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Mathematics of Computation

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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
MathSciNet review: 628714
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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.

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

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Keywords: Finite semigroups, involutions, antiautomorphisms, automated theoremproving
Article copyright: © Copyright 1981 American Mathematical Society