Congruences on posets, relatively pseudocomplemented and Boolean posets
Ivan Chajda, Helmut Länger
TL;DR
The paper extends lattice congruence theory to posets by defining congruences as relations compatible with $\mathrm{Max}\,L$ and $\mathrm{Min}\,U$, yielding convex classes and, under $ACC$ and $DCC$, interval-structured quotients that embed into the original poset. It develops a dedicated theory for relatively pseudocomplemented posets, introducing deductive systems and a Malcev operator to connect kernels with strong filters, and showing that strong filters induce congruences with those kernels. For Boolean posets, it reveals that congruences need not mirror Boolean algebra properties (permibility, regularity, uniformity), but establishes Pixley-type operators and kernel–filter relations, broadening the algebraic interpretation of classical and intuitionistic logics within poset frameworks. Overall, the work broadens congruence theory to posets with extra operations, linking logical semantics to order-theoretic structure and providing new tools for understanding quotients, kernels, and kernels-strong-filter correspondences in these settings.
Abstract
The aim of the present paper is to extend the concept of a congruence from lattices to posets. We use an approach different from that used by the first author and V. Snášel. By using our definition we show that congruence classes are convex. If the poset in question satisfies the Ascending Chain Condition as well as the Descending Chain Condition, then these classes turn out to be intervals. If the poset has a top element 1 then the 1-class of every congruence is a so-called strong filter. We study congruences on relatively pseudocomplemented posets which form a formalization of intuitionistic logic. For such posets we define so-called deductive systems and we show how they are connected with congruence kernels. We prove that every strong filter F of a relatively pseudocomplemented poset induces a congruence having F as its kernel. Finally, we consider Boolean posets which form a natural generalization of Boolean algebras. We show that congruences on Boolean posets in general do not share properties known from Boolean algebras, but congruence kernels of Boolean posets still have some interesting properties.