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 (shape P, ∨, ∧, 0,1) is a bounded lattice and shape f is a unary operator on shape P, determines a relational system (frame) $(X(P), \lesssim_1, \lesssim_2, R_f, S_f)$ with binary relations $\lesssim_1$, $\lesssim_2$, shape R _f , shape S _f , appropriately defined from shape P and shape f. Similarly, any frame of the form $(X, \lesssim_1, \lesssim_2, R, S)$ with two quasi-orders $\lesssim_1$ and $\lesssim_2$, and two binary relations shape R and shape S induces an algebra (shape L(shape X), ∨, ∧, 0,1, shape f _R,S ), where the operations ∨, ∧, and shape f _R,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 shape f in an algebra shape P correspond to the properties of relations shape R _f and shape 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 shape f _R,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.

This research was supported by COST Action 274 “Theory and Applications of Relational Structures as Knowledge Instruments” (http://www.tarski.org).