Abstract
For each monoid S that is an ideal extension of a nilpotent semigroup N by a group G we construct a group H such that S divides the direct product C x H for some cyclic aperiodic monoid C. This leads to an elementary proof and some refinements of certain join decomposition results by Almeida and Weil dealing with the pseudovariety of semigroups with central idempotents.
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References
J. Almeida, Finite Semigroups and Universal Algebra, World Scientific, Singapore, 1994.
J. Almeida, Some pseudovariety joins involving the pseudovariety of finite groups, Semigroup Forum, 37 (1988), 53–57.
J. Almeida and P. Weil, Reduced factorizations in free profinite groups and join decompositions of pseudovarieties, Internat. J. Algebra Comput., 4 (1994), 375–403.
H. Neumann, Varieties of Groups, Springer-Verlag, Berlin, Heidelberg, New York, 1967.
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Auinger, K. (2000). Semigroups with Central Idempotents. In: Birget, JC., Margolis, S., Meakin, J., Sapir, M. (eds) Algorithmic Problems in Groups and Semigroups. Trends in Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1388-8_2
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DOI: https://doi.org/10.1007/978-1-4612-1388-8_2
Publisher Name: Birkhäuser, Boston, MA
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