| Title:
|
Orthomodular lattices that are horizontal sums of Boolean algebras (English) |
| Author:
|
Chajda, Ivan |
| Author:
|
Länger, Helmut |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
61 |
| Issue:
|
1 |
| Year:
|
2020 |
| Pages:
|
11-20 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $\mathcal H$. (English) |
| Keyword:
|
orthomodular lattice |
| Keyword:
|
horizontal sum |
| Keyword:
|
commuting elements |
| Keyword:
|
Boolean algebra |
| MSC:
|
06C15 |
| MSC:
|
06C20 |
| MSC:
|
06E75 |
| idZBL:
|
Zbl 07217154 |
| idMR:
|
MR4093425 |
| DOI:
|
10.14712/1213-7243.2020.003 |
| . |
| Date available:
|
2020-04-30T11:12:19Z |
| Last updated:
|
2022-04-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148071 |
| . |
| Reference:
|
[1] Beran L.: Orthomodular Lattices, Algebraic Approach.Mathematics and Its Applications (East European Series), D. Reidel Publishing, Dordrecht, 1985. Zbl 0558.06008, MR 0784029 |
| Reference:
|
[2] Burris S., Sankappanavar H. P.: A Course in Universal Algebra.Graduate Texts in Mathematics, 78, Springer, New York, 1981. Zbl 0478.08001, MR 0648287, 10.1007/978-1-4613-8130-3_3 |
| Reference:
|
[3] Chajda I., Länger H., Padmanabhan R.: Single identities forcing lattices to be Boolean.Math. Slovaca 68 (2018), no. 4, 713–716. MR 3841901, 10.1515/ms-2017-0138 |
| Reference:
|
[4] Chajda I., Padmanabhan R.: Lattices with unique complementation.Acta Sci. Math. (Szeged) 83 (2017), no. 1–2, 31–34. MR 3701028, 10.14232/actasm-016-514-2 |
| Reference:
|
[5] Jónsson B.: Algebras whose congruence lattices are distributive.Math. Scand. 21 (1967), 110–121. MR 0237402, 10.7146/math.scand.a-10850 |
| Reference:
|
[6] Kalmbach G.: Orthomodular Lattices.London Mathematical Society Monographs, 18, Academic Press, London, 1983. Zbl 0554.06009, MR 0716496 |
| . |
