Address:
Corresponding author: Sabna K.S., Centre for Research in Mathematical Science, St. Joseph’s College (Autonomous), Irinjalakuda, Kerala, India, and Assistant Professor, K.K.T.M. Government College, Pullut, Kerala, India
Mangalambal N.R., Associate Professor, Centre for Research in Mathematical Sciences, St. Joseph’s College (Autonomous), Irinjalakuda, Kerala, India
E-mail:
sabnaks7@gmail.com
thottuvai@gmail.com
Abstract: 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.
AMSclassification: primary 06D22; secondary 06A12, 03G10.
Keywords: L-slice, L-slice homomorphism, subslice, fixed set and ideals.
DOI: 10.5817/AM2019-1-43