Duality for Quasilattices and Galois Connections

Issue title: Helena Rasiowa’s Heritage in Mathematical Logic. Centenary Special Issue

Guest editors: Damian Niwiński and Ewa Orłowska

Article type: Research Article

Authors: Romanowska, Anna B.^{a; *} | Smith, Jonathan D.H.^b

Affiliations: [a] Faculty of Mathematics and Information Sciences, Warsaw University of Technology, 00-661 Warszawa, Poland. [email protected] | [b] Department of Mathematics, Iowa State University, Ames, Iowa, 50011, USA. [email protected]

Correspondence: [*] Address for correspondence: Faculty of Mathematics and Information Sciences,Warsaw University of Technology, 00-661 Warszawa, Poland

Abstract: The primary goal of the paper is to establish a duality for quasilattices. The main ingredients are duality for semilattices and their representations, the structural analysis of quasilattices as Płonka sums of lattices, and the duality for lattices developed by Hartonas and Dunn. Lattice duality treats the identity function on a lattice as a Galois connection between its meet and join semilattice reducts, and then invokes a duality between Galois connections and polarities. A second goal of the paper is a further examination of this latter duality, using the concept of a pairing to provide an algebraic equivalent to the relational structure of a polarity.

Keywords: duality, Plonka sum, quasilattice, semilattice sum, Galois connection

DOI: 10.3233/FI-2017-1611

Journal: Fundamenta Informaticae, vol. 156, no. 3-4, pp. 331-359, 2017