Orthogonal relational systems

TL;DR

This paper explores orthogonal relational systems with involution, establishing their connection to groupoids, and demonstrating their algebraic properties like decomposition and amalgamation.

Contribution

It introduces the concept of orthogonal relational systems with involution and characterizes their relationship with groupoids, revealing their algebraic structure and properties.

Findings

Many properties of orthogonal relational systems are captured by associated groupoids.

Orthogonal relational systems admit direct decomposition representations.

They possess the strong amalgamation property.

Abstract

In this paper we discuss the concept of relational system with involution. This system is called orthogonal if, for every pair of non-zero orthogonal elements, there exists a supremal element in their upper cone and the upper cone of orthogonal elements $x,\,x'$ is a singleton (i.e. $x,\,x'$ are complements each other). To every orthogonal relational system can be assigned a groupoid with involution. The conditions under which a groupoid is assigned to an orthogonal relational systems are investigated. We will see that many properties of the relational system can be captured by the associated groupoid. Moreover, these structures enjoy several desirable algebraic features such as, e.g., a direct decomposition representation and the strong amalgamation property.