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Quasi-multiplicative maps on Baer semigroups

Published online by Cambridge University Press:  18 May 2009

C. S. Johnson Jr
C. S. Johnson Jr
Affiliation:
University of Massachusetts, Amherst, Massachusetts 01002
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A Baer semigroup is a semigroup 5 with 0 and 1 in which, for each x∈S, the left annihilator L(x) = {y∈S: yx = 0} of x is a principal left ideal generated by an idempotent and the right annihilator R(x) = {y∈S: xy = 0} of x is a principal right ideal generated by an idempotent. Baer semigroups are of interest because (see [5]) the left annihilators of the elements of a Baer semigroup S, ℒ(S) = {L(x): x∈S}, form a bounded lattice and (see [4]) every bounded lattice arises in this manner. In this note we look at a type of map φ on a Baer semigroup S which has the property that Sφ is a Baer semigroup. (The homomorphic image of a Baer semigroup need not be a Baer semigroup. For a case where it is, see [7].) When the Baer semigroup is specialized to a Boolean algebra, this type of map generalizes Halmos's notion of a quantifier.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 13 , Issue 2 , September 1972 , pp. 107 - 110
Copyright
Copyright © Glasgow Mathematical Journal Trust 1972

References

REFERENCES

1.Blyth, T. S. and Janowitz, M. F., On decreasing Baer semigroups, Bull. Soc. Roy. Sci. Liège 38 (1969), 414–423.Google Scholar
2.Foulis, D. J., Baer *-semigroups, Proc. Amer. Math. Soc. 11 (1960), 648–654.Google Scholar
3.Halmos, P. R., Algebraic logic I, monadic Boolean algebras, Compositio Math. 12 (1955), 217–249.Google Scholar
4.Janowitz, M. F., A semigroup approach to lattices, Canad. J. Math. 18 (1966), 1212–1223.CrossRefGoogle Scholar
5.Janowitz, M. F., Baer semigroups, Duke Math. J. 32 (1965), 85–95.CrossRefGoogle Scholar
6.Janowitz, M. F., Quantifiers and orthomodular lattices, Pacific J. Math. 13 (1963), 1241–1249.CrossRefGoogle Scholar
7.Thorne, B. J., A–P congruences on Baer semigroups, Pacific J. Math. 28 (1969), 681–698.CrossRefGoogle Scholar