内蕴左R--模范畴在Ω-集范畴中的表示

本文研究内蕴环范畴在集合范畴中的表示,内蕴Abel群范畴在集合范畴中的表示,以及内蕴左R-模范畴在集合范畴中的表示,进一步,研究内蕴左R-模范畴在Ω-集范畴中的表示,给出基于Ω-集范畴的内蕴左R-模范畴与Ω-左R-模范畴之间的同构关系.     

Heyting代数的不变滤子

综合运用泛代数与逻辑学的方法和原理对Heyting代数中滤子概念作进一步深入研究.引入了Heyting代数的关于其子集的不变滤子概念.给出了不变滤子的若干性质.证明了Heyting代数的关于一个给定子集的不变滤子全体之集构成一个完备Heyting代数.     

HEYTING代数与FUZZY蕴涵代数

Heyting代数是作为直觉主义命题逻辑的代数模型而引进的Fuzzy蕴涵代数是 [0 ,1]值逻辑的蕴函联结词的一种代数抽象 .本文给出Heyting代数的若干基本性质 ,并证明了Heyting代数是Fuzzy蕴涵代数 ,也是Heyting型Fuzzy蕴涵代数。     

完全分配格与点格

本文第一部分利用完备格上的上拓扑子基,给出完全分配格与点格的若干新刻划,并讨论其上的 Scott 拓扑与 Lawson 拓扑的基与子基的构造.第二部分讨论点格与代数格的关系,证明了 L 是点格当且仅当 L 为代数格且 L~(op)为完全 Heyting代数,并证明了代数偏序集范畴与点格范畴是等价的.     

Heyting代数的扩张滤子

运用泛代数与逻辑学的方法和原理对Heyting代数中滤子概念作进一步研究.在Heyting代数H中引入了滤子F关于H的子集A的扩张滤子概念并考察其性质.证明了一个滤子F关于H的所有子集的扩张滤子全体之集构成一个完备Heyting代数且构成一个Stone格.     

HEYTING代数与FUZZY蕴函代数

Heyting代数是作为直觉主义命题逻辑的代数模型而引起的，Fuzzy蕴涵代数是[0,1]值逻辑的蕴涵联结词的一种代数抽象。本文给出Heyting代数的若干基本性质，并证明了Heyting代数是Fuzzy蕴涵代数，也是Heyting型Fuzzy蕴涵代数。     

有界Heyting代数的交换理想与关联理想

运用泛代数和格理论的方法和原理进一步深入研究有界Heyting代数的理想问题。在有界Heyting代数中引入了交换理想、关联理想和正关联理想概念并讨论了它们的性质和相互关系。获得了各种理想的若干等价刻画。证明了在有界Heyting代数中,关联理想和正关联理想等价;在Ockham型有界Heyting代数中,理想和交换理想等价。同时,给出了有界Heyting代数的交换理想成为关联理想的一个充分必要条件。     

有界Heyting代数的⊕运算与模糊LI理想

对有界Heyting代数的模糊LI理想问题作进一步深入研究.首先,在有界Heyting代数上定义了⊕运算并讨论其运算性质.其次,借助运算⊕获得了模糊LI理想的若干等价刻画.最后,获得了由一个有界Heyting代数的全体模糊LI理想之集构成的完备Heyting代数中蕴涵算子的表示定理.     

有界Heyting代数上基于模糊LI理想的一致拓扑结构研究

为了从拓扑层面进一步揭示有界Heyting代数的内部特征,基于由模糊LI理想诱导的同余关系在有界Heyting代数上构造一致拓扑结构并研究其拓扑性质.证明了:1)一致拓扑空间是第一可数、零维、非连通、局部紧的完全正则空间;2)—致拓扑空间是T_1空间当且仅当是T_2空间;3)有界Heyting代数中格运算和蕴涵运算关于一致拓扑都是连续的,从而构成拓扑有界Heyting代数.同时,获得了一致拓扑空间是离散空间和紧致空间的充分必要条件.     

模糊蕴涵代数的结构特征

本文研究了模糊蕴涵代数的一些性质,给出了模糊蕴涵代数成为Heyting代数的一个条件,得到对模糊蕴涵代数的结构特征刻画,并给出了一个(2,0)型代数(X,→,0)成为模糊蕴涵代数的充分必要条件.     

On Bellissima’s construction of the finitely generated free Heyting algebras, and beyond

We study finitely generated free Heyting algebras from a topological and from a model theoretic point of view. We review Bellissima’s representation of the finitely generated free Heyting algebra; we prove that it yields an embedding in the profinite completion, which is also the completion with respect to a naturally defined metric. We give an algebraic interpretation of the Kripke model used by Bellissima as the principal ideal spectrum and show it to be first order interpretable in the Heyting algebra, from which several model theoretic and algebraic properties are derived. In particular, we prove that a free finitely generated Heyting algebra has only one set of free generators, which is definable in it. As a consequence its automorphism group is the permutation group over its generators.     

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.     

Characteristic Formulas of Partial Heyting Algebras

The goal of this paper is to generalize a notion of characteristic (or Jankov) formula by using finite partial Heyting algebras instead of the finite subdirectly irreducible algebras: with every finite partial Heyting algebra we associate a characteristic formula, and we study the properties of these formulas. We prove that any intermediate logic can be axiomatized by such formulas. We further discuss the correlations between characteristic formulas of finite partial algebras and canonical formulas. Then with every well-connected Heyting algebra we associate a set of characteristic formulas that correspond to each finite relative subalgebra of this algebra. Finally, we demonstrate that in many respects these sets enjoy the same properties as regular characteristic formulas. In the last section we outline an approach how to generalize these obtained results to the broad classes of algebras.     

Quotients of dimension effect algebras

We show that the quotient of a dimension effect algebra by its dimension equivalence relation is a unital bounded lattice-ordered positive partial abelian monoid that satisfies a version of the Riesz decomposition property. For a dimension effect algebra of finite type, the quotient is a centrally orthocomplete Stone–Heyting MV-effect algebra; moreover, an orthocomplete effect algebra in which equality is a dimension equivalence relation is the same thing as a complete Stone–Heyting MV-effect algebra.     

Grothendieck-like duality for Heyting algebras

In this paper we shall use the multipliers in order to define a sheaf on the prime spectrum of a Heyting algebra. This construction allows us to obtain a Grothendieck-style duality for Heyting algebras. Received March 3, 2000; accepted in final form February 16, 2001.     

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.     

Functional monadic Heyting algebras

We show every monadic Heyting algebra is isomorphic to a functional monadic Heyting algebra. This solves a 1957 problem of Monteiro and Varsavsky [9]. Received May 18, 2001; accepted in final form October 18, 2001.     

具有Heyting结构的Ockham代数

引入一个具有Heyting结构Ockham代数,简称HO-代数．所谓HO-代数,是指具有（2,2,2,1,0,0）类型的代数（L;∧,∨,→,f,0,1）．其中（L;f）是Ockham代数,（L;→）是Heyting代数,且运算f和→由恒等式f（x→y）=f^2（x）∧f（y）与f（x）→y=f^2（x）∨y所连结．主要讨论了HO-代数的同余关系的性质．并刻画了其次直不可约代数的某些性质．