Orthogonal relational systems

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 is a singleton (i.e. x; are complements of 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.