Adjointness of generalized Sasaki operations in posets

TL;DR

This paper extends the concept of Sasaki operations to bounded posets with unary operations, characterizing conditions for their adjointness and establishing their properties in saturated orthomodular posets.

Contribution

It introduces generalized Sasaki projections and operations in bounded posets, providing conditions for their well-definedness and adjointness, especially in saturated orthomodular posets.

Findings

Generalized Sasaki operations are well-defined only in orthogonal posets.

Adjointness of these operations requires the unary operation to be a complementation.

In saturated orthomodular posets, the generalized Sasaki operations form an adjoint pair.

Abstract

The Sasaki projection and its dual were introduced as a mapping from the lattice of closed subspaces of a Hilbert space onto one of its segments. In a previous paper the authors showed that the Sasaki operations induced by the Sasaki projection and its dual form an adjoint pair in every orthomodular lattice. Later on the authors described large classes of algebras in which Sasaki operations can be defined and form an adjoint pair. The aim of the present paper is to extend these investigations to bounded posets with a unary operation. We introduce the so-called generalized Sasaki projection and its dual as well as the so-called generalized Sasaki operations induced by them. When treating these projections and operations we consider only so-called saturated posets, i.e. posets having the property that above any lower bound of two elements there is at least one maximal lower bound and below any upper bound of two elements there is at least one minimal upper bound. We prove that the generalized Sasaki operations are well-defined if and only if the poset in question is orthogonal. We characterize adjointness of the generalized Sasaki operations in different ways and show that adjointness is possible only if the unary operation is a complementation. Finally, we prove that in every saturated orthomodular poset the generalized Sasaki operations form an adjoint pair.