Relational correspondences for lattices with operators

Abstract

In this paper we present some examples of relational correspondences for not necessarily distributive lattices with modal-like operators of possibility (normal and additive operators) and sufficiency (co-normal and co-additive operators). Each of the algebras (P, ∨, ∧, 0, 1, f), where (P, ∨, ∧, 0, 1) is a bounded lattice and / is a unary operator on P, determines a relational system (frame) (X(P), ≲1, ≲2, R f, Sf) with binary relations ≲1, ≲2, Rf, Sf, appropriately defined from P and f. Similarly, any frame of the form (X, ≲1, ≲2, R, S) with two quasi-orders ≲1 and ≲2, and two binary relations R and S induces an algebra (L(X), ∨, ∧, 0, 1, fR,S), where the operations ∨, ∧, and fR,S and constants 0 and 1 are defined from the resources of the frame. We investigate, on the one hand, how properties of an operator f in an algebra P correspond to the properties of relations Rf and S f in the induced frame and, on the other hand, how properties of relations in a frame relate to the properties of the operator fR, S of an induced algebra. The general observations and the examples of correspondences presented in this paper are a first step towards development of a correspondence theory for lattices with operators. © Springer-Verlag Berlin Heidelberg 2006.