Narrow Results

We will show that an antiautomprphism satisfies a Kishimoto type theorem. As an application, we can show Glimm type theorem for transpositions.

An antiautomorphismH 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 H²(x) = x for all x in S. In this paper the following question is answered: Does there exist a finite semigroup with an 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.