ORDER COMPACTIFICATIONS OF DISCRETE Semigroups

Given a partially ordered set (X,≤) one can construct the order compactification µX of X in the same fashion as Čech’s construction of the Stone-Čech compactification, using the order preserving functions from X into the unit interval [0, 1]. We consider a semigroup (S, ·) which has an ordering which the semigroup respects in the sense that x ≤ y implies that z · x ≤ z · y and x · z ≤ y · z for all x, y, z ∈ S. We show that the operation can be extended to µS making it into a right topological semigroup with S contained in the topological center such that both the left and right translations are order preserving. We then investigate the structure of µS for certain specific semigroups S