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Maximal cancellative subsemigroups and cancellative congruences


Author: Mohan S. Putcha
Journal: Proc. Amer. Math. Soc. 47 (1975), 49-52
DOI: https://doi.org/10.1090/S0002-9939-1975-0352308-1
MathSciNet review: 0352308
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Abstract | References | Additional Information

Abstract: A subsemigroup $ T$ of a commutative semigroup $ S$ is called a mild ideal if for any $ a \in S,aT \cap T \ne \phi $. It is shown here that any maximal cancellative subsemigroup $ T$ of a commutative, idempotent-free, archimedean semigroup $ S$ must be a mild ideal of $ S$. Maximal cancellative subsemigroups exist in abundance due to Zorn's lemma. It is also shown that if $ T$ is mild ideal of a commutative semigroup $ S$, then every cancellative congruence of $ T$ has a unique extension to a cancellative congruence of $ S$.


References [Enhancements On Off] (What's this?)

  • [1] M. S. Putcha, Positive quasi-orders on semigroups, Duke Math. J. 40(1973), 857-869. MR 0338232 (49:2998)
  • [2] T. Tamura, Commutative nonpotent archimedean semigroup with cancellation law. I, J. Gakugei Tokushima Univ. 8 (1957), 5-11. MR 20 #3224. MR 0096741 (20:3224)
  • [3] -, Construction of trees and commutative archimedean semigroups, Math. Nachr. 36 (1968), 257-287. MR 37 #6222. MR 0230662 (37:6222)
  • [4] -, $ \mathfrak{N}$-congruences of $ \mathfrak{N}$-semigroups, J. Algebra 27 (1973), 11-30. MR 0327956 (48:6298)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0352308-1
Keywords: Semigroups, commutative, archimedean, maximal cancellative congruence, mild ideal
Article copyright: © Copyright 1975 American Mathematical Society

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