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

Andrew P.K. Craig, Maria J. Gouveia, Miroslav Haviar

Research output: Contribution to journalArticlepeer-review

15 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)123-138
Number of pages16
JournalAlgebra Universalis
Issue number1-2
Publication statusPublished - 1 Sept 2015


  • 06B15
  • Primary: 06B23
  • Secondary: 06D50

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Logic


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