Join-semilattices whose principal filters are pseudocomplemented lattices

Miskolc Mathematical Notes

article

Ivan Chajda¹  Helmut Länger² 

[1] Palacky University Olomouc, Faculty of Science, Department of Algebra and Geometry;TU Wien, Faculty of Mathematics and Geoinformation, Institute of Discrete Mathematics and Geometry

关键词: join-semilattice;    pseudocomplemented lattice;    intuitionistic implication;    non-classical implication semilattice;    sectionally residuated join-semilattice;    relatively residuated join-semilattice;    I-algebra;    variety of I-algebras;    congruence distributive;    3-permutable;    weakly regular;    R-algebra;   

DOI  :  10.18514/MMN.2022.3854

学科分类：数学（综合）

来源: Miskolci Egyetem

【 摘 要 】

This paper deals with join-semilattices whose sections, i.e. principal filters, are pseudocomplemented lattices. The pseudocomplement of a\vee b in the section [b,1] is denoted by a\rightarrow b and can be considered as the connective implication in a certain kind of intuitionistic logic. Contrary to the case of Brouwerian semilattices, sections need not be distributive lattices. This essentially allows possible applications in non-classical logics. We present a connection of the semilattices mentioned in the beginning with so-called non-classical implication semilattices which can be converted into I-algebras having everywhere defined operations. Moreover, we relate our structures to sectionally and relatively residuated semilattices which means that our logical structures are closely connected with substructural logics. We show that I-algebras form a congruence distributive, 3-permutable and weakly regular variety.

【 授权许可】

CC BY