Relational Correspondences for Lattices with Operators
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.
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