A STUDY ON GRAPHS DEFINED ON L-SLICES

Abstract

Let L be a locale with top element 1L and J be a join semilattice with bottom element 0J. The L-slice (σ,J) is the action of the locale on join semilattice satisfying certain properties. The concept of L-slices were modelled in tune with the modules in algebra. The benefit of studying L-slices is that we can approach the structure algebraically as well as topologically. This paper deals with the graph theoretic approach to L-slices. The idea of relating graphs with algebraic structures was started by the work of Beck in [3]. The algebraic properties of L-slices prompted us to consider the possibility of various graphs that could be associated with L-slices. The article introduces two different graphs on L-slices. The total graph Γ((T(σ,J)) is defined. We derive a characterisation for such graphs to be nonempty. The structural properties of Γ((T(σ,J)) is studied. The weak Zariski Topology on (σ,J) gives us the graph GT(ω∗). The conditions under which the graph is nonempty is examined. Also some of the structural properties of GT(ω∗) is obtained.