, Pioro K, Rossello F (2004) On the coverings by tolerance classes. Inf Sci 166(1–4):193–211. https://doi.org/10.1016/j.ins.2003.12.002
Cattaneo G, Ciucci D (2002) A quantitative analysis of preclusivity vs. similarity based rough approximations, Rough Sets and Current Trends in Computing, vol. 2475 of LNCS, 2002, 69–76
Cavaliere D, Senatore S, Loia V (2017) Contextaware profiling of concepts from a semantic topological space. KnowlBased Syst 130:102–115. https://doi.org/10.1016/j.knosys.2017.05.008
Deng T, Chen Y, Xi W, Dai Q (2007) A novel approach to fuzzy rough sets based on a fuzzy covering. Inf Sci 177(11):2308–2326. https://doi.org/10.1016/j.ins.2006.11.013
Diker M (2010) Textural approach to generalized rough sets based on relations. Inf Sci 180(8):1418–1433. https://doi.org/10.1016/j.ins.2009.11.032
Edmonds J (1971) Matroids and the greedy algorithm. Math Program 1(1):127–136. https://doi.org/10.1007/BF01584082
El Atik AA, Nawar A, Atef M (2021) Rough approximation models via graphs based on neighborhood Systems, Granular. Computing 6:1025–1035. https://doi.org/10.1007/s4106602000245z
El Atik AA, Wahba AS (2020) Topological approaches of graphs and their applications by neighborhood systems and rough sets. J Intell Fuzzy Syst 39(5):6979–6992. https://doi.org/10.3233/JIFS200126
El Atik AA, Wahba AS (2022) Some betweenness relation topologies induced by simplicial complexes. Hacettepe J Math Statistics 51(4):981–994. https://doi.org/10.15672/hujms.787479
El Atik AA, Nasef AA (2020) Some topological structures of fractals and their related graphs. Filomat 34(1):1–24. https://doi.org/10.2298/FIL2001153A
El Atik AA (2020) Reduction based on similarity and decisionmaking. J Egyptian Math Soc 28(1):1–12. https://doi.org/10.1186/s42787020000784
El Atik AA, Hassan HZ (2020) Some nano topological structures via ideals and graphs. J Egyptian Math Soc 28(41):1–21. https://doi.org/10.1186/s42787020000935
Estrada E, Ross GJ (2018) Centralities in simplicail complexes, Applications to protein interaction. J Theor Biol 438:46–60. https://doi.org/10.1016/j.jtbi.2017.11.003
Herawan T, Deris M, Abawajy J (2010) Rough set approach for selecting clustering attribute. KnowlBased Syst 23:220–231. https://doi.org/10.1016/j.knosys.2009.12.003
Hu Q, Zhang L, Chen D, Pedrycz W, Yu D (2010) Gaussian kernel based fuzzy rough sets: model, uncertainty measures and applications. Int J Approximate Reason 51(4):453–471. https://doi.org/10.1016/j.ijar.2010.01.004
Huang KY, Chang TH, Chang TC (2011) Determination of the threshold value \(\beta \) of variable precision rough set by fuzzy algorithms. Int J Approximate Reason 52(7):1056–1072. https://doi.org/10.1016/j.ijar.2011.05.001
Gong Z, Sun B, Chen D (2008) Rough set theory for the intervalvalued fuzzy information system. Inf Sci 178(8):1968–1985. https://doi.org/10.1016/j.ins.2007.12.005
Jensen R, Shen Q (2004) Semanticspreserving dimensionality reduction: rough and fuzzyroughbased approaches. IEEE Trans Knowl Data Eng 16(12):1457–1471. https://doi.org/10.1109/TKDE.2004.96
Kondo M (2005) On the structure of generalized rough sets. Inf Sci 176(5):589–600. https://doi.org/10.1016/j.ins.2005.01.001
Lashin E, Kozae A, Khadra AA, Medhat T (2005) Rough set theory for topological spaces. Int J Approximate Reason 40:35–43. https://doi.org/10.1016/j.ijar.2004.11.007
Li X, Liu S (2012) Matroidal approaches to rough set theory via closure operators. Int J Approximate Reason 53(4):513–527. https://doi.org/10.1016/j.ijar.2011.12.005
Lai HJ (2002) Matroid theory, Higher Education Press
Liu G, Zhu W (2008) The algebraic structures of generalized rough set theory. Inf Sci 178(21):4105–4113. https://doi.org/10.1016/j.ins.2008.06.021
Mao H (2006) The relation between matroid and concept lattice. Adv Math 35(3):361–365
Nada S, El Atik AA, Atef M (2018) New types of topological structures via graphs. Math Methods Appl Sci 41(15):5801–5810. https://doi.org/10.1002/mma.4726
Nawar AS, ElBably MK, El Atik AA (2020) Certain types of coverings based rough sets with application. J Intel Fuzzy Syst 39(3):3085–3098. https://doi.org/10.3233/JIFS191542
Ouyang Y, Wang Z, Zhang H (2010) On fuzzy rough sets based on tolerance relations. Inf Sci 180(4):532–542. https://doi.org/10.1016/j.ins.2009.10.010
Oxley JG (1993) Matroid theory. Oxford University Press, New York
Pawlak Z (1982) Rough sets. Int J Comput Inform Sci 11:341–356
Pal S, Mitra P (2004) Case generation using rough sets with fuzzy representation. IEEE Trans Knowl Data Eng 16:293–300. https://doi.org/10.1109/TKDE.2003.1262181
Pawlak Z (1991) Rough sets: theoretical aspects of reasoning about data. Kluwer Academic Publishers, Boston
Qian Y, Liang J, Dang C (2009) Knowledge structure, knowledge granulation and knowledge distance in a knowledge base. Int J Approximate Reason 50(1):174–188. https://doi.org/10.1016/j.ijar.2008.08.004
Qin K, Yang J, Pei Z (2008) Generalized rough sets based on reflexive and transitive relations. Inform Sci 178(21):4138–4141. https://doi.org/10.1016/j.ins.2008.07.002
Serrano DH, Serrano J, Gómez DS (2020) Simplicial degree in complex networks. Applications of topological data analysis to network science, Chaos, Solitons and Fractals 137:109839. https://doi.org/10.1016/j.chaos.2020.109839
Skowron A, Stepaniuk J (1996) Tolerance approximation spaces. Fundamental Informaticae 27:245–253. https://doi.org/10.3233/FI1996272311
Skowron A, Stepaniuk J, Swiniarski R (2012) Modeling rough granular computing based on approximation spaces. Inf Sci 184:20–43. https://doi.org/10.1016/j.ins.2011.08.001
Slowinski R (2000) amd D. Vanderpooten, A generalized definition of rough approximations based on similarity, IEEE Transactions on Knowledge and Data Engineering 12(2):331–336. https://doi.org/10.1109/69.842271
Stolz B (2014) Computational topology in neuro science, M.Sc. thesis, University of Oxford, LondonEngland
Tsumoto S (2002) Rule and Matroid theory, Proceedings of the \(26^{\text{th}}\) Annual International Computer Software and Applications Conference, 1176–1181. https://doi.org/10.1109/CMPSAC.2002.1045171
Tsumoto S, Tanaka H (1994) Characterization of relevance and irrelevance in empirical learning methods based on rough sets and matroid theory, AAAI Technical Report FS9402, 183–186
Tsumoto S, Tanaka H (1996) A common algebraic framework of empirical learning methods based on rough sets and matroid theory. Fundamental Informaticae 27:273–288. https://doi.org/10.3233/FI1996272313
Wang S, Zhu W (2011) Matroidal structure of coveringbased rough sets through the upper approximation number. Int J Granular Comput, Rough Sets and Intell Syst 2(2):141–148
Wang S, Zhu Q, Xhu W, Min F (2012) Matroidal structure of rough sets and its characterization to attribute reduction. KnowlBased Syst 36:155–161. https://doi.org/10.1016/j.knosys.2012.06.006
Wang S, Zhu Q, Zhu W, Min F (2013) Quantitative analysis for coveringbased rough sets through the upper approximation number. Inf Sci 220:483–491. https://doi.org/10.1016/j.ins.2012.07.030
Wang S, Zhu Q, Zhu W, Min F (2014) Rough set characterization for 2circuit matroid. Fund Inform 129:377–393. https://doi.org/10.3233/FI2013977
Wang D, Zhao Y, Leng H, Small M (2020) A social communication model based on simplicial complexes. Phys Lett A 384:126895. https://doi.org/10.1016/j.physleta.2020.126895
Wang LXW (2021) Restricted intersecting families on simplicial complex. Adv Appl Math 124:102144. https://doi.org/10.1016/j.aam.2020.102144
Yang XP, Li TJ (2006) The minimization of axiom sets characterizing generalized approximation operators. Inf Sci 176:887–899. https://doi.org/10.1016/j.ins.2005.01.012
Yao Y (2011) Two semantic issues in probabilistic rough set model. Fund Inform 108(3–4):249–265. https://doi.org/10.3233/FI2011422
Yao Y (2010) Threeway decisions with probabilistic rough sets. Inf Sci 180(30):341–353. https://doi.org/10.1016/j.ins.2009.09.021
Yao Y (2011) The superiority of threeway decisions in probabilistic rough set models. Inf Sci 181:1080–1096. https://doi.org/10.1016/j.ins.2010.11.019
Yu Z, Bai X, Yun Z (2013) A study of rough sets based on \(1\)neighborhood systems. Inf Sci 248:103–113. https://doi.org/10.1016/j.ins.2013.06.031
Zhan J, Zhang X, Yao Y (2020) Covering based multigranulation fuzzy rough sets and corresponding applications. Artificial Intell Rev 53:1093–1126. https://doi.org/10.1007/s1046201909690y
Zhu W (2007) Generalized rough sets based on relations. Inf Sci 177(22):4997–5011. https://doi.org/10.1016/j.ins.2007.05.037
Zhu W (2007) Topological approaches to covering rough sets. Inf Sci 177(6):1499–1508. https://doi.org/10.1016/j.ins.2006.06.009
Zhu W (2009) Relationship among basic concepts in coveringbased rough sets. Inf Sci 179(14):2478–2486. https://doi.org/10.1016/j.ins.2009.02.013
Zhu W (2009) Relationship between generalized rough sets based on binary relation and covering. Inf Sci 179(3):210–225. https://doi.org/10.1016/j.ins.2008.09.015
Zhu W, Wang F (2007) On three types of covering rough sets. IEEE Trans Knowl Data Eng 19(8):1131–1144. https://doi.org/10.1109/TKDE.2007.1044
Zhu W, Wang S (2011) Matroidal approaches to generalized rough sets based on relations. Inform J Mach Learn Cybernet 2:273–279. https://doi.org/10.1007/s130420110027y
Zuffi L. Simplicial complexes from graphs towards graph persistence, M.Sc. thesis, Universita di Bologna, 2015/2016
Funding
Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB).
Author information
Authors and Affiliations
Contributions
The author jointly worked on the results, and He read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
El Atik, A.E.F. Approximation of simplicial complexes using matroids and rough sets. Soft Comput 27, 2217–2229 (2023). https://doi.org/10.1007/s00500022077746
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500022077746
Keywords
 Simplicial complexes
 Rough set theory
 Matroids
 2circuit matroids
 Graph theory