$\mathcal I$-ideals generated by a set in {\bf IS}-algebras

Bull. Korean Math. Soc. 1998 Vol. 35, No. 4, 615-622

Young Bae Jun, Eun Hwan Roh, and Xiao Long Xin Gyeongsang National University, Gyeongsang National University, Gyeongsang National University

Abstract : We consider a generalization of [1, Theorem 2.5]. We give a description of the element of $\langle A\cup B\rangle_{\Cal I}^l$ (resp. $\langle A\cup B\rangle_{\Cal I}^r)$, where $A$ and $B$ are left (resp. right) $\Cal I$-ideals of an {\bf IS}-algebra $X$. For a nonempty left (resp. right) stable subset $A$ of an {\bf IS}-algebra, we obtain a condition for $\langle A\rangle_{\Cal I}^l$ (resp. $\langle A\rangle_{\Cal I}^r$) to be closed. We give a characterization of a closed $\Cal I$-ideal in an {\bf IS}-algebra, and show that, in a finite {\bf IS}-algebra, every $\Cal I$-ideal is closed.

Keywords : {\bf IS}-algebra, stable set, (closed) $\Cal I$-ideal, $\Cal I$-ideal generated by a set