Abstract
Together with J. Paseka we introduced so-called sectionally pseudocomplemented lattices and posets and illuminated their role in algebraic constructions. We believe that—similar to relatively pseudocomplemented lattices—these structures can serve as an algebraic semantics of certain intuitionistic logics. The aim of the present paper is to define congruences and filters in these structures, derive mutual relationships between them and describe basic properties of congruences in strongly sectionally pseudocomplemented posets. For the description of filters in both sectionally pseudocomplemented lattices and posets, we use the tools introduced by A. Ursini, i.e., ideal terms and the closedness with respect to them. It seems to be of some interest that a similar machinery can be applied also for strongly sectionally pseudocomplemented posets in spite of the fact that the corresponding ideal terms are not everywhere defined.
Highlights
The concept of a relative pseudocomplemented lattice was introduced by R
In the present paper we focus on congruences and filters on sectionally pseudocomplemented lattices and posets
We have shown that every congruence on a strongly sectionally pseudocomplemented poset is fully determined by its 1-class [1]
Summary
The concept of a relative pseudocomplemented lattice was introduced by R. It was used in several branches of mathematics, e.g., as an algebraic axiomatization of intuitionistic logic (by Heyting and Brouwer) where the relative pseudocomplement is interpreted as the logical connective implication. Every relative pseudocomplemented lattice is distributive, see, e.g., Birkhoff (1979) and Lakser (1971). In the present paper we focus on congruences and filters on sectionally pseudocomplemented lattices and posets. For lattices we can use the machinery of universal algebra (see, e.g., Chajda et al (2012)) because sectionally pseudocomplemented lattices form a variety which is congruence permutable, congruence distributive and weakly regular. Pseudocomplemented lattices having 0 and their ideals will be the topic of one of our studies
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