Matoušek, Milan ; Pták, Pavel
Orthocomplemented difference lattices with few generators. (English). Kybernetika, vol. 47 (2011), issue 1, pp. 60-73
MSC: 03G12, 06C15, 81B10 | MR 2807864 | Zbl 1221.06011
orthomodular lattice; quantum logic; symmetric difference; Gödel's coding; Boolean algebra; free algebra
The algebraic theory of quantum logics overlaps in places with certain areas of cybernetics, notably with the field of artificial intelligence (see, e. g., [19, 20]). Recently an effort has been exercised to advance with logics that possess a symmetric difference ([13, 14]) - with so called orthocomplemented difference lattices (ODLs). This paper further contributes to this effort. In [13] 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 ${\rm MO}_3 \times 2^4$.
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