| Title:
|
c-ideals in complemented posets (English) |
| Author:
|
Chajda, Ivan |
| Author:
|
Kolařík, Miroslav |
| Author:
|
Länger, Helmut |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
149 |
| Issue:
|
3 |
| Year:
|
2024 |
| Pages:
|
305-316 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In their recent paper on posets with a pseudocomplementation denoted by $*$ the first and the third author introduced the concept of a $*$-ideal. This concept is in fact an extension of a similar concept introduced in distributive pseudocomplemented lattices and semilattices by several authors, see References. Now we apply this concept of a c-ideal (dually, c-filter) to complemented posets where the complementation need neither be antitone nor an involution, but still satisfies some weak conditions. We show when an ideal or filter in such a poset is a c-ideal or c-filter, and we prove basic properties of them. Finally, we prove the so-called separation theorems for c-ideals. The text is illustrated by several examples. (English) |
| Keyword:
|
complemented poset |
| Keyword:
|
antitone involution |
| Keyword:
|
ideal |
| Keyword:
|
filter |
| Keyword:
|
ultrafilter |
| Keyword:
|
c-ideal |
| Keyword:
|
c-filter |
| Keyword:
|
c-condition |
| Keyword:
|
separation theorem |
| MSC:
|
06A11 |
| MSC:
|
06C15 |
| DOI:
|
10.21136/MB.2023.0108-22 |
| . |
| Date available:
|
2024-09-11T13:45:05Z |
| Last updated:
|
2024-09-11 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152536 |
| . |
| Reference:
|
[1] Birkhoff, G.: Lattice Theory.American Mathematical Society Colloquium Publications 25. AMS, Providence (1979). Zbl 0505.06001, MR 0598630 |
| Reference:
|
[2] Chajda, I., Länger, H.: Filters and congruences in sectionally pseudocomplemented lattices and posets.Soft Comput. 25 (2021), 8827-8837. Zbl 1498.06020, 10.1007/s00500-021-05900-4 |
| Reference:
|
[3] Chajda, I., Länger, H.: Filters and ideals in pseudocomplemented posets.Available at https://arxiv.org/abs/2202.03166 (2022), 14 pages. |
| Reference:
|
[4] Grätzer, G.: Lattice Theory: Foundation.Birkhäuser, Basel (2011). Zbl 1233.06001, MR 2768581, 10.1007/978-3-0348-0018-1 |
| Reference:
|
[5] Larmerová, J., Rachůnek, J.: Translations of distributive and modular ordered sets.Acta Univ. Palacki. Olomuc., Fac. Rerum Nat. Math. 27 (1988), 13-23. Zbl 0693.06003, MR 1039879 |
| Reference:
|
[6] Nimbhorkar, S. K., Nehete, J. Y.: $\delta$-ideals in pseudo-complemented distributive join-semilattices.Asian-Eur. J. Math. 14 (2021), Article ID 2150106, 7 pages. Zbl 1483.06007, MR 4280926, 10.1142/S1793557121501060 |
| Reference:
|
[7] Rao, M. S.: $\delta$-ideals in pseudo-complemented distributive lattices.Arch. Math., Brno 48 (2012), 97-105. Zbl 1274.06036, MR 2946209, 10.5817/AM2012-2-97 |
| Reference:
|
[8] Talukder, M. R., Chakraborty, H. S., Begum, S. N.: $\delta$-ideals of a pseudocomplemented semilattice.Afr. Mat. 32 (2021), 419-429. Zbl 1488.06008, MR 4259344, 10.1007/s13370-020-00834-w |
| . |
