Relational correspondences for lattices with operators

• Authors: Jouni Järvinen;Ewa Orłowska
• Venue: RelMiCS'05 Proceedings of the 8th international conference on Relational Methods in Computer Science, Proceedings of the 3rd international conference on Applications of Kleene Algebra
• Year: 2005

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 Rf, shape Sf, 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 fR,S), where the operations ∨, ∧, and shape 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 shape f in an algebra shape P correspond to the properties of relations shape Rf and shape Sf in the induced frame and, on the other hand, how properties of relations in a frame relate to the properties of the operator shape 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.