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Relational Correspondences for Lattices with Operators

  • Jouni Järvinen
  • Ewa Orłowska
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3929)

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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Jouni Järvinen
    • 1
  • Ewa Orłowska
    • 2
  1. 1.Turku Centre for Computer ScienceTurkuFinland
  2. 2.National Institute of TelecommunicationsWarszawaPoland

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