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

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- by S. K. Winker, L. Wos and E. L. Lusk PDF
- Math. Comp.
**37**(1981), 533-545 Request permission

## 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

- © Copyright 1981 American Mathematical Society
- 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