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 (P, ∨, ∧, 0, 1) is a bounded lattice and f is a unary operator on P, determines a relational system (frame) (X(P), 1, 2, Rf , 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 Sf in the induced frame and, on the other hand, how properties of relations in a frame relate to th...

• Type: Conference
• Year: 2005
• Where: RELMICS
• Authors: Jouni Järvinen, Ewa Orlowska
• Keywords: Algebra, Binary Relations, Operator, RELMICS 2005