J. Aust. Math. Soc.
77 (2004), 269-296
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Optimal natural dualities: the role of endomorphisms
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M. J. Saramago
Departamento de Matemática
Faculdade de Ciencias
Universidade de Lisboa
1749--016 Lisboa
Portugal
matjoao@ptmat.fc.ul.pt
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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
, where
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.
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