Abstract
We consider properties of the graphs that arise as duals of bounded lattices in Ploščica’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 Ploščica’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.
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Presented by R. Quackenbush.
Dedicated to Professor Brian A. Davey on his 65th birthday
The first author gratefully acknowledges funding from the Claude Leon Foundation during the writing of this paper. He also acknowledges the support of the Rhodes Trust during his DPhil studies at Oxford when some of the preliminary work was done. The second author acknowledges support from Portuguese Project PEst-OE/MAT/UI0143/2014 of CAUL financed by FCT. The third author acknowledges support from Slovak grants VEGA 1/0212/13 and APVV-0223-10.
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Craig, A.P.K., Gouveia, M.J. & Haviar, M. TiRS graphs and TiRS frames: a new setting for duals of canonical extensions. Algebra Univers. 74, 123–138 (2015). https://doi.org/10.1007/s00012-015-0335-2
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DOI: https://doi.org/10.1007/s00012-015-0335-2