Residuation in twist products and pseudo-Kleene posets Abstract

TL;DR

This paper generalizes residuation concepts from lattice-ordered structures to non-lattice left-residuated groupoids, introducing operator residuated posets and exploring conditions for pseudo-Kleene structures.

Contribution

It introduces new constructions for residuation in twist products without requiring lattice order, expanding the theoretical framework to broader algebraic structures.

Findings

New construction for residuation in non-lattice groupoids

Operator residuated posets that preserve key properties

Conditions for pseudo-Kleene operator residuated posets

Abstract

M. Busaniche, R. Cignoli, C. Tsinakis and A. M. Wille 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, for the full twist product we cannot use the same construction. We present another appropriate construction which, however, does not preserve commutativity and associativity of multiplication. Hence we introduce 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.