| Title:
|
Quasimodules over bounded lattices (English) |
| Author:
|
Chajda, Ivan |
| Author:
|
Länger, Helmut |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0011-4642 |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
151 |
| Issue:
|
4 |
| Year:
|
2026 |
| Pages:
|
537-552 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We define a quasimodule $\bf Q$ over a bounded lattice $\bf L$ in an analogous way as a module over a semiring is defined. The essential difference is that $\bf L$ need not be distributive. Also for quasimodules there can be introduced the concepts of inner product, orthogonal elements, orthogonal subsets, bases and closed subquasimodules. We show that the set of all closed subquasimodules forms a complete lattice having orthogonality as an antitone involution. We describe important properties of closed subquasimodules. We prove that every canonical quasimodule has a nontrivial subquasimodule having a basis and we show that orthogonality can be introduced via the inner product. We call a subquasimodule $\bf P$ of a quasimodule $\bf Q$ splitting if the sum of $P$ and its orthogonal companion is the whole set $Q$ and the intersection of $P$ and its orthogonal companion is trivial. We show that every splitting subquasimodule is closed and that its orthogonal companion is splitting, too. Our results are illuminated by several examples. (English) |
| Keyword:
|
quasimodule |
| Keyword:
|
canonical quasimodule |
| Keyword:
|
lattice |
| Keyword:
|
$0$-distributive lattice |
| Keyword:
|
orthogonal set |
| Keyword:
|
basis |
| Keyword:
|
closed subquasimodule |
| Keyword:
|
splitting subquasimodule |
| MSC:
|
06B05 |
| MSC:
|
06D99 |
| MSC:
|
13C13 |
| DOI:
|
10.21136/MB.2025.0143-24 |
| . |
| Date available:
|
2026-09-23T08:22:11Z |
| Last updated:
|
2026-09-23 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153765 |
| . |
| Reference:
|
[1] Birkhoff, G.: Lattice Theory.American Mathematical Society Colloquium Publications 25. AMS, Providence (1979). Zbl 0505.06001, MR 0598630, 10.1090/coll/025 |
| Reference:
|
[2] Chajda, I., Länger, H.: The lattice of subspaces of a vector space over a finite field.Soft Comput. 23 (2019), 3261-3267. Zbl 1475.06003, 10.1007/s00500-019-03866-y |
| Reference:
|
[3] Chajda, I., Länger, H.: Orthogonality and complementation in the lattice of subspaces of a finite vector space.Math. Bohem. 147 (2022), 141-153. Zbl 1513.06018, MR 4407348, 10.21136/MB.2021.0042-20 |
| Reference:
|
[4] Chajda, I., Länger, H.: Semimodules over commutative semirings and modules over unitary commutative rings.Linear Multilinear Algebra 70 (2022), 1329-1344. Zbl 1547.16045, MR 4413126, 10.1080/03081087.2020.1760192 |
| Reference:
|
[5] Kruml, D., Paseka, J.: Algebraic and categorical aspects of quantales.Handbook of Algebra. Volume 5 Elsevier, Amsterdam (2008), 323-362. Zbl 1219.06016, MR 2523454, 10.1016/S1570-7954(07)05006-1 |
| Reference:
|
[6] Rao, G. N., Raju, R. V. A.: 0-distributive almost lattices.Bull. Int. Math. Virtual Inst. 10 (2020), 239-248. Zbl 1474.06050, MR 4021350 |
| . |
