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Department of Mathematics

Rings and Modules Seminar
~ Abstracts ~

R. Padmanabhan
ranganathan(dot)padmanabhan(at)umanitoba(dot)ca
© 2023, The Author

University of Manitoba

Tuesday, November 21, 2023

Implications among laws in semigroups
Abstract:

In this talk, we address the following question:

Let a semigroup law \( a = b \) imply a semigroup law \( u = v \) in groups.
Does the same implication hold in semigroups with cancellation?
This problem was first raised around the middle of 1980's in George Grätzer's University of Manitoba Seminar on Universal Algebra and Lattice Theory. Several illustrative examples where the answer to the above question was positive were given.
Independently, in 1986 G.M. Bergman [1] raised the following related question in the Berkeley Algebra Seminar.
Let a semigroup \(S\) generating \(G\) satisfy a law.
Must \( G\) satisfy the same law?
See [2] for the inter-relation between these two problems and also some interesting non-trivial examples. It is generally felt among group theorists that the answer to the first question should be negative but no counter-examples are known. In 2001, B. H. Neumann gave several examples of such implications in group theory suggesting that some of these may provide the required counter examples. However, the problem still remains open.

This research was done in collaboration with W. McCune, G. I. Moghaddam and Yang Zhang.

References:

  1. G. M. Bergman, Question in Algebra, Preprint, UC Berkeley, (1986).
  2. O. Macedonska, Two questions on semigroup laws, Bull. Austral, Math.Soc, 65 (2002), 431-437.
  3. B. H. Neumann, Some semigroup laws in groups, Canad. Math. Bull. 44 (2001), 93-96.
  4. G. I. Moghaddam, R. Padmanabhan, and Yang Zhang, Automated reasoning with power maps, Journal of Automated Reasoning 64 (2020) 689-697.

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