Residuation in twist products and pseudo-Kleene posets

Ivan Chajda; Helmut Länger

Mathematica Bohemica (2022)

  • Volume: 147, Issue: 3, page 369-383
  • ISSN: 0862-7959

Abstract

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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.

How to cite

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Chajda, Ivan, and Länger, Helmut. "Residuation in twist products and pseudo-Kleene posets." Mathematica Bohemica 147.3 (2022): 369-383. <http://eudml.org/doc/298479>.

@article{Chajda2022,
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.},
author = {Chajda, Ivan, Länger, Helmut},
journal = {Mathematica Bohemica},
keywords = {left-residuated poset; operator residuated poset; twist product; pseudo-Kleene poset; Kleene poset},
language = {eng},
number = {3},
pages = {369-383},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Residuation in twist products and pseudo-Kleene posets},
url = {http://eudml.org/doc/298479},
volume = {147},
year = {2022},
}

TY - JOUR
AU - Chajda, Ivan
AU - Länger, Helmut
TI - Residuation in twist products and pseudo-Kleene posets
JO - Mathematica Bohemica
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 147
IS - 3
SP - 369
EP - 383
AB - 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.
LA - eng
KW - left-residuated poset; operator residuated poset; twist product; pseudo-Kleene poset; Kleene poset
UR - http://eudml.org/doc/298479
ER -

References

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  1. Busaniche, M., Cignoli, R., 10.1007/s00012-014-0265-4, Algebra Univers. 71 (2014), 5-22. (2014) Zbl1303.03092MR3162417DOI10.1007/s00012-014-0265-4
  2. Chajda, I., A note on pseudo-Kleene algebras, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 55 (2016), 39-45. (2016) Zbl1431.06003MR3674598
  3. Chajda, I., Länger, H., Kleene posets and pseudo-Kleene posets, Available at { https://arxiv.org/abs/2006.04417} (2020), 18 pages. (2020) MR4440554
  4. Cignoli, R., 10.1090/S0002-9939-1975-0357259-4, Proc. Am. Math. Soc. 47 (1975), 269-278. (1975) Zbl0301.06009MR0357259DOI10.1090/S0002-9939-1975-0357259-4
  5. Kalman, J. A., 10.1090/S0002-9947-1958-0095135-X, Trans. Am. Math. Soc. 87 (1958), 485-491. (1958) Zbl0228.06003MR0095135DOI10.1090/S0002-9947-1958-0095135-X
  6. Tsinakis, C., Wille, A. M., 10.1007/s11225-006-8311-7, Stud. Log. 83 (2006), 407-423 9999DOI99999 10.1007/s11225-006-8311-7 . (2006) Zbl1101.06010MR2250118DOI10.1007/s11225-006-8311-7

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