Completions of semilattices of cancellative semigroups
W. D. Burgess · Glasgow Mathematical Journal · 1980
A semilattice of cancellative semigroupsSis a p.o. semigroup with the order relationa≤biffab=a2. IfSis a strong semilattice of cancellative semigroups (i.e., multiplication inSis given by structure maps ϕe,f(f≤einE)), for each supremum-preserving completion Ē of the semilatticeEthere is a strong semilattice of cancellative semigroupsTover Ē which is a supremum-preserving completion ofSin ≤. Given Ē,Tis constructed directly. In this paper it is shown that multiplication by an element ofSdistributes over suprema in ≤ ifEhas this property (called strong distributivity). Next it is shown that the completion construction also applies to a semilattice of cancellative semigroups which is not strong ifSis commutative and Ē is strongly distributive. Finally, it is shown that for semilattices of cancellative monoids a completion is completely determined, up to isomorphism overS, by completions ofE.