Heyting Algebras with a Dual Lattice Endomorphism
Heyting Algebras with a Dual Lattice Endomorphism
复制标题
具有双格自同态的 Heyting 代数
DOI：
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发表时间：
1987
影响因子：
0.3
通讯作者：
H. P. Sankappanavar
中科院分区：
文献类型：
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作者：
H. P. Sankappanavar
在文[14]中，我们刻画了非正则次直接不可约伪补de Morgan代数，而正则代数的相应问题则是公开的。为了解决这个问题，我们自然而然地考虑了具有de Morgan运算的Heyting代数。本文研究了具有对偶格自同态的Heyting代数(简称OH-代数)，特别是具有de Morgan运算的Heyting代数(即de Morgan-Heyting代数)。利用[14]中引入的正则滤子的概念，证明了OH代数上的同余是由正则滤子决定的。然后利用这一基本结果刻画了De Morgan-Heyting代数簇中的直接不可分、单、有限次直接不可约和次直不可约。作为推论，我们得到了一系列为鉴别簇的子簇，并证明了有限单De Morgan-Heyting代数是拟素的。证明了De Morgan-Heyting代数簇不具有等式可定义的主同余。还刻画了具有同余代换性质的OH-代数上的函数。这些结果也适用于正则伪补De Morgan代数。
In [14] we characterized non-regular subdirectly irreducible pseudocomplemented De Morgan algcbras, while the corresponding problem for regular algebras was left open. Our attempt to solve this problem led us naturally to consider Heyting algebras with a De Morgan operation. In this paper we examine Heyting algebras with a dual lattice endomorphism (OH-algebras, for short), with special emphasis on Heyting algebras with a De Morgan operation (i.e. De Morgan-Heyting algebras). Using the notion of a regular filter introduced in [14], i t is shown that the congruences on OHalgebras are determined by regular filters. This basic result is then applied to characterize the directly indecomposables, simples, finitely subdirectly irreducibles and subdirectly irreducibles in the variety of De Morgan-Heyting algebras. As consequences, we obtain a sequence of subvarieties which are discriminator varieties, and the #esult that finite simple De Morgan-Heyting algebras are quasiprimal. It turns out that the variety of De Morgan-Heyting algebras does not have equationally definable principal congruences. The functions on OH-algebras having congruence substitution property are also characterized. These results also hold for regular pseudocomplemented De Morgan algebras.