# Algebra Seminar talk

2013-09-10

Miroslav **Haviar***Canonical extensions of lattices - a new construction*

Abstract:

Canonical extensions of algebras have their origin in famous 1951-52 papers of
B. Jónsson and A. Tarski on Boolean algebras with operators. When the members
of a variety of lattice-based algebras are algebraic models of a logic,
canonicity (meaning that algebraic identities are preserved when constructing
canonical extensions of algebras) leads to completeness for the associated
logic. The concept has been intensively studied for distributive lattice
expansions, and more generally to lattice and even poset expansions, in the
last two decades (M. Gehrke, J. Harding, B. Jónsson, A. Palmigiano, H.A. Priestley and others).

A new construction of the canonical extensions of bounded lattices will be presented which is in the spirit of the theory of natural dualities. At the level of objects, this can be achieved similarly as in the distributive case, where Priestley duality is used. In the non-distributive case, a dual topological representation of bounded lattices due to M. Ploščica (1995) will be exploited which is less-known than the classical one due to A. Urquhart (1978). At the level of morphisms, a duality due to Allwein and Hartonas (1993) will be recast and applied.

This is joint work with Andrew P.K. Craig and Hilary A. Priestley.