Intersections of Maximal Ideals in Semigroups

For a completely regular frame L , the ring \mathcal RL of real-valued continuous functions on L is equipped with the uniform topology. The closed ideals of \mathcal RL in this topology are studied, and a new, merely algebraic characterization of these ideals is given. This result is used to describe the real ideals of \mathcal RL , and to characterize pseudocompact frames and Lindelöf frames. It is shown that a frame L is finite if and only if every ideal of \mathcal RL is closed. Finally, we prove that every closed ideal in \mathcal RL is an intersection of maximal ideals.

Maximal ideals in compact semigroups

It is well known that in a distributive lattice L, every filter is an intersection of prime filters, and dually. Moreover, L is relatively complemented if and only if every prime filter (ideal) is maximal. In 1947, A. Monteiro [3] posed the question of determining whether a distributive lattice, in which every filter is an intersection of maximal filters, and dually, is necessarily relatively complemented. This question also appears as Problem 39 in G. Gr/itzer [2; p. 156]. Monteiro states that J. Dieudonn6 and L. Nachbin have independently shown that neither the condition that every filter is an intersection of maximal filters, nor its dual, implies the other. We will solve this problem in the negative by constructing a non-Boolean distributive lattice L which satisfies the above conditions on the filters and ideals L. The referee was most helpful in simplifying the explanation of the construction of L. This lattice is necessarily infinite for if a finite distributive lattice is such that each of its filters is an intersection of maximal filters then it is a Boolean algebra. Indeed, this follows from the fact that if F, F1,..., F, are prime filters and F = Of="1 Fi then F=Fto for some io~ {1,..., n}.

In this paper, semisimple semigroups, duo semigroups, right (left) zero semigroups, right (left) simple semigroups, semilattice of left (right) simple semigroups, semilattice of left (right) groups and semilattice of groups are characterized in terms of soft intersection semigroups, soft intersection ideals of semigroups. Moreover, soft normal semigroups are defined and some characterizations of semigroups with soft normality are given.