Abstract
The concept of operator left residuation has been introduced by the authors in their previous paper (Chajda and Länger in Asian Eur J Math 11:1850097, 2018). Modifications of so-called quantum structures, in particular orthomodular posets, like pseudo-orthomodular, pseudo-Boolean and Boolean posets are investigated here in order to show that they are operator left residuated or even operator residuated. In fact, they satisfy more general sufficient conditions for operator residuation assumed for bounded posets equipped with a unary operation. It is shown that these conditions may be also necessary if a generalized version using subsets instead of single elements is considered. The above-listed posets can serve as an algebraic semantics for the logic of quantum mechanics in a broad sense. Moreover, our approach shows connections to substructural logics via the considered residuation.
Highlights
Keywords Operator residuation · Operator left adjointness · Boolean poset · Pseudo-Boolean poset · Pseudo-orthomodular poset · Generalized operator residuation. It was shown by Birkhoff and von Neumann (1936) and, independently, by Husimi (1937) that orthomodular lattices can serve as an algebraic semantics of the logic of quantum mechanics
The class of event-state systems in quantum mechanics is usually identified with the set of projection operators on a Hilbert space H and this set is in a bijective correspondence with the set of all closed linear subspaces of H
Certain doubts concerning the relevance of this representation arose when it was shown that the class of orthomodular lattices arising in this way does not generate the variety of orthomodular lattices
Summary
It was shown by Birkhoff and von Neumann (1936) and, independently, by Husimi (1937) that orthomodular lattices can serve as an algebraic semantics of the logic of quantum mechanics. The authors proved in Chajda and Länger (2017a, b) that every orthomodular lattice can be converted into a so-called left residuated l-groupoid. They showed in Chajda and Länger (submitted) that this result can be extended to a certain class of bounded lattices with a unary operation which, contains the variety of orthomodular lattices. The aim of the present paper is to provide several simple conditions under which a bounded poset with a unary operation can be organized into an operator left residuated poset As it was done for lattices in Chajda and Länger (submitted), we ask whether these conditions are sufficient and necessary. It is shown that if subsets instead of single elements are considered, these generalized conditions characterize the class of posets which can be converted into operator residuated ones
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