Orthomodular lattices that are horizontal sums of Boolean algebras

Ivan Chajda; Helmut Länger

Commentationes Mathematicae Universitatis Carolinae (2020)

  • Volume: 61, Issue: 1, page 11-20
  • ISSN: 0010-2628

Abstract

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The paper deals with orthomodular lattices which are so-called horizontal sums of Boolean algebras. It is elementary that every such orthomodular lattice is simple and its blocks are just these Boolean algebras. Hence, the commutativity relation plays a key role and enables us to classify these orthomodular lattices. Moreover, this relation is closely related to the binary commutator which is a term function. Using the class of horizontal sums of Boolean algebras, we establish an identity which is satisfied in the variety generated by but not in the variety of all orthomodular lattices. The concept of ternary discriminator can be generalized for the class in a modified version. Finally, we present several results on varieties generated by finite subsets of finite members of .

How to cite

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Chajda, Ivan, and Länger, Helmut. "Orthomodular lattices that are horizontal sums of Boolean algebras." Commentationes Mathematicae Universitatis Carolinae 61.1 (2020): 11-20. <http://eudml.org/doc/297325>.

@article{Chajda2020,
abstract = {The paper deals with orthomodular lattices which are so-called horizontal sums of Boolean algebras. It is elementary that every such orthomodular lattice is simple and its blocks are just these Boolean algebras. Hence, the commutativity relation plays a key role and enables us to classify these orthomodular lattices. Moreover, this relation is closely related to the binary commutator which is a term function. Using the class $\mathcal \{H\}$ of horizontal sums of Boolean algebras, we establish an identity which is satisfied in the variety generated by $\mathcal \{H\}$ but not in the variety of all orthomodular lattices. The concept of ternary discriminator can be generalized for the class $\mathcal \{H\}$ in a modified version. Finally, we present several results on varieties generated by finite subsets of finite members of $\mathcal \{H\}$.},
author = {Chajda, Ivan, Länger, Helmut},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {orthomodular lattice; horizontal sum; commuting elements; Boolean algebra},
language = {eng},
number = {1},
pages = {11-20},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Orthomodular lattices that are horizontal sums of Boolean algebras},
url = {http://eudml.org/doc/297325},
volume = {61},
year = {2020},
}

TY - JOUR
AU - Chajda, Ivan
AU - Länger, Helmut
TI - Orthomodular lattices that are horizontal sums of Boolean algebras
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2020
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 61
IS - 1
SP - 11
EP - 20
AB - The paper deals with orthomodular lattices which are so-called horizontal sums of Boolean algebras. It is elementary that every such orthomodular lattice is simple and its blocks are just these Boolean algebras. Hence, the commutativity relation plays a key role and enables us to classify these orthomodular lattices. Moreover, this relation is closely related to the binary commutator which is a term function. Using the class $\mathcal {H}$ of horizontal sums of Boolean algebras, we establish an identity which is satisfied in the variety generated by $\mathcal {H}$ but not in the variety of all orthomodular lattices. The concept of ternary discriminator can be generalized for the class $\mathcal {H}$ in a modified version. Finally, we present several results on varieties generated by finite subsets of finite members of $\mathcal {H}$.
LA - eng
KW - orthomodular lattice; horizontal sum; commuting elements; Boolean algebra
UR - http://eudml.org/doc/297325
ER -

References

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  1. Beran L., Orthomodular Lattices, Algebraic Approach, Mathematics and Its Applications (East European Series), D. Reidel Publishing, Dordrecht, 1985. Zbl0558.06008MR0784029
  2. Burris S., Sankappanavar H. P., 10.1007/978-1-4613-8130-3_3, Graduate Texts in Mathematics, 78, Springer, New York, 1981. Zbl0478.08001MR0648287DOI10.1007/978-1-4613-8130-3_3
  3. Chajda I., Länger H., Padmanabhan R., 10.1515/ms-2017-0138, Math. Slovaca 68 (2018), no. 4, 713–716. MR3841901DOI10.1515/ms-2017-0138
  4. Chajda I., Padmanabhan R., 10.14232/actasm-016-514-2, Acta Sci. Math. (Szeged) 83 (2017), no. 1–2, 31–34. MR3701028DOI10.14232/actasm-016-514-2
  5. Jónsson B., 10.7146/math.scand.a-10850, Math. Scand. 21 (1967), 110–121. MR0237402DOI10.7146/math.scand.a-10850
  6. Kalmbach G., Orthomodular Lattices, London Mathematical Society Monographs, 18, Academic Press, London, 1983. Zbl0554.06009MR0716496

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