Kybernetika 47 no. 1, 60-73, 2011

Orthocomplemented difference lattices with few generators

Milan Matoušek and Pavel Pták


The algebraic theory of quantum logics overlaps in places with certain areas of cybernetics, notably with the field of artificial intelligence (see, e. g., \cite{pp:book,wata}). Recently an effort has been exercised to advance with logics that possess a symmetric difference (\cite{matODL,MP1}) - with so called orthocomplemented difference lattices (ODLs). This paper further contributes to this effort. In \cite{matODL} the author constructs an ODL that is not set-representable. This example is quite elaborate. A main result of this paper somewhat economizes on this construction: There is an ODL with 3 generators that is not set-representable (and so the free ODL with 3 generators cannot be set-representable). The result is based on a specific technique of embedding orthomodular lattices into ODLs. The ODLs with 2 generators are always set-representable as we show by characterizing the free ODL with 2 generators - this ODL is ${\rm MO}_3 \times 2^4$.


orthomodular lattice, quantum logic, symmetric difference, Gödel's coding, Boolean algebra, free algebra


06C15, 03G12, 81B10


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