Abstract
In this paper, we introduce and study $ \mathcal{H}_Y $-s.m.c. and strong $ \mathcal{H}_Y $-s.m.c. sets and give some connections between them and lattice ideals of $ \mathcal{H}_Y $. Also, we introduce an ideal $ R_S $, for each subset set $ S $ of a ring $ R $. We prove a ring $ R $ is a Gelfand ring if and only if $ R_S $ is an intersection of maximal ideals, for every s.m.c. set $ S $ of $ R $.
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