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Details of talk

TitleExpansions of dually pseudocomplemented Heyting algebras
PresenterChristopher Taylor (La Trobe University)
Author(s)Mr Christopher Taylor
Time14:30:00 2016-12-06

Fundamental to the theory of Heyting algebras is the fact that the lattice of
congruences on a Heyting algebra is isomorphic to the lattice of filters of the
underlying lattice. We investigate the consequences of including further
operations in the signature. Filters are still involved, and this leads to
natural interest in characterising the filters that correspond to congruences on
the expanded algebra. We will call such a filter a \emph{normal filter}.

We utilise a method of Hasimoto to produce, in certain cases, a unary term that
determines normal filters, which we call a \emph{normal filter term}. We extend
his existence conditions to apply to a larger class of algebras, with particular
emphasis on the dual pseudocomplement operation. The presence of a normal filter
term and the dual pseudocomplement operation is powerful enough to yield rather
strong results. In particular we characterise varieties with equationally
definable principal congruences, and prove that a variety of these algebras is
semisimple if and only if it is a discriminator variety.