Abstract
D. R. Brown and M. Friedberg have conjectured that each compact abelian semigroup can be embedded in a compact divisible semigroup. V. R. Hancock proved that each abelian algebraic semigroup can be embedded in a divisible abelian algebraic semigroup. In this paper we provide a partial solution to the conjecture of Brown and Friedberg by employing a topological version of Hancock's method as part of our construction. A theorem giving sufficient conditions for the Bohr compactification of weakly reductive semigroups to be injective is proved and used in the proof of our main result.
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References
Brown, D. R., and M. Friedberg,A, new notion of semicharacters, Trans. Am. Math. Soc., 141(1969), 387–401.
Hancock, V. R.,Commutative Schreier Extensions of Semigroups, Doctoral Dissertation; Tulane University (1960).
Hildebrant, J. A. and J. D. Lawson,On semigroup compactifications, (submitted for publication).
Lawson, J. D. and B. L. Madison,On congruences and cones, Math Zeit., 120(1971), 18–24.
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Hildebrant, J.A., Lawson, J.D. Embedding in compact uniquely divisible semigroups. Semigroup Forum 4, 295–300 (1972). https://doi.org/10.1007/BF02570801
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DOI: https://doi.org/10.1007/BF02570801