Abstract

Let $ L $ be a locale with top element $ 1_{L} $ and $ J $ be a join semilattice with bottom element $ 0_{J} $. The L-slice $ (\sigma,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 $\Gamma((T(\sigma,J))$ is defined. We derive a characterisation for such graphs to be nonempty. The structural properties of $\Gamma((T(\sigma,J))$ is studied. The weak Zariski Topology on $(\sigma,J)$ gives us the graph $G_{T} (\omega^{*})$. The conditions under which the graph is nonempty is examined. Also some of the structural properties of $G_{T} (\omega^{*})$ is obtained.

Keywords and Phrases

Locale, L-slices, total graph.

A.M.S. subject classification

06D22, 05C25.

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