Mathematica Bohemica, first online, pp. 1-15


Residuation in twist products and pseudo-Kleene posets

Ivan Chajda, Helmut Länger

Received November 29, 2020.   Published online August 26, 2021.

Abstract:  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.
Keywords:  left-residuated poset; operator residuated poset; twist product; pseudo-Kleene poset; Kleene poset
Classification MSC:  06A11, 06D30, 03G25, 03B47
DOI:  10.21136/MB.2021.0182-20

PDF available at:  Institute of Mathematics CAS

References:
[1] M. Busaniche, R. Cignoli: The subvariety of commutative residuated lattices represented by twist-products. Algebra Univers. 71 (2014), 5-22. DOI 10.1007/s00012-014-0265-4 | MR 3162417 | Zbl 1303.03092
[2] I. Chajda: A note on pseudo-Kleene algebras. Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 55 (2016), 39-45. MR 3674598 | Zbl 1431.06003
[3] I. Chajda, H. Länger: Kleene posets and pseudo-Kleene posets. Available at {https://arxiv.org/abs/2006.04417} (2020), 18 pages.
[4] R. Cignoli: Injective De Morgan and Kleene algebras. Proc. Am. Math. Soc. 47 (1975), 269-278. DOI 10.1090/S0002-9939-1975-0357259-4 | MR 0357259 | Zbl 0301.06009
[5] J. A. Kalman: Lattices with involution. Trans. Am. Math. Soc. 87 (1958), 485-491. DOI 10.1090/S0002-9947-1958-0095135-X | MR 0095135 | Zbl 0228.06003
[6] C. Tsinakis, A. M. Wille: Minimal varieties of involutive residuated lattices. Stud. Log. 83 (2006), 407-423. DOI 10.1007/s11225-006-8311-7 \vfil | MR 2250118 | Zbl 1101.06010

Affiliations:   Ivan Chajda, Palacký University Olomouc, Faculty of Science, Department of Algebra and Geometry, 17. listopadu 12, 771 46 Olomouc, Czech Republic, e-mail: ivan.chajda@upol.cz; Helmut Länger, TU Wien, Fakultät für Mathematik und Geoinformation, Institut für Diskrete Mathematik und Geometrie, Wiedner Hauptstrasse 8-10, 1040 Wien, Austria, and Palacký University Olomouc, Faculty of Science, Department of Algebra and Geometry, 17. listopadu 12, 771 46 Olomouc, Czech Republic, e-mail: helmut.laenger@tuwien.ac.at


 
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