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# Details of talk

Title Expansions of dually pseudocomplemented Heyting algebras Christopher Taylor (La Trobe University) Mr Christopher Taylor Algebra 14:30:00 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.