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

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.