Algebra Seminar talk
Miroslav Haviar (Univerzita Mateja Bela, Banská Bystrica)
The canonical extension of lattices and lattice duals
Canonical extensions of lattice-based algebras originated in 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, during the last 25 years (M. Gehrke, J. Harding, B. Jónsson, A. Palmigiano, H.A. Priestley, and others). We present a brief overview of the studies of the canonical extensions of lattices.
We then present results of our recent work where we consider properties of the graphs that arise as duals of bounded lattices in their Ploščica's representation (1995) via maximal partial maps into the two-element set (this recasts Urquhart’s representation from 1978 in the spirit of the natural dualities). We introduce TiRS graphs which abstract the considered lattice duals. We demonstrate a one-to-one correspondence of TiRS graphs with so-called TiRS frames which are a subclass of the class of RS frames introduced by Gehrke (2006) to represent perfect lattices. This yields a dual representation of finite lattices via finite TiRS frames, or equivalently finite TiRS graphs, which generalises the well-known Birkhoff dual representation of finite distributive lattices via finite posets from the 1930s. By using both Ploščica's and Gehrke's representations in tandem we present a new construction of the canonical extensions of lattices. We also present open problems concerning the canonical extensions of lattices and lattice duals that can be of interest to researchers in this area.
(This is joint work with Andrew P.K. Craig and Maria J. Gouveia)