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In this paper,for a distributive lattice $\mathcal L$, we study and compare some lattice theoretic features of $\mathcal L$ and topological properties of the Stone spaces ${\rm Spec}(\mathcal L)$ and ${\rm Max}(\mathcal L)$ with the corresponding graph theoretical aspects of the zero-divisor graph $\Gamma(\mathcal L)$.Among other things,we show that the Goldie dimension of $\mathcal L$ is equal to the cellularity of the topological space ${\rm Spec}(\mathcal L)$ which is also equal to the clique number of the zero-divisor graph $\Gamma(\mathcal L)$. Moreover, the domination number of $\Gamma(\mathcal L)$ will be compared with the density and the weight of the topological space ${\rm Spec}(\mathcal L)$. For a $0$-distributive lattice $\mathcal L$, we investigate the compressed subgraph $\Gamma_E(\mathcal L)$ of the zero-divisor graph $\Gamma(\mathcal L)$ and determine some properties of this subgraph in terms of some lattice theoretic objects such as associated prime ideals of $\mathcal L$.

Any lattice embedding i : 핃 → M between two lattices 핃 ⊆ M induces a Banach lattice homomorphism î : FBL<핃> → FBL-- between the corresponding free Banach lattices generated by these lattices. We show that this mapping î might not be an isomorphic embedding. Sufficient conditions for ˆı to be an isometric embedding are provided by considering a notion of locally complemented lattices. As a consequence, we obtain that every free Banach lattice generated by a lattice is Banach lattice isomorphic to an AM-space. Furthermore, we prove that î is an isomorphic embedding if and only if it is injective, which in turn is equivalent to the fact that every lattice homomorphism x* : 핃 → [--1,1] extends to a lattice homomorphism x* : M→ [--1,1] . Using this characterization we provide an example of lattices 핃 ⊆M for which ˆı is an isomorphic not isometric embedding. [ABSTRACT FROM AUTHOR]

We discuss initial value problems for time evolution equations in one dimensional space which are expressed by the lattice operators and propose some new equations to which complexity of solutions is of polynomial class. Novel type of expressions using shift operators and binary trees are applied for the derivation of solution. [ABSTRACT FROM AUTHOR]

In this paper, we present a rough set model based on fuzzy ideals of distributive lattices. In fact, we consider a distributive lattice as a universal set and we apply the concept of a fuzzy ideal for definitions of the lower and upper approximations in a distributive lattice. A novel congruence relation induced by a fuzzy ideal of a distributive lattice is introduced. Moreover, we study the special properties of rough sets which can be constructed by means of the congruence relations determined by fuzzy ideals in distributive lattices. Finally, the properties of the generalized rough sets with respect to fuzzy ideals in distributive lattices are also investigated.

Let L be a restricted Lie algebra over a field of characteristic p>0. We investigate the structure of L when its lattice S (L) of restricted subalgebras satisfies some prescribed properties. In particular, we establish when S (L) is distributive. The special form that this result takes when F is either the field with p elements or an algebraically closed field is also discussed. Furthermore, we establish when S (L) is Boolean or the two-elements lattice. [ABSTRACT FROM AUTHOR]

EQ-algebras were introduced by Novak (2006) as an algebraic structure of truth values for fuzzy-type theory (FFT). Novák and De Baets (2009) introduced various kinds of EQ-algebras such as good, residuated, and IEQ-algebras. In this paper, we define the notion of (pre)ideal in bounded EQ-algebras (BEQ-algebras) and investigate some properties. Then, we introduce a congruence relation on good BEQ-algebras by using ideals, and then, we solve an open problem in Paad (2019). Moreover, we show that in IEQ-algebras, there is a one-to-one correspondence between congruence relations and the set of ideals. In the following, we characterize the generated preideal in BEQ-algebras, and by using this, we prove that the family of all preideals of a BEQ-algebra is a complete lattice. Then, we show that the family of all preideals of a prelinear IEQ-algebras is a distributive lattice and becomes a Heyting algebra. Finally, we show that we can construct an MV-algebra from the family of all preideals of a prelinear IEQ-algebra. [ABSTRACT FROM AUTHOR]

The existing generating methods of fuzzy implications are closed in the set of all fuzzy implications, but not when applied to some families of these operators. In this paper, some binary operations are defined on some well-established families of fuzzy implications. Namely, the families of (S , N) -implications with continuous t-conorms, R -implications obtained from continuous t-norms, Yager's f - and g -generated implications, h - and generalized (h , e) -implications and k -implications are considered and it is proved that with these operations, they are lattices. Moreover, some other lattice substructures are defined in some of the families when some restrictions on the underlying generators of the implications are imposed. Furthermore, it is emphasized that these generating methods preserve important properties like the exchange principle and the law of importation. [ABSTRACT FROM AUTHOR]

A discrete Sugeno integral on a bounded distributive lattice L is defined as an idempotent weighted lattice polynomial. Another possibility for axiomatization of Sugeno integrals is to consider compatible aggregation functions, uniquely extending the L-valued fuzzy measures. This paper aims to study the mentioned unique extension property concerning the possible extension of a Sugeno integral to non-distributive lattices. We show that this property is equivalent to the distributivity of the underlying bounded lattice. As a byproduct, an alternative proof of Iseki’s result, stating that a lattice having prime ideal separation property for every pair of distinct elements is distributive, is provided.

We study centralizer clones of finite lattices and semilattices. For semilattices, we give two characterizations of the centralizer and also derive formulas for the number of operations of a given essential arity in the centralizer. We also characterize operations in the centralizer clone of a distributive lattice, and we prove that the essential arity of operations in the centralizer is bounded for every finite (possibly nondistributive) lattice. Using these results, we present a simple derivation for the centralizers of clones of Boolean functions. [ABSTRACT FROM AUTHOR]

Let D be a distributive lattice, and let L be a frame. In this article, we introduce the notion of L-ideals of D. We show that the set of all L-ideals of D is a distributive lattice, and some essential properties of this lattice are studied. Also, we discuss some special elements of this lattice. Moreover, we define a novel congruence relation for the concept of L-ideal of D. Finally, we study some properties of rough sets inherited from the congruence relation. [ABSTRACT FROM AUTHOR]

Distributive lattices form an important, well-behaved class of lattices. They are instances of two larger classes of lattices: congruence-uniform and semidistributive lattices. Congruence-uniform lattices allow for a remarkable second order of their elements: the core label order; semidistributive lattices naturally possess an associated flag simplicial complex: the canonical join complex. In this article we present a characterization of finite distributive lattices in terms of the core label order and the canonical join complex, and we show that the core label order of a finite distributive lattice is always a meet-semilattice. [ABSTRACT FROM AUTHOR]

If R ⊆ S is an extension of commutative rings, we consider the lattice ([R, S], ⊆) of all the R-subalgebras of S. We assume that the poset [R, S] is both Artinian and Noetherian; that is, R ⊆ S is an FCP extension. The Loewy series of such lattices are studied. Most of main results are gotten in case these posets are distributive, which occurs for integrally closed extensions. In general, the situation is much more complicated. We give a discussion for finite field extensions. [ABSTRACT FROM AUTHOR]

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This paper introduces the concept of lattice valued inner product space. A natural inner product structure for the space of n-tuples over Boolean algebras is generalised to n-tuples over a distributive lattice with 0 and 1. A dimension theorem for orthonormal basis of a Boolean vector space is generalised to a lattice vector space. Further, we prove that for any naturals m; n with m < n, any set of m orthogonal unit ortho vectors in a lattice vector space can be extended to an orthonormal basis (known in the Boolean algebra case). [ABSTRACT FROM AUTHOR]

This paper introduces the concept of lattice valued inner product space. A natural inner product structure for the space of n-tuples over Boolean algebras is generalised to n-tuples over a distributive lattice with 0 and 1. A dimension theorem for orthonormal basis of a Boolean vector space is generalised to a lattice vector space. Further, we prove that for any naturals m; n with m < n, any set of m orthogonal unit ortho vectors in a lattice vector space can be extended to an orthonormal basis (known in the Boolean algebra case). [ABSTRACT FROM AUTHOR]

We give a full description of associative algebras over an arbitrary field, whose subalgebra lattice is distributive. All such algebras are commutative, their nil-radical is at most two-dimensional, and the factor algebra with respect to the nil-radical is an algebraic extension of the base field. [ABSTRACT FROM AUTHOR]

Let G be a plane elementary bipartite graph with more than two vertices. Then its resonance graph Z (G) is a median graph and the set M (G) of all perfect matchings of G with a specific partial order is a finite distributive lattice. In this paper, we prove that Z (G) is cube-free if and only if it can be obtained from an edge by a sequence of convex path expansions with respect to a reducible face decomposition of G. As a corollary, a structure characterization is provided for G whose Z (G) is cube-free. Furthermore, Z (G) is cube-free if and only if the Clar number of G is at most two, and sharp lower bounds on the number of perfect matchings of G can be expressed by the number of finite faces of G and the number of Clar formulas of G. It is known that a cube-free median graph is not necessarily planar. Using the lattice structure on M (G) , we show that Z (G) is cube-free if and only if Z (G) is planar if and only if M (G) is an irreducible sublattice of m × n. We raise a question on how to characterize irreducible sublattices of m × n that are M (G). [ABSTRACT FROM AUTHOR]

Let D be a finite distributive lattice with n join-irreducible elements. It is well-known that D can be represented as the congruence lattice of a rectangular lattice L which is a special planer semimodular lattice. In this paper, we shall give a better upper bound for the size of L by a function of n, improving a 2009 result of G. Grätzer and E. Knapp. [ABSTRACT FROM AUTHOR]

Let D be a finite distributive lattice with n join-irreducible elements. It is well-known that D can be represented as the congruence lattice of a rectangular lattice L which is a special planer semimodular lattice. In this paper, we shall give a better upper bound for the size of L by a function of n, improving a 2009 result of G. Grätzer and E. Knapp. [ABSTRACT FROM AUTHOR]

Solution sets of systems of homogeneous linear equations over fields are characterized as being subspaces, i.e., sets that are closed under linear combinations. Our goal is to characterize solution sets of systems of equations over arbitrary finite algebras by a similar closure condition. We show that solution sets are always closed under the centralizer of the clone of term operations of the given algebra; moreover, the centralizer is the only clone that could characterize solution sets. If every centralizer-closed set is the set of all solutions of a system of equations over a finite algebra, then we say that the algebra has Property (SDC) . Our main result is the description of finite lattices and semilattices with Property (SDC) : we prove that a finite lattice has Property (SDC) if and only if it is a Boolean lattice, and a finite semilattice has Property (SDC) if and only if it is distributive. [ABSTRACT FROM AUTHOR]

It was recently proved that the dualization in lattices given by implicational bases is impossible in output-polynomial time unless P = NP. In this paper, we show that this result holds even when the premises in the implicational base are of size at most two. Then we show using hypergraph dualization that the problem can be solved in output quasi-polynomial time whenever the implicational base has bounded independent-width, defined as the size of a maximum set of implications having independent conclusions. Lattices that share this property include distributive lattices coded by the ideals of an interval order, when both the independent-width and the size of the premises equal one. [ABSTRACT FROM AUTHOR]

The present paper is part of the research on the description of rings with a given property of the lattice of left (right) annihilators. The anti-isomorphism of lattices of left and right annihilators in any ring gives some kind of symmetry: the lattice of left annihilators is Boolean (complemented, distributive) if and only if the lattice of right annihilators is such. This allows us to restrict our investigations mainly to the left side. For a unital associative ring R, we prove that the lattice of left annihilators in R is Boolean if and only if R is a reduced ring. We also prove that the lattice of left annihilators of R being two-sided ideals is complemented if and only if this lattice is Boolean. The last statement, in turn, is known to be equivalent to the semiprimeness of R. On the other hand, for any complete lattice L, we construct a nilpotent ring whose lattice of left annihilators coincides with its sublattice of left annihilators being two-sided ideals and is isomorphic to L. This construction shows that the assumption of R being unital cannot be dropped in any of the above two results. Some additional results on rings with distributive or complemented lattices of left annihilators are obtained.

In this paper, we prove new versions of Stone Duality. The main version is the following: the category of Kolmogorov locally small spaces and bounded continuous mappings is equivalent to the category of spectral spaces with decent lumps and with bornologies in the lattices of (quasi-) compact open sets as objects and spectral mappings respecting those decent lumps and satisfying a boundedness condition as morphisms. Furthermore, it is dually equivalent to the category of bounded distributive lattices with bornologies and with decent lumps of prime filters as objects and homomorphisms of bounded lattices respecting those decent lumps and satisfying a domination condition as morphisms. This helps to understand Kolmogorov locally small spaces and morphisms between them. We comment also on spectralifications of topological spaces.

In this paper the authors develop a logic of concepts within a mathematical linguistic theory. In the set of concepts defined in a belief system, the order relationship and Boolean algebra of the concepts are considered. This study is designed to obtain a tool, which is the metatheoretical base of this type of theory. [ABSTRACT FROM AUTHOR]