Zariski topology on L-slices

Abstract

The action of a locale L on a join semilattice J gives us the newly introduced notion of L-slices (σ, J).We have tried to extend the idea of Zariski topology on modules to L-slices. Given a locale L and a L-slice (σ, J), for m ∈ (σ, J)and r ∈ L,we have constructed (σ, J) ideals r ↳ m = n ∈ σ, J σ r, n ≤ m}. Their properties and characteristics are studied. Similarly, for a given L-slice (σ, J) and n, m ∈ σ, J, we examine the properties of L-ideals [r ↳ n] = r ∈ L σ r, n ≤ m}. We introduce the notion of L-prime elements on ( σ, J ) and their properties are discussed. The collection of L-prime elements is defined as Spec((σ, J) and we examine the possibility of existence of Zariski topology on it.