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$L$-slice; $L$-slice homomorphism; subslice; fixed set and ideals
Given a locale $L$ and a join semilattice $J$ with bottom element $0_{J}$, a new concept $(\sigma ,J)$ called $L$-slice is defined,where $\sigma $ is as an action of the locale $L$ on the join semilattice $J$. The $L$-slice $(\sigma ,J)$ adopts topological properties of the locale $L$ through the action $\sigma $. It is shown that for each $a\in L$, $\sigma _{a} $ is an interior operator on $(\sigma ,J)$.The collection $M=\lbrace \sigma _{a};a \in L\rbrace $ is a Priestly space and a subslice of $L$-$\operatorname{Hom}(J,J)$. If the locale $L$ is spatial we establish an isomorphism between the $L$-slices $(\sigma ,L) $ and $(\delta ,M) $. We have shown that the fixed set of $\sigma _{a}$, $a\in L $ is a subslice of $(\sigma ,J)$ and prove some equivalent properties.
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