Kybernetika 47 no. 1, 60-73, 2011

Orthocomplemented difference lattices with few generators

Milan Matoušek and Pavel Pták

Abstract:

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 ${\rm MO}_3 \times 2^4$.

Keywords:

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

Classification:

06C15, 03G12, 81B10

paper.pdf

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