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Completions of semilattices of cancellative semigroups

Published online by Cambridge University Press:  18 May 2009

W. D. Burgess
W. D. Burgess
Affiliation:
University of Ottawa, Ottawa, Canada K1N 6N5
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A semilattice of cancellative semigroups S is a p.o. semigroup with the order relation a ≤ b iff ab = a2. If S is a strong semilattice of cancellative semigroups (i.e., multiplication in S is given by structure maps ϕe,f (f ≤ e in E)), for each supremum-preserving completion Ē of the semilattice E there is a strong semilattice of cancellative semigroups T over Ē which is a supremum-preserving completion of S in ≤. Given Ē, T is constructed directly. In this paper it is shown that multiplication by an element of S distributes over suprema in ≤ if E has 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 if S is commutative and Ē is strongly distributive. Finally, it is shown that for semilattices of cancellative monoids a completion is completely determined, up to isomorphism over S, by completions of E.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 21 , Issue 1 , January 1980 , pp. 29 - 37
Copyright
Copyright © Glasgow Mathematical Journal Trust 1980

References

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