L-congruences on Posets, Relatively Pseudocomplemented and Boolean posets

Abstract In the present paper we introduce and investigate a new kind of generalization of the notion of a lattice congruence to posets. Other such generalizations can be found e.g. in [5, 7] and [9]. In those papers congruences on posets are defined as kernels of some kind of homomorphisms. But in case of lattices these homomorphisms need not be lattice homomorphisms and hence the corresponding poset congruences need not be lattice congruences. This disadvantage disappears when using our approach. We show that the classes of our so-called L-congruences 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 an L-congruence is a so-called strong filter. We study special L-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 kernels of L-congruences. We prove that every strong filter F of a relatively pseudocomplemented poset induces an L-congruence having F as its kernel. Finally, we consider Boolean posets which form a natural generalization of Boolean algebras. We show that special L-congruences on Boolean posets in general do not share properties known from Boolean algebras, but kernels of L-congruences on Boolean posets still have some interesting properties.