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Department of Mathematics

Rings and Modules Seminar
~ Abstracts ~

R. Padmanabhan
Ranganathan(dot)Padmanabhan(at)umanitoba(dot)ca
© 2024, The Author

University of Manitoba

Tuesday, October 22, 2024

B.H. Neumann's Semigroup Laws
Abstract:

In response to my friendly challenge made in 1999 to the veteran group theorist Professor B. H. Neumann, he proposed some semigroup implications as potential counter-examples to my CS-Conjecture - shown to be equivalent to the more well-known as George Bergman's problem ([1], [2]: Let a semigroup law \( a=b \) imply a semigroup law \( u=v \) in groups. Does the same identity hold in cancellation semigroups? Here the``challenge'' was to prove these theorems while staying within the first order theory of semigroups (i.e. without using the unary inverse ``\(-1\)'' or the nullity ``\(e\)''. Here is an example of such a semigroup implication proposed by Neumann (Theorem 17, [4 ]). \[ x^{1+t} \ast y^2 \ast x^t = y \ast x^{2t+1} \ast y \mbox{ implies } x \ast y = y \ast x \] where \( t \) is a positive integer. Neumann proves that the above implication is valid for all groups and wondered whether this will be valid for the class of all cancellation semigroups. Here we add one ``new'' semigroup implication M valid for all groups such that the above commutativity theorem is valid for all cancellation M-semigroups. This works for all the other examples proposed by Professor Neumann in [4].

References:

  1. Bergman G.; Questions in Algebra, Preprint, Berkeley, U.D. 1986.
  2. Olga Macedonska and Piotr Slanina; On identities satisfied by cancellative semigroups and their groups of fractions, (Preprint)
  3. Olga Macedonska; Two questiosn on Semigroup Laws, Bull. Australian Math. Soc. 65 (2002), 431-437.
  4. B. H. Neumann; SSome Semigroup Laws in Groups, Canad. Math. Bull., 44 (1), 2001, pp. 93–96.
  5. R. Padmanabhan and Yang Zhang; Commutativity Theorems in Groups with Power-like Maps, J. Formalized Reasoning 12 (2019) No, 1, 1-10.


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