Some remarks concerning the infinite distributivity of lattices

Theorems are proved concerning the regular imbedding of an infinitely distributive lattice in least decomposable lattices and in completely decomposable lattices. A relation is established between decomposable lattices, Stone lattices, and completely decomposable lattices.Translated from Matematicheskie Zametki, Vol. 8, No. 1, pp. 95–103, July, 1970.We use this opportunity to thank A. G. Pinsker for his valuable advice concerning this work.     

Simplicial elimination schemes,extremal lattices and maximal antichain lattices

Fix a partial order P=(X, <). We first show that bipartite orders are sufficient to study structural properties of the lattice of maximal antichains. We show that all orders having the same lattice of maximal antichains can be reduced to one representative order (called the poset of irreducibles by Markowsky [14]). We then define the strong simplicial elimination scheme to characterize orders which have distributive lattice of maximal antichains. The notion of simplicial elimination corresponds to the decomposition process described in [14] for extremal lattices. This notion leads to simple greedy algorithms for distributivity checking, lattice recognition and jump number computation. In the last section, we give several algorithms for lattices and orders.     

On the Dual Ramsey Property for Finite Distributive Lattices

The class of finite distributive lattices, as many other natural classes of structures, does not have the Ramsey property. It is quite common, though, that after expanding the structures with appropriately chosen linear orders the resulting class has the Ramsey property. So, one might expect that a similar result holds for the class of all finite distributive lattices. Surprisingly, Kechris and Soki? have proved in 2012 that this is not the case: no expansion of the class of finite distributive lattices by linear orders satisfies the Ramsey property. In this paper we prove that the class of finite distributive lattices does not have the dual Ramsey property either. However, we are able to derive a dual Ramsey theorem for finite distributive lattices endowed with a particular linear order. Both results are consequences of the recently observed fact that categorical equivalence preserves the Ramsey property.     

Coalgebraic representations of distributive lattices with operators

We present a framework for extending Stone's representation theorem for distributive lattices to representation theorems for distributive lattices with operators. We proceed by introducing the definition of algebraic theory of operators over distributive lattices. Each such theory induces a functor on the category of distributive lattices such that its algebras are exactly the distributive lattices with operators in the original theory. We characterize the topological counterpart of these algebras in terms of suitable coalgebras on spectral spaces. We work out some of these coalgebraic representations, including a new representation theorem for distributive lattices with monotone operators.     

Computational complexity for bounded distributive lattices with negation

We study the computational complexity of the universal and quasi-equational theories of classes of bounded distributive lattices with a negation operation, i.e., a unary operation satisfying a subset of the properties of the Boolean negation. The upper bounds are obtained through the use of partial algebras. The lower bounds are either inherited from the equational theory of bounded distributive lattices or obtained through a reduction of a global satisfiability problem for a suitable system of propositional modal logic.     

幂等扩张的分配格的同余可换性

本文研究了幂等扩张的有界分配格的同余可换性问题.利用幂等扩张的有界分配格的对偶理论,得到了同余可换的幂等扩张的有界分配格的一个充分必要条件,推广了Davey和Priestley关于有界分配格的一些结果.     

The structure of orders all of whose representations are completely decomposable

The paper describes orders all of whose representations are completely decomposable, lying in a matrix algebra over the quotient field of a complete local Dedekind ring. For Morita-reduced orders an isomorphism criterion is provided in terms of the lattices of irreducible modules.Translated from Matematicheskie Zametki, Vol. 13, No. 2, pp. 325–335, February, 1973.The author wishes to thank V. V. Kirichenko, and also the participants in the Kiev seminar on representation theory, for their constant attention as well as for a host of valuable suggestions.     

Compact spaces and distributive lattices

This note presents a general construction connecting compact locales and distributive lattices, that allows us to reduce results about compactness of locales to theorems about distributive lattices. Two applications are given. One noteworthy feature of our arguments is that they can be formulated both in topos theory and in a predicative theory such as CZF.     

Decomposition of submodular functions

A decomposition theory for submodular functions is described. Any such function is shown to have a unique decomposition consisting of indecomposable functions and certain highly decomposable functions, and the latter are completely characterized. Applications include decompositions of hypergraphs based on edge and vertex connectivity, the decomposition of matroids based on three-connectivity, the Gomory—Hu decomposition of flow networks, and Fujishige’s decomposition of symmetric submodular functions. Efficient decomposition algorithms are also discussed.     

Distributive lattices with a decidable monadic second order theory

A class of finite distributive lattices has a decidable monadic second order theory iff (a) the join irreducible elements of its members have a decidable monadic second order theory, and (b) the width of the lattices is bounded. Similar results are obtained for the monadic chain and the monadic antichain theory where quantification is restricted to chains and antichains, resp. Furthermore, there is no (up to finite difference) maximal set of finite distributive lattices with a decidable monadic (chain or antichain, resp.) theory. Received December 6, 2000; accepted in final form May 30, 2002.     

Computability of Distributive Lattices

The class of (not necessarily distributive) countable lattices is HKSS-universal, and it is also known that the class of countable linear orders is not universal with respect to degree spectra neither to computable categoricity. We investigate the intermediate class of distributive lattices and construct a distributive lattice with degree spectrum {d: d ≠ 0}. It is not known whether a linear order with this property exists. We show that there is a computably categorical distributive lattice that is not relatively Δ₂⁰-categorical. It is well known that no linear order can have this property. The question of the universality of countable distributive lattices remains open.     

分配格与模格的另一二条件等价定义

针对分配格与模格的格等式定义问题,得知了二条件是定义分配格与模格的最少条件,并进一步证明了Sholander's basis是定义分配格的最短最少变量格等式,最后又从分配格和模格的基本定义出发给出了新的分配格的二条件和三条件等价定义等式及模格的二条件与三条件等价定义等式.     

Constructing simple residuated lattices

A method of constructing residuated lattices is presented. As an application, examples of simple, integral, cancellative, distributive residuated lattices are given that are not linearly ordered. This settles a problem raised in [5] and [2].     

幂等扩张的有界分配格

考察了扩张的有界分配格类eD即带有自同态k的有界分配格,研究了具有幂等性的eD-代数的表示、同余关系以及次直不可约性,证明了这样的代数类有5个互不同构的次直不可约的幂等扩张的有界分配格。     

Abelian-regular positive semirings

The theory of Abelian-regular positive semirings (arp-semirings) is developed here. This class contains all semiskewfields and all bounded distributive lattices. Bibliography: 11 titles. The second author received partial financial support from the Russian Foundation for Basic Research. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 20, pp. 282–309, 1997.     

完全分配格上拓扑生成序的加细及其性质

在文[1]中,作者建立了完全分配格上的共生结构的一般理论,研究了余拓扑,拟一致和T结构的一致化结构。本文引进了完全分配格L上半拓扑生成序加细的概念,研究了其性质,得到了一系列重要的结论,为进一步研究完全分配格上拓扑共生结构的加细奠定了基础。     

The strong endomorphism kernel property for modular p-algebras and for distributive lattices

We study the strong endomorphism kernel property (SEKP) for some classes of universal algebras. Using the Katriňák-Mederly triple construction, we prove a universal equivalent condition under which a modular p-algebra has SEKP. As a consequence, we characterize distributive lattices with top element that enjoy SEKP. Using Priestley duality, we also characterize unbounded distributive lattices that have SEKP.     

Representations of finite partially ordered sets over commutative artinian uniserial rings

Categories of representations of finite partially ordered sets over commutative artinian uniserial rings arise naturally from categories of lattices over orders and abelian groups. By a series of functorial reductions and a combinatorial analysis, the representation type of a category of representations of a finite partially ordered set S over a commutative artinian uniserial ring R is characterized in terms of S and the index of nilpotency of the Jacobson radical of R. These reductions induce isomorphisms of Auslander-Reiten quivers and preserve and reflect Auslander-Reiten sequences. Included, as an application, is the completion of a partial characterization of representation type of a category of representations arising from pairs of finite rank completely decomposable abelian groups.     

Boolean Topological Distributive Lattices and Canonical Extensions

This paper presents a unified account of a number of dual category equivalences of relevance to the theory of canonical extensions of distributive lattices. Each of the categories involved is generated by an object having a two-element underlying set; additional structure may be algebraic (lattice or complete lattice operations) or relational (order) and, in either case, topology may or may not be included. Among the dualities considered is that due to B. Banaschewski between the categories of Boolean topological bounded distributive lattices and the category of ordered sets. By combining these dualities we obtain new insights into canonical extensions of distributive lattices. The second author was supported by Slovak grants VEGA 1/3026/06 and APVV-51-009605.     

Combinatorial representation and convex dimension of convex geometries

We develop a representation theory for convex geometries and meet distributive lattices in the spirit of Birkhoff's theorem characterizing distributive lattices. The results imply that every convex geometry on a set X has a canonical representation as a poset labelled by elements of X. These results are related to recent work of Korte and Lovász on antimatroids. We also compute the convex dimension of a convex geometry.Supported in part by NSF grant no. DMS-8501948.