On strongly symmetric skew lattices

Skew lattices are a non-commutative generalization of lattices. In the past 20?years, several varieties of skew lattices have been introduced. In the present paper we study the variety of strongly symmetric skew lattices.     

Internal Decompositions of Skew Lattices

In the present note we study internal decompositions of skew lattices. This provides further insight into the structure of skew lattices. Special attention is devoted to skew lattices in rings.     

On the coset laws for skew lattices

Skew lattices are a noncommutative generalization of lattices. In the paper we study the varieties of symmetric, strongly symmetric and cancellative skew lattices, and characterize them in terms of certain laws regarding the coset structure of a skew lattice. As a consequence, combinatorial results connecting powers of D{\mathcal{D}}-classes and indices are derived.     

Skew lattices of matrices in rings

In [6] J. Leech introduced skew lattices in rings. In the present paper we study skew lattices in rings of matrices. We prove that every symmetric, normal skew lattice with a finite, distributive maximal lattice image can be embedded in a skew lattice of upper-triangular matrices.Received September 5, 2003; accepted in final form October 12, 2004.     

A non-commutative Priestley duality

We prove that the category of left-handed strongly distributive skew lattices with zero and proper homomorphisms is dually equivalent to a category of sheaves over local Priestley spaces. Our result thus provides a non-commutative version of classical Priestley duality for distributive lattices and generalizes the recent development of Stone duality for skew Boolean algebras.     

A New Proof of Spinks' Theorem

Skew lattices form a class of non-commutative lattices. Spinks' Theorem [Matthew Spinks, On middle distributivity for skew lattices, ] states that for symmetric skew lattices the two distributive identities and are equivalent. Up to now only computer proofs of this theorem have been known. In the present paper the author presents a direct proof of Spinks' Theorem. In addition, a new result is proved showing that the assumption of symmetry can be omitted for cancellative skew lattices.     

Stone duality for spectral sheaves and the patch monad

We establish a duality between global sheaves on spectral spaces and right distributive bands. This is a sheaf-theoretical extension of classical Stone duality between spectral spaces and bounded distributive lattices.The topology of a spectral space admits a refinement, the so-called patch topology, giving rise to a patch monad on sheaves over a fixed spectral space. Under the duality just mentioned the algebras of this patch monad are shown to correspond to distributive skew lattices.     

On Middle Distributivity for Skew Lattices

In [3, Section 2.5] Jonathan Leech asked if the middle distributive identities for skew lattices are independent. This note exhibits a nine element counterexample, found with the model generating program SEM [7].     

Algebras with implication and fusion: a different point of view

This work uses well-known results on tensor products of lattices and semilattices developed by Fraser and Grätzer et al., and the duality for bounded distributive lattices introduced by Cignoli et al., in order to develop dual categorical equivalences involving bounded distributive lattices with fusion and implication, respectively. We show that these equivalences are essentially those developed by Cabrer and Celani as part of the PhD thesis of the former.     

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.     

分子格范畴中的极限

本文通过对完备格上“”关系的抽象分析，引入了　完备格及　下集的概念。讨论了下集的一系列性质。证明了一个完备格中所有下集之集构成一个分子格，由此再结合范畴论知识，构造出了分子格范畴中的极限结构。解决了分子格范畴性质这一研究领域中最困难的问题之一。     

Ordered groups with a conucleus

Our work proposes a new paradigm for the study of various classes of cancellative residuated lattices by viewing these structures as lattice-ordered groups with a suitable operator (a conucleus). One consequence of our approach is the categorical equivalence between the variety of cancellative commutative residuated lattices and the category of abelian lattice-ordered groups endowed with a conucleus whose image generates the underlying group of the lattice-ordered group. In addition, we extend our methods to obtain a categorical equivalence between -algebras and product algebras with a conucleus. Among the other results of the paper, we single out the introduction of a categorical framework for making precise the view that some of the most interesting algebras arising in algebraic logic are related to lattice-ordered groups. More specifically, we show that these algebras are subobjects and quotients of lattice-ordered groups in a “quantale like” category of algebras.     

A survey of recent results on congruence lattices of lattices

We review recent results on congruence lattices of (infinite) lattices. We discuss results obtained with box products, as well as categorical, ring-theoretical, and topological results. Received March 13, 2002; accepted in final form September 24, 2002. RID="h1" ID="h1"     

Envelopes of geometric lattices

A categorical embedding theorem is proved for geometric lattices. This states roughly that, if one wants to consider only those embeddings into projective spaces having a suitable universal property, then the existence of such an embedding can be checked by seeing whether corresponding properties hold for many small intervals. Tutte's embedding theorem for binary geometric lattices is a consequence of this result.     

Characteristic triangles of closure operators with applications in general algebra

Our aim is to investigate groups and their weak congruence lattices in the abstract setting of lattices L with (local) closure operators C in the categorical sense, where L is regarded as a small category and C is a family of closure maps on the principal ideals of L. A useful tool for structural investigations of such “lattices with closure” is the so-called characteristic triangle, a certain substructure of the square L ². For example, a purely order-theoretical investigation of the characteristic triangle shows that the Dedekind groups (alias Hamiltonian groups) are precisely those with modular weak congruence lattices; similar results are obtained for other classes of algebras.     

A categorical equivalence for bounded distributive quasi lattices satisfying: x ∨ 0 = 0 ⇒ x = 0

In this work, we investigate a categorical equivalence between the class of bounded distributive quasi lattices that satisfy the quasiequation x∨0 = 0 =? x = 0, and a category whose objects are sheaves over Priestley spaces.     

Pure Skew Lattices in Rings

Given a ring $R$, let $S\subseteq R$ be a pure multiplicative band that is closed under the cubic join operation $x\nabla y = x+y+yx-xyx-yxy.$ We show that $\left( S,\cdot,\nabla\right) $ forms a pure skew lattice if and only if $S$ satisfies the polynomial identity $\left(xy-yx\right)^{2}z = z\left(xy-yx\right)^{2}$. We also examine properties of pure skew lattices in rings.     

单调概念格性质研究

概念格是知识表示和数据分析的重要工具,单调概念格是概念格的推广。本文就Deogun等提出的单调概念格进行了两方面的研究:一,指出Deogun等提出的单调概念格性质的错误并加以修正;二,证明单调概念格就是闭格,从而找到用拓扑闭包算子和拓扑交结构来表示单调概念格的两种格表示方法,并建立起单调概念格与有上界的拓扑交结构的范畴等价。本文所建立的单调概念格的拓扑表示方法将方便我们进一步研究单调概念格的构造算法、约简算法和实际应用,具有理论和实际的双重意义。     

Order-enriched categorical models of the classical sequent calculus

It is well-known that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. Starting from a convenient formulation of the well-known categorical semantics of linear classical sequent proofs, we give models of weakening and contraction that do not collapse. Cut-reduction is interpreted by a partial order between morphisms. Our models make no commitment to any translation of classical logic into intuitionistic logic and distinguish non-deterministic choices of cut-elimination. We show soundness and completeness via initial models built from proof nets, and describe models built from sets and relations.