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TiRS graphs and TiRS frames: a new setting for duals of canonical extensions

Craig, A.; Gouveia, M. J.; Haviar, M.

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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 nite lattices via nite TiRS frames, or equivalently nite TiRS graphs, which generalises the well-known Birkho dual representation of nite distributive lattices via nite 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 will be
of interest to researchers working in this area.