Stellenbosch University
Dual representations of
ortholattices
Andrew Craig\(^*\), University of
Johannesburg
Gerhard Dorfer, Technical University
of Vienna, Austria
Miroslav Haviar, Matej Bel University,
Slovakia & University of Johannesburg
Klarise Marais, University of
Johannesburg
SAMS Subject Classification Number: 34
An ortholattice is a bounded lattice with a complement operation (i.e. a unary map \('\) satisfying \(x\vee x'=1\) and \(x \wedge x'=0\)) that is both involutive and order-reversing. This operation is known as the orthocomplement. We use digraphs with topology, due to Ploščica [3], to provide a representation of the underlying lattice of an ortholattice. We then equip these digraphs with an additional unary map to represent the orthocomplement. We will contrast our dual representation with the approaches of Goldblatt [2] and Dzik et al. [1].
References
[1] W. Dzik, E. Orlowska, C. van Alten, Relational representation theorems for general lattices with negations, In: RelMiCS 2006 (R.A. Schmidt ed.), LNCS 4136 (2006), 162-176.
[2] R. Goldblatt, The Stone space of an ortholattice, Bull. London Math. Soc. 7 (1975), 45–48.
[3] M. Ploščica, A natural representation of bounded lattices, Tarta Mountain Math. Publ. 5 (1995), 75–88.
