Abstract
The related properties of derivations in lattices are investigated. We show that the set of all isotone derivations in a distributive lattice can form a distributive lattice. Moreover, we introduce the fixed set of derivations in lattices and prove that the fixed set of a derivation is an ideal in lattices. Using the fixed sets of isotone derivations, we establish characterizations of a chain, a distributive lattice, a modular lattice and a relatively pseudo-complemented lattice, respectively. Furthermore, we discuss the relations among derivations, ideals and fixed sets in lattices.
MSC:06B35, 06B99.
Similar content being viewed by others
1 Introduction
The system of lattice algebra plays a significant role in information theory [1], information retrieval [2], information access controls [3] and cryptanalysis [4]. In [1], Bell described the co-information lattice, used it to show how to express the probability density under a general hypergraphical model, and then used this to derive the lattice of dependent component analysis algorithms. In [2], Carpineto and Romano applied lattices to information retrieval. They introduced the bound facility and the integration of this and several other useful features, such as automatic indexing, fisheye view browser for lattice, and the use of thesaurus into a basic lattice framework. In [3], Sandhu showed that lattice-based mandatory access controls can be enforced by appropriate configuration of RBAC components. His constructions demonstrated that role hierarchies and constraints were required to effectively achieve this result. In [4], Durfee applied tools from the geometry of numbers to solve several problems in cryptanalysis. They used algebraic techniques to cryptanalyze several public key cryptosystems. They focused on RSA and RSA-like schemes and used tools from the theory of integer lattices to get some results.
The notion of derivation, introduced from the analytic theory, is helpful for the research of structure and property in an algebraic system. Recently, analytic and algebraic properties of lattices have been widely researched [5–7]. Several authors [8–12] studied derivations in rings and near-rings. Jun and Xin [13] applied the notion of derivation in ring and near-ring theory to BCI-algebras.
In [14], Xin et al. introduced the concept of derivation in a lattice and investigated some properties. They gave some equivalent conditions, under which a derivation is isotone for lattices with a greatest element, modular lattices and distributive lattices, respectively. They characterized modular lattices and distributive lattices by isotone derivations. But the relations among derivations, ideals and fixed sets were not investigated in that paper. We will discuss when an ideal can appear as this ‘fixed set’ for a derivation in this paper. This paper is a continuation to the paper [14].
The remainder of this paper is organized as follows. In Section 2, we recall some definitions and some properties of lattice theory. In Section 3, we investigate further related properties of derivations in lattices and show a structural theorem of all isotone derivations in distributive lattices. In Section 4, we introduce the fixed set of derivations and get some interesting properties of them. Especially, using the fixed set of isotone derivations, we establish characterizations for some kinds of lattices. Furthermore, we discuss the relations among derivations, ideals and fixed sets in lattices. Finally, some concluding remarks are made in Section 5.
2 Preliminaries
Definition 2.1 [15]
Let L be a nonempty set endowed with operations ‘∧’ and ‘∨’. If satisfies the following conditions: for all ,
-
(A)
, ;
-
(B)
, ;
-
(C)
, ;
-
(D)
, ,
then L is called a lattice.
Definition 2.2 [15]
A lattice L is distributive if the identity (E) or (F) holds.
-
(E)
,
-
(F)
.
In any lattice, the conditions (E) and (F) are equivalent.
Definition 2.3 [16]
A lattice L is modular if the identity (M) holds.
-
(M)
If , then .
Definition 2.4 [15]
A relatively pseudo-complemented lattice (or Brouwerian lattice) is a lattice L in which, for any given elements , the set of all such that contains a greatest element , the relative pseudo-complement of a in b.
Lemma 2.5 [15]
Any relatively pseudo-complemented lattice is distributive.
Definition 2.6 [15]
A Boolean algebra is an algebra with two binary operations ∨, ∧, one unary operation ′, and two nullary operations 0, 1, such that the following conditions are satisfied:
-
(1)
is a distributive lattice;
-
(2)
for all , , ;
-
(3)
for all , there is such that , .
Definition 2.7 [15]
Let be a lattice. A binary relation ‘≤’ is defined by if and only if and .
Lemma 2.8 [15]
Let be a lattice. Define the binary relation ‘≤’ as in Definition 2.7. Then is a poset and for any , is the g.l.b. of , and is the l.u.b. of .
From Lemma 2.8, we can see that a lattice is not only an algebraic system, but also an order structure.
Definition 2.9 [15]
Let be a function from a lattice L to a lattice M. Then θ is a lattice-homomorphism (or homomorphism) when
and
for all .
As always, a homomorphism is called an isomorphism if it is a bijection, an epimorphism if onto, a monomorphism if one-to-one.
Definition 2.10 [15]
An ideal is a non-void subset I of a lattice L with the properties
-
(1)
, ,
-
(2)
, for all . Moreover, an ideal I of a lattice L is called a prime ideal if I satisfies the following condition:
-
(3)
implies or for all .
Note that if and are ideals of a lattice L, so is .
3 The derivations in lattices
In this section, we recall some definitions and results of the paper [14].
The following definition introduces the notion of derivation for a lattice, which comes in analogy with Leibniz’s formula for derivations in a ring.
Definition 3.1 [14]
Let L be a lattice and be a function. We call d a derivation on L if it satisfies the condition .
We often abbreviate to dx.
Now we give some examples and present some properties for the derivations in lattices.
Example 3.2 Let L be the lattice of Figure 1, and define functions and on L by
Then we can see that is not a derivation but is a derivation on L.
Proposition 3.3 [14]
Let L be a lattice and d be a derivation on L. Then the following hold:
-
(1)
;
-
(2)
;
-
(3)
If I is an ideal of L, then , where ;
-
(4)
If L has a least element 0, then .
Remark 3.4 In Proposition 3.3, we get an interesting property of derivation, i.e., . This means that any derivation in lattices is a contraction mapping. By the principle of a contraction mapping, any derivation in lattices must have fixed points. We will discuss the structures and properties of the fixed point set of a derivation for a lattice later.
Definition 3.5 [14]
Let L be a lattice and d be a derivation on L.
-
(1)
If implies , we call d an isotone derivation.
-
(2)
If d is one-to-one, we call d a monomorphic derivation.
-
(3)
If d is onto, we call d an epic derivation.
By analogy with principal ideals, we introduce a principal derivation in lattices as follows.
Definition 3.6 Let L be a lattice and . Define a function on L by for all . Then we can see that is a derivation on L. In the following, we refer to such derivations as principal.
Proposition 3.7 Every principal derivation of a lattice L is an isotone derivation of L.
Proof Let be a principal derivation of a lattice L. Since for any and , we have and hence is isotone. □
Proposition 3.8 [14]
Let L be a lattice and d be a derivation on L. If and , then .
Proposition 3.9 [14]
Let L be a lattice and d be a derivation on L. Define for all . Then we have .
Theorem 3.10 Let L be a lattice and be a derivation. Then the following are equivalent:
-
(1)
d is an isotone derivation;
-
(2)
.
Proof (1) ⇒ (2). Assume d is isotone. Then . Conversely, since and , we can get and . Then . Therefore, .
-
(2)
⇒ (1). Assume for all in L. Then . Then . Furthermore, if , since , then . Therefore, . We can get . □
Theorem 3.11 Let L be a lattice and be a derivation. Then the following are equivalent:
-
(1)
;
-
(2)
.
Proof (1) ⇒ (2). Obversely, we have . By (1), . Since and , we can get .
-
(2)
⇒ (1). Assume for all in L. If , then . We can get . This shows that d is an isotone derivation. From Theorem 3.10, we know (1) holds. □
From the Theorem 3.10 and Theorem 3.11, we have the following theorem.
Theorem 3.12 Let L be a lattice and be a derivation. Then the following are equivalent:
-
(1)
d is an isotone derivation;
-
(2)
;
-
(3)
.
However, derivations of distributive lattices have stronger properties.
Theorem 3.13 [14]
Let L be a distributive lattice and d be a derivation on L. Then the following are equivalent:
-
(1)
d is isotone;
-
(2)
;
-
(3)
.
Theorem 3.14 Let L be a distributive lattice and and be two isotone derivations on L. Define
Then and are also isotone derivations on L.
Proof We first prove is an isotone derivation on L.
By Theorem 3.13, we have
Similarly, we can get .
Combining the above arguments, we have
So, is a derivation on L by Definition 3.1.
Moveover, , so is isotone by Theorem 3.13.
Similar to the above process, we can prove is an isotone derivation on L and we omit it. □
Theorem 3.15 Let L be a distributive lattice and be a set of all isotone derivations on L. Then is a distributive lattice.
Proof From Theorem 3.14, ∨ and ∧ are binary operators on . Define a binary relation ‘≤’ on by iff . Then ‘≤’ is a partial order relation on and , . Therefore, is a lattice.
In addition, for any and any ,
Therefore, . This shows that is a distributive lattice. □
4 The fixed set of a derivation in lattices
Theorem 4.1 Let L be a lattice and d be an isotone derivation on L. Denote . Then is an ideal of L.
Proof By Proposition 3.8 we can see that and imply . This means that satisfies the condition (1) of Definition 2.10. For the condition (2) of Definition 2.10, we consider . By the isotoneness of d, we have and so . This means that satisfies Definition 2.10. It follows that is an ideal of L. □
In the following proposition, we can see that an isotone derivation d is determined by the ideal .
Proposition 4.2 Let L be a lattice and and be two isotone derivations on L. Then if and only if .
Proof It is obvious that implies . Inversely, let and . By Proposition 3.9, and so . Similarly, we can get . Since and are isotone, we have and so . Symmetrically, we can also get , this shows that . It follows that , that is, . □
Theorem 4.3 Let L be a lattice. Then the following are equivalent:
-
(1)
L is a chain;
-
(2)
For every isotone derivation d, is a prime ideal.
Proof (1) ⇒ (2). Let L be a chain and d be an isotone derivation on L. Then is an ideal of L by Theorem 4.1. Moreover, let . Since L is a chain, then or . Assume , then and so . It follows that . This shows that is a prime ideal.
-
(2)
⇒ (1). Let, for every isotone derivation d, be a prime ideal. For , consider the principal derivation , which is induced by . Then is a prime ideal by hypothesis. Note that . Hence, or . Assume , then . So, . This means that L is a chain. □
To get a characterization of distributive lattices using the fixed set of a derivation, we introduce the following concept.
Let L be a lattice and I be an ideal of L. Define a relation ‘≡’ in L by if and only if and for some . We can easily see that this relation is an equivalent relation.
Definition 4.4 [15]
Let L be a lattice and I be an ideal of L. We call I a standard ideal if it satisfies the following condition: implies and for all or, equivalently, the relation ‘≡’ is a congruence relation.
Theorem 4.5 Let L be a lattice. Then the following are equivalent:
-
(1)
L is distributive;
-
(2)
For every isotone derivation d, is a standard ideal of L.
Proof (1) ⇒ (2). Let L be a distributive lattice and d be an isotone derivation. Now we claim that this relation is a congruence relation. In fact, let . If , then and for some , and so and . Similarly, we can get and . This shows that and . It follows that the relation is a congruence relation. Thus, is a standard ideal of L.
-
(2)
⇒ (1). Assume that holds. For any , consider the derivation , which is induced by a, that is, for all . Note that is a standard ideal of L and . Hence, the relation ‘≡’, which is defined by if and only if and for some , is a congruence relation on L by hypothesis. Notice that and , , we have . Similarly, we can get . Moreover, . It follows that for some . From , we have , and then we get . Hence, . It follows that L is distributive. □
In order to discuss the structural properties of the fixed set of isotone derivations in modular lattices, we introduce a semi-standard ideal in a lattice.
Let L be a lattice and I be a principal ideal of L generated by , that is, . Define a relation ‘∼’ in L by if and only if for all . Then we can see that the relation ∼ is an equivalent relation on L.
Definition 4.6 Let L be a lattice and be a principal ideal of L. We call I a semi-standard ideal if it satisfies the following condition: implies for all .
In the following, we give a property of principal ideals in a modular lattice.
Proposition 4.7 In a modular lattice, every principal ideal is a semi-standard ideal.
Proof Let L be a modular lattice and be a principal ideal of L. Assume and . Then . Taking , then . Notice that
since L is modular. It follows that and so I is a semi-standard ideal. □
Now, using fixed sets of derivations, we give a condition by which a lattice becomes a modular lattice.
Proposition 4.8 Let L be a lattice. If d is a principal derivation of L, then is a principal ideal.
Proof Assume that d is a principal derivation of L, that is, for some . We claim that . In fact, for any , we have and hence . This means that . Conversely, let , that is, . Then and hence . By the above arguments, we have , and so is a principal ideal. □
Proposition 4.9 Let L be a lattice. If for every principal derivation d of L, the ideal is semi-standard, then L is modular.
Proof Assume that for every principal derivation d of L, the ideal is semi-standard. Let and . Consider the derivation induced by a, that is, for all . Since is a principal derivation, then the fixed set is a principal ideal by Proposition 4.8 and hence it is semi-standard by Proposition 4.7. Notice that and , we have . Moreover, since I is semi-standard. This means that . Since , we have . Hence, and so L is modular. □
Combining Proposition 4.7 and Proposition 4.9, we can get a characterization of a modular lattice by the fixed set of a derivation.
Theorem 4.10 Let L be a lattice. Then the following are equivalent:
-
(1)
L is modular;
-
(2)
For every principal derivation d of L, the ideal is semi-standard.
Now we discuss a characterization of relatively pseudo-complemented lattices by the fixed set of isotone derivations.
Theorem 4.11 Let L be a lattice. Then the following are equivalent:
-
(1)
L is a relatively pseudo-complemented lattice.
-
(2)
Every principal derivation d of L satisfies that the set has a greatest element for any .
-
(3)
Every principal derivation d of L satisfies that the set has a greatest element for any .
-
(4)
Every principal derivation d of L satisfies that the set is a principal ideal of L for any .
Proof (1) ⇒ (2). Let L be a relatively pseudo-complemented lattice and d be a principal derivation. Then there is such that . Assume that and . Then and hence since L is a relatively pseudo-complemented lattice. On the other hand, . It follows that . So, we have that has a greatest element .
-
(2)
⇒ (3). Straightforward.
-
(3)
⇒ (4). Let (3) hold. Let be the greatest element of for . Then , where is the ideal generated by . In fact, for , we have and so . Conversely, let , then . It follows that , this means . So, .
-
(4)
⇒ (1). Let (4) hold and . Consider a principal derivation , induced by a. By Proposition 3.7, is isotone. Note that and so . By hypothesis, the set is a principal ideal of L. Let , where is a principal ideal generated by . Therefore, for any , . It follows that and hence . So, . On the other hand, from , we have and . This shows that the set has a greatest element . It follows that exists. □
In the following, we discuss the relation between principal derivations and principal ideals in lattices.
Theorem 4.12 Let L be a lattice.
-
(1)
If d is a principal derivation of L, then is a principal ideal.
-
(2)
If I is a principal ideal of L, then there exists a unique isotone derivation d such that .
Proof (1) It follows from Proposition 4.8.
-
(2)
Let be a principal ideal of L. Consider the derivation d induced by a, that is, for all . Then if and only if . It follows that . In order to prove the uniqueness, we assume that there exist two derivations and , such that and . So, and hence by Proposition 4.2. □
Theorem 4.13 Let L be a lattice and I be a non-void prime ideal of L. Then there exists a derivation d such that .
Proof Define a function d as follows:
where . We claim that d is a derivation. In fact, if , then we can see that . If , , then and so . Hence, . This shows that . If , then since I is prime. Hence, . By the above argument, we can get that d is a derivation. Clearly, . □
Example 4.14 Let and , then is a lattice and I is an ideal of L, where ≤ is the ordinary order. Moreover, we can see that there is not any isotone derivation d such that .
We now determine some classes of lattices all of whose ideals are principle ideals.
Definition 4.15 A poset P is said to satisfy the ascending chain condition (A.C.C.) if every non-void subset of P has a maximal element. A poset P is said to satisfy the descending chain condition (D.C.C.) if every non-void subset of P has a minimal element.
Theorem 4.16 Let L be a lattice. If L satisfies A.C.C., then every ideal of L is a principal ideal.
Proof Let I be an ideal of L. By assumption, I has a maximal element . Therefore, for any , . Note that and is a maximal element of I, we have . Hence, . This shows that . □
By Theorem 4.12 and Theorem 4.16, we have the following theorem.
Theorem 4.17 Let L be a lattice satisfying A.C.C. Then for every ideal of L, there exists a unique isotone derivation d such that .
Finally, we can see that the set of fixed sets of isotone derivations has the same structure as the set of isotone derivations in distributive lattices.
Theorem 4.18 Let L be a distributive lattice and be a set of isotone derivations on L. Denote . Define
Then is a distributive lattice.
Proof By Theorem 3.13, for any , we have and . This shows that the operations ‘∧’ and ‘∨’ are closed on ℱ. We can easily show that is a lattice. Consider the function defined by . Then we can see that f is an isomorphism from to ℱ. It follows from the distributivity of that is a distributive lattice. □
From the proof of Theorem 4.18, we can get the following corollary.
Corollary 4.19 Let L be a distributive lattice. Then the lattice is isomorphic to the lattice .
5 Conclusions
In this paper, we investigate further related properties of derivations in lattices. We show that the set of all isotone derivations in a distributive lattice forms a distributive lattice under suitable binary operations. Moreover, we introduce the fixed set of a derivation and prove that the fixed set of a derivation is an ideal in lattices. Using the fixed sets of isotone derivations, we establish characterizations of a chain, a distributive lattice, a modular lattice and a relatively pseudo-complemented lattice, respectively. Furthermore, we discuss the relation between ideals and fixed sets of derivations in lattices. We get that for every principal ideal I and every prime ideal I, there exists a derivation d such that the fixed set of d is I.
We have seen that in some situations like lattices satisfying A.C.C., for every ideal of L, there exists an isotone derivation d such that . The question whether or not this property holds in general lattices remains unsolved. We will discuss this question on general ideals in further work.
References
Bell AJ: The co-information lattice. 4th Int. Symposium on Independent Component Analysis and Blind Signal Separation (ICA2003) 2003, 921–926. Nara
Carpineto C, Romano G: Information retrieval through hybrid navigation of lattice representations. Int. J. Human-Comput. Stud. 1996, 45: 553–578. 10.1006/ijhc.1996.0067
Sandhu RS: Role hierarchies and constraints for lattice-based access controls. Proceedings of the 4th European Symposium on Research in Computer Security 1996, 65–79. Rome
Durfee, G: Cryptanalysis of RSA using algebraic and lattice methods. A dissertation submitted to the department of computer science and the committee on graduate studies of stanford university, pp. 1–114 (2002)
Degang C, Wenxiu Z, Yeung D, Tsang ECC: Rough approximations on a complete distributive lattice with applications to generalized rough sets. Inf. Sci. 2006, 176: 1829–1848. 10.1016/j.ins.2005.05.009
Honda A, Grabisch M: Entropy of capacities on lattices and set systems. Inf. Sci. 2006, 176: 3472–3489. 10.1016/j.ins.2006.02.011
Karacal F: On the direct decomposability of strong negations and S-implication operators on product lattices. Inf. Sci. 2006, 176: 3011–3025. 10.1016/j.ins.2005.12.010
Bell HE, Kappe LC: Rings in which derivations satisfy certain algebraic conditions. Acta Math. Hung. 1989, 53(3–4):339–346. 10.1007/BF01953371
Bell HE, Mason G: On derivations in near-rings and near-fields. North-Holl. Math. Stud. 1987, 137: 31–35.
Hvala B: Generalized derivations in prime rings. Commun. Algebra 1998, 26: 1147–1166. 10.1080/00927879808826190
Kaya K: Prime rings with α derivations. Bull. Mater. Sci. 1987–1988, 16–17: 63–71.
Posner E: Derivations in prime rings. Proc. Am. Math. Soc. 1957, 8: 1093–1100. 10.1090/S0002-9939-1957-0095863-0
Jun YB, Xin XL: On derivations of BCI-algebras. Inf. Sci. 2004, 159: 167–176. 10.1016/j.ins.2003.03.001
Xin XL, Li TY, Lu JH: On derivations of lattices. Inf. Sci. 2008, 178: 307–316. 10.1016/j.ins.2007.08.018
Birkhoff G Colloquium Publications. In Lattice Theory. Am. Math. Soc., New York; 1940.
Balbes R, Dwinger P: Distributive Lattices. University of Missouri Press, Columbia; 1974.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Xin, X.L. The fixed set of a derivation in lattices. Fixed Point Theory Appl 2012, 218 (2012). https://doi.org/10.1186/1687-1812-2012-218
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2012-218