Geodesic
Residuation in twist products and pseudo-Kleene posets
Mathematica Bohemica, Tome 147 (2022) no. 3, pp. 369-383 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.21136/MB.2021.0182-20
Classification : 03B47, 03G25, 06A11, 06D30
Keywords: left-residuated poset; operator residuated poset; twist product; pseudo-Kleene poset; Kleene poset 
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Chajda, Ivan; Länger, Helmut. Residuation in twist products and pseudo-Kleene posets. Mathematica Bohemica, Tome 147 (2022) no. 3, pp. 369-383. doi: 10.21136/MB.2021.0182-20

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