Continuity of the operation in a semilattice
Joseph T. Borrego · Colloquium Mathematicum · 1970
By a semilattice we shall mean, a set X together with an associative, commutative, idempotent operation A defined on X.In a natural way A induces a partial order on X, i.e., x < y iitx Ay = x.A. topological semilattice is a semilattice X, where X is a Hausdorff space and A is continuous.Topological semilattices are similar to topo logical lattices [6] and have attracted the attention of various authors, e.g., [1] and [2].Am example of a lattice in which A is continuous but not V is given in [1].In [1] and [7] various continuity properties of < in a topological semilattice were obtained.(Continuous relations have been studied in [4], [5], and [7].)In [3] sufficient conditions on the continuity of < to insure the continuity of A in a compact Hausdorff space were given.The purpose of this note is to extend the results of [3] by giving necessary and sufficient conditions on the continuity of < to insure the continuity of A in a compact Hausdorff space.The previously mentioned example in [1] shows that these conditions cannot be selfdual.However, it is clear that the results given here may be extended to the lattice case by using the conditions given here and the dual of these conditions.