Approximation of simplicial complexes using matroids and rough sets

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 2-circuit simplicial matroids.

1 Introduction

The simplicial complex is a collection of some simplices or simplexes. A zero-dimensional simplex is a vertex $(s^0)$. A one-dimensional simplex is a line segment $(s^1)$. A two- dimensional simplex is a field triangle $(s^2)$. A three-dimensional 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 n-cliques which contain n-vertices 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 n-vertices are connected all to all like the case in ordinary clique in an undirected graph, but where in every sub-clique there is a unique vertex that is a source, i.e., all directed edges in the sub-clique are pointing out of it, and unique vertex that is a sink, that is, all directed edges in the sub-clique 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 n-clique 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), covering-based 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 well-founded 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 2-circuit 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 $0-1$ 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

(k-simplex Cavaliere et al. (2017)). A k-simplex $s^k= (s_1^0$, $s_2^0$, $s_3^0$, $\ldots$, $s_{k+1}^0)$, 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

(h-face Cavaliere et al. (2017)). Let $s^k=$ $(s_1^0$, $s_2^0$, $s_3^0$, $\ldots$, $s_{k+1}^0)$ be a k-simplex. An h-face is an h-simplex 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 k-simplices.

Definition 4

(boundary Estrada and Ross (2018)). The boundary of a k-simplex consists of $(k+1)$-simplices with dimension $k-1$; for instance, the boundary of a 1-simplex is two vertices and the boundary of a 2-simplex is three edges.

Definition 5

(directly connected Cavaliere et al. (2017)). If the intersection of two simplices generates a nonempty h-face with $h\le k$, then they are directly linked.

Definition 6

(h-connected Cavaliere et al. (2017)). Let $A=s_1^k, s_2^h,\ldots , s_{m-1}^h, s_m^k= B$ be nonempty k-simplices. Then, A and B are h-connected if each pair of successive $s_i^h$ and $s_{i+1}^h$ shares an h-face with $i=0, 1, 2,$ $\ldots ,$ $m-1$, where $h\le k$.

From Definitions 5 and 6, it is noted that each directly connected is h-connected.

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 $\exists$ $b\in B-A$ 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 $x\in A-B$, then $\exists$ $y\in B-A$ 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

(2-circuit matroids Wang et al. (2014)). ${\mathcal {M}}$ is a 2-circuit 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 0-face;

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

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

• $S^3=$ $\{s_m^3: m\in I_4\}$, in which each point is called a 3-face, 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 $s_i^{n-1}$ be distinct simplices in $\sigma$ with dimensions n and $n-1$, respectively. A matrix ${\mathcal {D}}$ is defined by

$$\begin{aligned}&{\mathcal {D}}= [d_{ij}] = {\left\{ \begin{array}{ll} 1, &{}{\text {if}} s_i^{n-1} {\text {is a boundary for}}\,\, s_j^n \\ 0, &{}{\text {otherwise}}. \end{array}\right. }&\end{aligned}$$

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 2-simplicial complex $\sigma _1$ with just one 2-face $s_1^2$, three 1-faces $s_1^1, s_2^1, s_3^1$ and three 0-faces $s_1^0, s_2^0, s_3^0$. 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.

Fig. 1

A 2-simplicial complex $\sigma _1$

Example 2

Figure 2 shows a 2-simplicial complex $\sigma _2$ with two 2-faces $s_1^2, s_2^2$, five 1-faces $s_1^1, s_2^1, s_3^1, s_4^1, s_5^1$, and four 0-faces $s_1^0, s_2^0, s_3^0, s_4^0$. Because the dimension of $\sigma _2$ is 2, there are two matrices.

Fig. 2

A 2-simplicial complex $\sigma _2$

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 $j \in J-I$. 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 $j\in J-I$, $J-I$ $\subseteq$ Span I and $I\subseteq$ Span I. Hence, $(J-I)\cup I$ $\subseteq$ 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 $A, B\in {\mathcal {B}}$, $a\in A-B$, and $b\in B-A$, $A-\{a\}\in {\mathcal {W}}$, $B\in {\mathcal {W}}$, 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 $a \in S_1-S_2$, then there is $b\in S_2-S_1$ 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. $a\in B-S_1$ (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 $a\in A-B$, $b\in B-A$. 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, $S_1\cup \{b\}\in {\mathcal {W}}$. $\square$

Fig. 3

$a\in B- S_1$

Fig. 4

$S_1\subseteq A-\{a\}$

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 $a\in B-A$. 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:

$$\begin{aligned} R_\sigma = \{(s_i^n, s_j^n): s_i^n \text { and } s_j^n \text { are } h\text {-connected}. \end{aligned}$$

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 h-connected (directly connected) with itself, $R_\sigma$ is reflexive. Since $s_i^n R_\sigma s_j^n$, then $s_i^n$ is h-connected with $s_j^n$, that is, $s_i^n=s_1^n, s_2^h,$ $\ldots ,$ $s_{m-1}^h, s_m^n =s_j^n$ and can also be reformulated to $s_j^n=s_m^n, s_{m-1}^h, \ldots , s_2^h, s_1^n=s_i^n$. So, $s_j^n$ and $s_i^n$ are h-connected, $s_j^n R_\sigma s_i^n$, 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, $s_i^n = s_1^n, s_2^h,$ $\ldots$ $, s_{m-1}^h, s_m^n = s_a^n$, 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 h-connected. 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) 0-faces 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 3-face 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,

$$\begin{aligned} {\mathcal {W}}(R_\sigma )&= \left\{ S\subseteq {\mathcal {U}}_\sigma : \forall s_i^n, s_j^n \right. \\ \in S, s_i^n&\left. \ne s_j^n ~{ implies} ~{ that} ~(s_i^n, s_j^n)\notin R_\sigma \right\} . \end{aligned}$$

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:

$$\begin{aligned} Cl_{{\mathcal {M}}_\sigma (R_\sigma )}(A)&=A \cup \{s_i^n\in {\mathcal {U}}_\sigma : \exists ~C\in C(R_\sigma )\,\, {\text{ such } \text{ that }}\,\, \\&\quad s_i^n\in C\subseteq A\cup \{s_i^n\}\}\\&= A\cup \{s_i^n \in {\mathcal {U}}_\sigma : \exists ~s_j^n\in A , s_i^n R_\sigma s_j^n\,\, {\text {such that}}\,\,\\&\quad \{s_i^n, s_j^n\}\subseteq A\cup \{s_i^n\}\}\\&= \{s_i^n\in {\mathcal {U}}_\sigma : \exists ~ s_j^n\in A, s_i^n R_\sigma s_j^n\}&\\&=\{s_i^n\in {\mathcal {U}}_\sigma : [s_i^n]_{R_\sigma }\cap A\ne \phi \}&\\&={\overline{R}}_\sigma (A).&\end{aligned}$$

$\square$

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, 2-circuit 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 2-circuit 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 2-circuit 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 2-circuit matroid.

Proof

Let ${\mathcal {M}}_\sigma =$ ${\mathcal {M}}_\sigma (R_1)$. Then, by Proposition 19, ${\mathcal {M}}_\sigma (R_1)$ is a 2-circuit matroid and hence ${\mathcal {M}}_\sigma$ is so. Conversely, let ${\mathcal {M}}_\sigma$ be a 2-circuit 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 2-circuit 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 2-circuit 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 2-circuit 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 2-circuit simplicial matroid. $\square$

Proposition 22

Let ${\mathcal {M}}_\sigma =$ $({\mathcal {V}}_\sigma , {\mathcal {W}})$ be a simplicial matroid. Then, ${\mathcal {M}}_\sigma$ is a 2-circuit 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 2-circuit 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 2-circuit matroid. $\square$

Proposition 23

Let ${\mathcal {M}}_\sigma =$ $({\mathcal {V}}_\sigma , {\mathcal {W}})$ be a 2-circuit 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 2-circuit 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 2-circuit 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 2-circuit 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 2-circuit 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 $R_{\sigma _3}$.

Fig. 5

A 2-directed simplicial complex $\sigma _3$ and their matrices

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 self-similar 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.