Derived operators on skew orthomodular and strong skew orthomodular posets
Ivan Chajda, Helmut Länger
TL;DR
The paper extends the sharp and flat derived operators from orthomodular lattices to skew orthomodular and strong skew orthomodular posets by defining a generalized compatibility relation $a \mathrel{\mathop{\mathrm{\rm C}}\nolimits} b$ and a commutator $c(a,b)$. It introduces $S$, $F$, and $c$ on these posets, analyzes their interrelations, and establishes key identities that generalize the lattice case, including behavior in Boolean posets via a Pixley-like ternary operator. Furthermore, it develops an adjointness framework for a derived conjunction and implication in orthoposets, proving an adjunction $a \odot b \le c$ iff $a \le b \to c$ under ACC plus two conditions, with Boolean posets yielding a classical adjoint pair and a Modus Ponens-like principle. These results contribute to a richer algebraic and logical toolkit for quantum-logic-inspired structures where joins/meets may be undefined, enhancing potential applications in quantum reasoning and poset-based logical frameworks.
Abstract
It is well-known that in the logic of quantum mechanics disjunctions and conjunctions can be represented by joins and meets, respectively, in an orthomodular lattice provided their entries commute. This was the reason why J. Pykacz introduced new derived operations called ''sharp'' and ''flat'' coinciding with joins and meets, respectively, for commuting elements but sharing some appropriate properties with disjunction and conjunction, respectively, in the whole orthomodular lattice in question. The problem is that orthomodular lattices need not formalize the logic of quantum mechanics since joins may not be defined provided their entries are neither comparable nor orthogonal. A corresponding fact holds for meets. Therefore, orthomodular posets are more accepted as an algebraic formalization of such a logic. The aim of the present paper is to extend the concepts of ''sharp'' and ''flat'' operations to operators in skew orthomodular and strong skew orthomodular posets. We generalize the relation of commuting elements as well as the commutator to such posets and we present some important properties of these operators and their mutual relationships. Moreover, we show that if the poset in question is even Boolean then there can be defined a ternary operator sharing the identities of a Pixley term. Finally, under some weak conditions which are automatically satisfied in Boolean algebras we show some kind of adjointness for operators formalizing conjunction and implication, respectively.