关于Robbins代数与Heyting代数的类似性质

Ｓ．Ｗｉｎｋｅｒ在文〔１〕中给出了Ｒｏｂｂｉｎｓ代数成为Ｂｏｏｌｅａｎ代数的一个较弱的条件。本文将讨论Ｈｅｙｔｉｎｇ代数成为Ｂｏｏｌｅａｎ代数的条件，我们发现Ｈｅｙｉｎｇ代数成为Ｂｏｏｌｅａｎ代数的条件与Ｒｏｂｂｉｎｓ代数成为Ｂｏｏｌｅａｎ代数的条件相似，从而提示了Ｈｅｙｔｉｎｇ代数与Ｒｏｂｂｉｎｓ代数之间具有一定的关系。     

拓扑分子格的极不连通性和S－闭性

利用完备余ｃｏ－Ｈｅｙｔｉｎｇ代数上的伪补运算，在拓扑分子格上引入极不连通性和Ｓ－闭性，得到了每个Ｓ－闭的正则拓扑分子格是极不连通的以及同胚的广义序同态保持极不连通性和Ｓ－闭性．     

关于格的极小生成元集

本文证明了格的极小生成元集一定是最小生成元集且只能是非零完全并既约元全体，证明了分配格具有最小生成元集的必要条件是它满足并无限分配律．本文还证明了完全Ｈｅｙｔｉｎｇ代数具有最小生成元集当且仅当它是强代数格，证明了完备格是强代数格当且仅当它和它的对偶格均是具有最小生成元集的分配格．     

拓扑分子格的极不连通性和S—闭性

利用完备余ｃｏ－Ｈｅｙｔｉｎｇ代数上的伪补运算，在拓扑分子格上引入极不连通性和Ｓ－闭性，得到了每个Ｓ－闭的正则拓扑分子格是极不连通的以及同胚的广义序同态保持极不连通性和Ｓ－闭性。     

近性Heyting代数

本文在有最小元０的Ｈｅｙｔｉｎｇ代数上引入了对偶等价的（半）近性关系和（半）近性序的概念,它们是一般拓扑学中全正则空间上对偶等价的近性关系和近性序的合理推广．本文的工作弥补了Ｇｉｅｒｚ和Ｋｅｉｍｅｌ相应工作的缺陷,更正了他们的一个错误结果。     

可积代数正规类中半素代数类及半素一致代数类确定的上根

利用代数正规类中的理想乘积公理，引入可积代数正规类及可积代数正规类中素代数、半素代数类及一致代数类概念，讨论了可积代数正规类中半素代数类及半素一致代数类确定的上根性质。     

李代数W(2,2)上的Hom-李代数结构

李代数W（2,2）是一类重要的无限维李代数,它是在研究权为2的向量生成的顶点算子代数的过程当中提出来的.Hom-李代数是指同时具备代数结构和李代数结构的一类代数,并且乘法与李代数乘法运算满足Leibniz法则.本文确定了李代数W（2,2）上的Hom-李代数结构.主要结论是李代数W（2,2）上没有非平凡的Hom-李代数结构.本文的研究结果对于W（2,2）代数的进一步研究有一定的帮助作用.     

李代数W(2,2)上的Hom-李代数结构

李代数W(2,2)是一类重要的无限维李代数,它是在研究权为2的向量生成的顶点算子代数的过程当中提出来的.Hom-李代数是指同时具备代数结构和李代数结构的一类代数,并且乘法与李代数乘法运算满足Leibniz法则.本文确定了李代数W(2,2)上的Hom-李代数结构.主要结论是李代数W(2,2)上没有非平凡的Hom-李代数结构.本文的研究结果对于W(2,2)代数的进一步研究有一定的帮助作用.     

函数恒等式理论在泊松超代数上的应用(英文)

泊松超代数既是结合超代数又是李超代数.本文利用结合超代数上的函数恒等式理论研究泊松超代数上的映射,将李超代数上的一类映射转化为结合超代数上的映射.     

坡代数区间值模糊子坡代数

模糊子代数是模糊代数的一个重要研究内容.为了进一步了解坡代数的模糊子坡代数的特性,在坡代数中引入了区间值模糊子坡代数概念.讨论了坡代数的区间值模糊子坡代数的性质.证明了坡代数的区间值模糊子坡代数的交,直积以及同态像也是区间值模糊子坡代数.     

On self-distributive weak Heyting algebras

We use the left self-distributive axiom to introduce and study a special class of weak Heyting algebras, called self-distributive weak Heyting algebras (SDWH-algebras). We present some useful properties of SDWH-algebras and obtain some equivalent conditions of them. A characteristic of SDWH-algebras of orders 3 and 4 is given. Finally, we study the relation between the variety of SDWH-algebras and some of the known subvarieties of weak Heyting algebras such as the variety of Heyting algebras, the variety of basic algebras, the variety of subresiduated lattices, the variety of reflexive WH-algebras (RWH-algebras), and the variety of transitive WH-algebras (TWH-algebras).     

Topo-canonical completions of closure algebras and Heyting algebras

We introduce and investigate topo-canonical completions of closure algebras and Heyting algebras. We develop a duality theory that is an alternative to Esakia’s duality, describe duals of topo-canonical completions in terms of the Salbany and Banaschewski compactifications, and characterize topo-canonical varieties of closure algebras and Heyting algebras. Consequently, we show that ideal completions preserve no identities of Heyting algebras. We also characterize definable classes of topological spaces. Received January 20, 2006; accepted in final form September 12, 2006.     

Projective algebras and primitive subquasivarieties in varieties with factor congruences

We prove that in the varieties where every compact congruence is a factor congruence and every nontrivial algebra contains a minimal subalgebra, a finitely presented algebra is projective if and only if it has every minimal algebra as its homomorphic image. Using this criterion of projectivity, we describe the primitive subquasivarieties of discriminator varieties that have a finite minimal algebra embedded in every nontrivial algebra from this variety. In particular, we describe the primitive quasivarieties of discriminator varieties of monadic Heyting algebras, Heyting algebras with regular involution, Heyting algebras with a dual pseudocomplement, and double-Heyting algebras.     

Lattice subordinations and Priestley duality

There is a well-known correspondence between Heyting algebras and S4-algebras. Our aim is to extend this correspondence to distributive lattices by defining analogues of S4-algebras for them. For this purpose, we introduce binary relations on Boolean algebras that resemble de Vries proximities. We term such binary relations lattice subordinations. We show that the correspondence between Heyting algebras and S4-algebras extends naturally to distributive lattices and Boolean algebras with a lattice subordination. We also introduce Heyting lattice subordinations and prove that the category of Boolean algebras with a Heyting lattice subordination is isomorphic to the category of S4-algebras, thus obtaining the correspondence between Heyting algebras and S4-algebras as a particular case of our approach. In addition, we provide a uniform approach to dualities for these classes of algebras. Namely, we generalize Priestley spaces to quasi-ordered Priestley spaces and show that lattice subordinations on a Boolean algebra B correspond to Priestley quasiorders on the Stone space of B. This results in a duality between the category of Boolean algebras with a lattice subordination and the category of quasi-ordered Priestley spaces that restricts to Priestley duality for distributive lattices. We also prove that Heyting lattice subordinations on B correspond to Esakia quasi-orders on the Stone space of B. This yields Esakia duality for S4-algebras, which restricts to Esakia duality for Heyting algebras.     

并素元有限生成格的弱直积分解

本文建立了并素元有限生成格的弱直积分解,并给出一个解决并素元生成的完全Heyting代数的直积分解问题的新方法;作为弱直积分解的应用,证明了并素元有限生成的完全Heyting代数必然同构于有限个既约的完全Heyting代数的直积,证明了并素元有限生成格是Boole代数的充要条件是它同构于某有限集的幂集格.     

Bounded distributive lattices with strict implication

The present paper introduces and studies the variety WH of weakly Heyting algebras. It corresponds to the strict implication fragment of the normal modal logic K which is also known as the subintuitionistic local consequence of the class of all Kripke models. The tools developed in the paper can be applied to the study of the subvarieties of WH; among them are the varieties determined by the strict implication fragments of normal modal logics as well as varieties that do not arise in this way as the variety of Basic algebras or the variety of Heyting algebras. Apart from WH itself the paper studies the subvarieties of WH that naturally correspond to subintuitionistic logics, namely the variety of R‐weakly Heyting algebras, the variety of T‐weakly Heyting algebras and the varieties of Basic algebras and subresiduated lattices. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Endomorphisms and homomorphisms of Heyting algebras

For a non-trivial Heyting algebraH, the cardinality of its endomorphism monoid is always at least two. If it is exactly two thenH is called 0-endomorphism rigid. It is shown that there exists a proper class of non-isomorphic 0-endomorphism rigid Heyting algebras. This is a consequence of a more general result: the variety of Heyting algebras is 0-universal.The support of NSERC is gratefully acknowledged.Presented by J. Berman.     

Decidability of the Projective Beth Property in Varieties of Heyting Algebras

Previously, we proved that there are only finitely many varieties of Heyting algebras possessing the projective Beth property and gave an exhaustive list of these. The projective Beth property is equivalent to strong epimorphisms surjectivity (SES). Here, we prove that the projective Beth property and SES are base-decidable on a class of varieties of Heyting algebras.     

Esakia Style Duality for Implicative Semilattices

We develop a new duality for implicative semilattices, generalizing Esakia duality for Heyting algebras. Our duality is a restricted version of generalized Priestley duality for distributive semilattices, and provides an improvement of Vrancken-Mawet and Celani dualities. We also show that Heyting algebra homomorphisms can be characterized by means of special partial functions between Esakia spaces. On the one hand, this yields a new duality for Heyting algebras, which is an alternative to Esakia duality. On the other hand, it provides a natural generalization of Köhler’s partial functions between finite posets to the infinite case.     

Endomorphisms of complete heyting algebras

Each monoid can be represented as the endomorphism monoid of a complete Heyting algebra. More generally, the category of soberT ₁-spaces and their open continuous maps, and consequently the category of complete Heyting algebras are universal. On the other hand, it is often impossible to represent monoids using more special (Hausdorff) complete Heyting algebras: for instance, each finite commutative endomorphism monoid of such a Heyting algebra is weakly idempotent. Support of the Natural Sciences and Engineering Research Council of Canada and the Grant Agency of the Czech Republic under Grant 201/93/950 is gratefully acknowledged. Support of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.