Conference abstracts

Session S09 - Logic and Universal Algebra

July 25, 16:30 ~ 16:55

A new setting for duals of canonical extensions of lattices

Miroslav Haviar

Matej Bel University, Eslovaquia   -

We consider properties of the graphs that arise as duals of bounded lattices in Ploscica’s representation via maximal partial maps into the two-element set. We introduce TiRS graphs which abstract those duals of bounded lattices. We demonstrate their one-to-one correspondence with so–called TiRS frames which are a subclass of the class of RS frames introduced by Gehrke 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. By using both Ploscica’s and Gehrke’s representations in tandem we present a new construction of the canonical extension of a bounded lattice. We present two open problems that can be of interest to researchers working in this area.

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