Distributive Semilattices
U. Maddana Swamy · Institutional Repositories DataBase (IRDB) · 1979
Distributive semilattices were first introduced by Gratzer and Schmidt (see [5], [6], [7] and [15]}.In the present paper, we introduce the concept of an ideal in a (meet} semilattice as a directed (above} initial segment which is motivated towards the result that a subset P of a lattice L is a prime ideal if and only if its set theoretic complement is a prime filter.Previously Frink [4] and Venkatanarasimhan [19], [20] have introduced different concepts of an ideal in a semilattice (poset} which are easily seen to be different from the present one.Also, our definition coincides with the usual ideal concept in lattices and extends almost all known results on ideals of a distributive lattice to the case of a distributive semilattice.The proofs in most cases are not routine, though not difficult, since the ideals of a semilattice don't form a lattice in general under set inclusion; in fact, the ideals of a semilattice S form a lattice under set inclusion if and only if S is a lattice under the partial order induced by the semilattice operation on S.A major part of this paper is a base for the investigations undertaken by the author [9]-[11] on the representation of distributive semilattices