2023/11/10 by Davide Fazio, Fazio, Davide, Raffaele Mascella +1
Computer Science · #Advanced Algebra and Logic #FOS: Mathematics #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2311.06109
openalex publication_date 2023/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we provide a preliminary investigation of subclasses of bounded posets with antitone involution which are "pastings" of their maximal Kleene sub-lattices. Specifically, we introduce super-paraorthomodular lattices, namely paraothomodular lattices whose order determines, and it is fully determined by, the order of their maximal Kleene sub-algebras. It will turn out that the (spectral) paraorthomodular lattice of effects over a separable Hilbert space can be considered as a prominent example of such. Therefore, it arguably provides an algebraic/order theoretical rendering of complementarity phenomena between unsharp observables. A number of examples, properties and characterization theorems for structures we deal with will be outlined. For example, we prove a forbidden configuration theorem and we investigate the notion of commutativity for modular pseudo-Kleene lattices, examples of which are (spectral) paraorthomodular lattices of effects over finite-dimensional Hilbert spaces. Finally, we show that structures introduced in this paper yield paraconsistent partial referential matrices, the latter being generalizations of J. Czelakowski's partial referential matrices. As a consequence, a link between some classes of posets with antitone involution and algebras of partial "unsharp" propositions is established.