Canonical extensions of lattices are more than perfect
09/09/2019 FCUL, 17:00  6.2.33
Miroslav Haviar (Matej Bel University, Banská Bystrica, Slovakia)
Faculty of Sciences, University of Lisbon
Canonical extensions of latticebased algebras originated in the 195152 papers of B. Jónsson and A. Tarski on Boolean algebras with operators. When the members of a variety of latticebased 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 for lattice and even poset expansions, during the last 25 years (M. Gehrke, J. Harding, B. Jónsson, A. Palmigiano, H.A. Priestley, Y. Venema, and others). We present a brief overview of the studies of the canonical extensions of lattices.
We then present results of our work where we consider properties of the graphs that arise as duals of bounded lattices in Ploš?ica's representation (1995) via maximal partial maps into the twoelement 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 onetoone correspondence of TiRS graphs with socalled 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 wellknown Birkhoff dual representation of finite distributive lattices via finite posets from the 1930s.
We indicate our recent investigations in this topic. We show that the canonical extensions of lattices are more than perfect  they are perfect lattices satisfying an extra (PTi) condition. Thus they belong to a newly introduced class of (PTi) lattices that correspond to our TiRS frames (resp. TiRS graphs). We also show that there are (PTi) lattices that are not the canonical extension of any bounded lattice. On the side of the relational structures this extends the notion of a nonrepresentable poset to the duals of nondistributive lattices. We present questions that can be of interest to researchers in this and related areas.
*a joint work with Andrew P.K. Craig (Johannesburg) and Maria J. Gouveia (Lisbon)
