Abstract
This paper constructs from the homogeneous quotients of an arbitrary semigroupS a universal group (G(S), γ) onS. If S is left reversible and cancellative, thenG(S) coincides with the embedding group of quotients of S due to Ore. If S is an inverse semigroup, G(S) coincides with the maximum group homomorphic image of S due to Munn. In these cases, γ coincides with the embedding and canonical homomorphism respectively ofS intoG(S).
In general (G(S), γ) is equivalent to the universal group on S due to N. Bouleau. A universal group constructed from the set of Lambek ratios had earlier been exhibited by A.H. Clifford and G.B. Preston for cancellative semigroups satisfying the condition Z of Malcev. No previous construction has, however, emerged as a direct generalisation of both the work of Ore and Munn as does the present.
Elementary properties of homogeneous quotients are employed to illuminate Bouleau's counter-example on why certain Malcev conditions are insufficient to guarantee the embeddability of a semigroup in a group.
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Communicated by Gerard Lallement
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Osondu, K.E. The universal group of homogeneous quotients. Semigroup Forum 21, 143–152 (1980). https://doi.org/10.1007/BF02572545
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DOI: https://doi.org/10.1007/BF02572545