Orthocomplemented difference lattices with few generators

Milan Matoušek; Pavel Pták

Kybernetika (2011)

  • Volume: 47, Issue: 1, page 60-73
  • ISSN: 0023-5954

Abstract

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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 MO 3 × 2 4 .

How to cite

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Matoušek, Milan, and Pták, Pavel. "Orthocomplemented difference lattices with few generators." Kybernetika 47.1 (2011): 60-73. <http://eudml.org/doc/196848>.

@article{Matoušek2011,
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., [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 $\{\rm MO\}_3 \times 2^4$.},
author = {Matoušek, Milan, Pták, Pavel},
journal = {Kybernetika},
keywords = {orthomodular lattice; quantum logic; symmetric difference; Gödel's coding; Boolean algebra; free algebra; orthomodular lattice; quantum logic; symmetric difference; Boolean algebra; free algebra; orthocomplemented difference lattice},
language = {eng},
number = {1},
pages = {60-73},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Orthocomplemented difference lattices with few generators},
url = {http://eudml.org/doc/196848},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Matoušek, Milan
AU - Pták, Pavel
TI - Orthocomplemented difference lattices with few generators
JO - Kybernetika
PY - 2011
PB - Institute of Information Theory and Automation AS CR
VL - 47
IS - 1
SP - 60
EP - 73
AB - 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 ${\rm MO}_3 \times 2^4$.
LA - eng
KW - orthomodular lattice; quantum logic; symmetric difference; Gödel's coding; Boolean algebra; free algebra; orthomodular lattice; quantum logic; symmetric difference; Boolean algebra; free algebra; orthocomplemented difference lattice
UR - http://eudml.org/doc/196848
ER -

References

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