| Title:
|
Residuation in twist products and pseudo-Kleene posets (English) |
| Author:
|
Chajda, Ivan |
| Author:
|
Länger, Helmut |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
147 |
| Issue:
|
3 |
| Year:
|
2022 |
| Pages:
|
369-383 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
M. Busaniche, R. Cignoli (2014), C. Tsinakis and A. M. Wille (2006) showed that every residuated lattice induces a residuation on its full twist product. For their construction they used also lattice operations. We generalize this problem to left-residuated groupoids which need not be lattice-ordered. Hence, we cannot use the same construction for the full twist product. We present another appropriate construction which, however, does not preserve commutativity and associativity of multiplication. Hence we introduce the so-called operator residuated posets to obtain another construction which preserves the mentioned properties, but the results of operators on the full twist product need not be elements, but may be subsets. We apply this construction also to restricted twist products and present necessary and sufficient conditions under which we obtain a pseudo-Kleene operator residuated poset. (English) |
| Keyword:
|
left-residuated poset |
| Keyword:
|
operator residuated poset |
| Keyword:
|
twist product |
| Keyword:
|
pseudo-Kleene poset |
| Keyword:
|
Kleene poset |
| MSC:
|
03B47 |
| MSC:
|
03G25 |
| MSC:
|
06A11 |
| MSC:
|
06D30 |
| idZBL:
|
Zbl 07584131 |
| idMR:
|
MR4482312 |
| DOI:
|
10.21136/MB.2021.0182-20 |
| . |
| Date available:
|
2022-09-05T09:39:15Z |
| Last updated:
|
2022-12-27 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151014 |
| . |
| Reference:
|
[1] Busaniche, M., Cignoli, R.: The subvariety of commutative residuated lattices represented by twist-products.Algebra Univers. 71 (2014), 5-22. Zbl 1303.03092, MR 3162417, 10.1007/s00012-014-0265-4 |
| Reference:
|
[2] Chajda, I.: A note on pseudo-Kleene algebras.Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 55 (2016), 39-45. Zbl 1431.06003, MR 3674598 |
| Reference:
|
[3] Chajda, I., Länger, H.: Kleene posets and pseudo-Kleene posets.Available at { https://arxiv.org/abs/2006.04417} (2020), 18 pages. MR 4440554 |
| Reference:
|
[4] Cignoli, R.: Injective De Morgan and Kleene algebras.Proc. Am. Math. Soc. 47 (1975), 269-278. Zbl 0301.06009, MR 0357259, 10.1090/S0002-9939-1975-0357259-4 |
| Reference:
|
[5] Kalman, J. A.: Lattices with involution.Trans. Am. Math. Soc. 87 (1958), 485-491. Zbl 0228.06003, MR 0095135, 10.1090/S0002-9947-1958-0095135-X |
| Reference:
|
[6] Tsinakis, C., Wille, A. M.: Minimal varieties of involutive residuated lattices.Stud. Log. 83 (2006), 407-423 \99999DOI99999 10.1007/s11225-006-8311-7 \vfil. Zbl 1101.06010, MR 2250118, 10.1007/s11225-006-8311-7 |
| . |
