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

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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 $\mathscr{H}$ of horizontal sums of Boolean algebras, we establish an identity which is satisfied in the variety generated by $\mathscr{H}$ but not in the variety of all orthomodular lattices. The concept of ternary discriminator can be generalized for the class $\mathscr{H}$ in a modified version. Finally, we present several results on varieties generated by finite subsets of finite members of $\mathscr{H}$.

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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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