带有同态运算的分配格（英文）

本文研究了扩展的有界分配格(即带有一个自同态运算的有界分配格)的次直不可约问题.利用不动点集和同余的方法,刻画了e_(p,q)D代数类中次直不可约代数,获得了此类代数中有限次直不可约性和次直不可约性两者等价的结果,推广了Balbes以及Dwinger关于半格、分配格和布尔代数的相关结果.     

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.     

对称扩展的有界分配格的同余关系及其应用

给出了对称扩展的有界分配格的定义,即带有满足一定条件的一元运算的有界分配格.然后给出了这种分配格上的主同余的等式刻划及其可补性.最后,讨论了对称扩展的有界分配格的次直不可约性。     

群在分配格上的作用

本文将分配格的自同构群对分配格的作用推广成抽象的群对分配格的作用(即G-分配格),建立了G-有界分配格范畴和G-Priestley空间范畴的对偶等价性,并在此基础上刻画了G-同余关系的对偶以及G-分配格的次直不可约性和同余可换性。所得结论丰富了分配格、格群等代数理论。     

交换双重Ockham代数

研究一类双重Ockham代数 $(L;∧,V, f, k), 即赋予一对可交换一元运算f和k的有界分配格, 其中f和k是偶格同态. 刻画了其次直不可约代数, 并研究当(L,f)和(L;k)都是de Morgan代数的特殊情形. 利用Priestley对偶理论, 证明这类代数中 仅有9个非同构的次直不可约代数, 而且这些次直不可约代数都是单纯的.     

分配格上关于三角模的幂等矩阵和幂零矩阵

介绍了分配格上的T-幂等矩阵和S-幂等矩阵的反自反性、自反性和幂等性,给出了分配格上反自反矩阵是T-幂零矩阵和S-幂零矩阵的充分条件,并用自己的方法和改进的方法予以证明。     

两类新域上的Bergman核函数的显表达式

在有界对称域上引入两类蛋型域, 得到其上Bergman 核函数的显表达式. 作为辅助工具, 利用著名的Selberg积分, 计算有界对称域上一般范数(generic norm)下复幂积分. 推广了华罗庚的矩阵积分, 得到不可约有界对称域的一个整系数或半整系数特殊多项式.     

一类逆半群的亚直可约性

本文利用逆半群上的同余扩张，讨论了一类逆半群的亚直可约性，并刻划了这类逆半群的幂等元集的特征。     

一类平坦半环生成的簇

平坦半环是一类重要的加法幂等元半环,它在半环簇理论的研究中扮演着重要的角色.主要研究了次直不可约的平坦半环,以及一类平坦半环生成的簇.给出了次直不可约的nil-平坦半环的等价刻画,证明了当n小于4时,平坦半环S(x1x2…xn)均是有限基底的.     

李color代数的T~*-扩张

介绍了李color代数的T*-扩张的定义,并证明李color代数的很多性质,如幂零性、可解性和可分解性,都可以提升到它的T*-扩张上.还证明在特征不等于2的代数闭域上,有限维幂零二次李color代数A等距同构于一个幂零李color代数B的T*-扩张,并且B的幂零长度不超过A的一半.此外,用上同调的方法研究了李color代数的T*-扩张的等价类.     

Subdirectly irreducible commutative multiplicatively idempotent semirings

Commutative multiplicatively idempotent semirings were studied by the authors and F. ?vr?ek, where the connections to distributive lattices and unitary Boolean rings were established. The variety of these semirings has nice algebraic properties and hence there arose the question to describe this variety, possibly by its subdirectly irreducible members. For the subvariety of so-called Boolean semirings, the subdirectly irreducible members were described by F. Guzmán. He showed that there were just two subdirectly irreducible members, which are the 2-element distributive lattice and the 2-element Boolean ring. We are going to show that although commutative multiplicatively idempotent semirings are at first glance a slight modification of Boolean semirings, for each cardinal n > 1, there exist at least two subdirectly irreducible members of cardinality n and at least 2ⁿ such members if n is infinite. For \({n \in \{2, 3, 4\}}\) the number of subdirectly irreducible members of cardinality n is exactly 2.     

A duality theory for bilattices

Recent studies of the algebraic properties of bilattices have provided insight into their internal strucutres, and have led to practical results, especially in reducing the computational complexity of bilattice-based multi-valued logic programs. In this paper the representation theorem for interlaced bilattices without negation found in [19] and extended to arbitrary interlaced bilattices without negation in [2] is presented. A natural equivalence is then established between the category of interlaced bilattices and the cartesian square of the category of bounded lattices. As a consequence a dual natural equivalence is obtained between the category of distributive bilattices and the coproduct of the category of bounded Priestley spaces with itself. Some applications of these equivalences are given. The subdirectly irreducible interlaced bilattices are characterized in terms of subdirectly irreducible lattices. A known characterization of the join-irreducible elements of the "knowledge" lattice of an interlaced bilattice is used to establish a natural equivalence between the category of finite, distributive bilattices and the category of posets of the form . Received February 2, 1998; accepted in final form September 2, 1999.     

Commuting double Ockham algebras

In this paper,we study a certain class of double Ockham algebras (L;∧,∨,f,k,0,1), namely the bounded distributive lattices (L;∧,∨,0,1) endowed with a commuting pair of unary op- erations f and k,both of which are dual endomorphisms.We characterize the subdirectly irreducible members,and also consider the special case when both (L;f) and (L;k) are de Morgan algebras.We show via Priestley duality that there are precisely nine non-isomorphic subdirectly irreducible members, all of which are simple.     

Some subdirectly irreducible idempotent semigroups

Subdirectly irreducible idempotent semigroups were characterized in [3], and in that paper, their connection with the various equational classes of idempotent semigroups was discussed. All these results are in terms of identities, so that examples of subdirectly irreducibles in the equational classes are explicitly known only for small classes. It is easy to show from general considerations (see the last section of the present paper) that every proper equational subclass of the class of idempotent semigroups is generated (as an equational class) by one or two subdirectly irreducibles. In this paper we give an example of a subdirectly irreducible for each join irreducible equational class of idempotent semigroups, which generates the class. This list, together with known results, gives explicit examples of one or two finite subdirectly irreducibles which generate the various equational classes. Research supported by the National Research Council of Canada.     

Congruences on near-Heyting algebras

A near-Heyting algebra is a join-semilattice with a top element such that every principal upset is a Heyting algebra. We establish a one-to-one correspondence between the lattices of filters and congruences of a near-Heyting algebra. To attain this aim, we first show an embedding from the lattice of filters to the lattice of congruences of a distributive nearlattice. Then, we describe the subdirectly irreducible and simple near-Heyting algebras. Finally, we fully characterize the principal congruences of distributive nearlattices and near-Heyting algebras. We conclude that the varieties of distributive nearlattices and near-Heyting algebras have equationally definable principal congruences.     

Congruence distributive quasivarieties whose finitely subdirectly irreducible members form a universal class

By a congruence distributive quasivariety we mean any quasivarietyK of algebras having the property that the lattices of those congruences of members ofK which determine quotient algebras belonging toK are distributive. This paper is an attempt to study congruence distributive quasivarieties with the additional property that their classes of relatively finitely subdirectly irreducible members are axiomatized by sets of universal sentences. We deal with the problem of characterizing such quasivarieties and the problem of their finite axiomatizability.Presented by Joel Berman.To the memory of Basia Czelakowska.     

Separation in distributive congruence lattices

We define separable sets in algebraic lattices. For a finitely generated congruence distributive variety V \mathcal{V} , we show a close connection between non-separable sets in congruence lattices of algebras in V \mathcal{V} and the structure of subdirectly irreducible algebras in V \mathcal{V} . We apply the general results to some lattice varieties.