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Optimal natural dualities: the role of endomorphisms

Part of: Boolean algebras (Boolean rings) Distributive lattices Other classes of algebra

Published online by Cambridge University Press:  09 April 2009

M. J. Saramago
M. J. Saramago
Affiliation:
Departmento de Matemática, Faculdade de Ciências, Universidade de Lisboa, 1749–016 Lisboa, Portugal e-mail: matjoao@ptmat.fc.ul.pt Centro de Álgebra, Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649–003 Lisboa, Portugal
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Abstract

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The optimality of dualities on a quasivariety , generated by a finite algebra , has been introduced by Davey and Priestley in the 1990s. Since every optimal duality is determined by a transversal of a certain family of subsets of Ω, where Ω is a given set of relations yielding a duality on , an understanding of the structures of these subsets—known as globally minimal failsets—was required. A complete description of globally minimal failsets which do not contain partial endomorphisms has recently been given by the author and H. A. Priestley. Here we are concerned with globally minimal failsets containing endomorphisms. We aim to explain what seems to be a pattern in the way endomorphisms belong to these failsets. This paper also gives a complete description of globally minimal failsets whose minimal elements are automorphisms, when is a subdirectly irreducible lattice-structured algebra.

Keywords

natural dualityoptimal dualityfailset

MSC classification

Secondary: 06D05: Structure and representation theory 06D30: De Morgan algebras, L ukasiewicz algebras 08C15: Quasivarieties
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 77 , Issue 2 , October 2004 , pp. 269 - 296
Copyright
Copyright © Australian Mathematical Society 2004

References

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