New Families of Finite Coherent Orthoalgebras without Bivaluations

TL;DR

This paper introduces a new method for constructing finite coherent orthoalgebras that lack bivaluations, with implications for quantum logic and Bell-Kochen-Specker theory, including a new infinite family of such structures.

Contribution

It provides a general construction method for finite orthoalgebras and presents a novel infinite family without bivaluations, expanding the understanding of their structure and symmetries.

Findings

Developed a general method for constructing finite orthoalgebras.

Created an infinite family of coherent orthoalgebras without bivaluations.

Described the symmetry groups of these orthoalgebras.

Abstract

In the present paper we study the following problem: how to construct a coherent orthoalgebra which has only a finite number of elements, but at the same time does not admit a bivaluation (i.e. a morphism with a codomain being an orthoalgebra with just two elements). This problem is important in the perspective of Bell-Kochen-Specker theory, since one can associate such an orthoalgebra to every saturated non-colorable finite configuration of projective lines. The first result obtained in this paper provides a general method for constructing finite orthoalgebras. This method is then applied to obtain a new infinite family of finite coherent orthoalgebras that do not admit bivaluations. The corresponding proof is combinatorial and yields a description of the groups of symmetries for these orthoalgebras.