Semigroups, antiautomorphisms, and involutions: a computer solution to an open problem. I

Authors:
S. K. Winker, L. Wos and E. L. Lusk

Journal:
Math. Comp. **37** (1981), 533-545

MSC:
Primary 20M15; Secondary 03B35, 20-04, 68G20

DOI:
https://doi.org/10.1090/S0025-5718-1981-0628714-5

MathSciNet review:
628714

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Abstract | References | Similar Articles | Additional Information

Abstract: An *antiautomorphism H* of a semigroup *S* is a 1-1 mapping of *S* onto itself such that $H(xy) = H(y)H(x)$ for all *x, y* in *S*. An antiautomorphism *H* is an *involution* if ${H^2}(x) = x$ for all *x* in *S*. In this paper the following question is answered: Does there exist a finite semigroup with antiautomorphism but no involution? This question, suggested by I. Kaplansky, was answered in the affirmative with the aid of an automated theorem-proving program. More precisely, there are exactly four such semigroups of order seven and none of smaller order. The program was a completely general one, and did not calculate the solution directly, but rather rendered invaluable assistance to the mathematicians investigating the question by helping to generate and examine various models. A detailed discussion of the approach is presented, with the intention of demonstrating the usefulness of a theorem prover in carrying out certain types of mathematical research.

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Additional Information

Keywords:
Finite semigroups,
involutions,
antiautomorphisms,
automated theoremproving

Article copyright:
© Copyright 1981
American Mathematical Society