Quasi-Semilattices on Networks

This paper introduces a representation of subnetworks of a network Γ consisting of a set of vertices and a set of relations, where relations are the primitive structures of a network. It is proven that all connected subnetworks of a network Γ form a quasi-semilattice L(Γ), namely a network quasi-semilattice.Two equivalences σ and δ are defined on L(Γ). Each δ class forms a semilattice and also has an order structure with the maximum element and minimum elements. Here, the minimum elements correspond to spanning trees in graph theory. Finally, we show how graph inverse semigroups, Leavitt path algebras and Cuntz–Krieger graph C*-algebras are constructed in terms of relations.