Subject Classiication (ams) 06f05 20m14

Two classes of linear involutory semigroups We construct two classes of involutory semigroups, which together with two already known generating rules yield n linear involutory semigroups of order n, where 2n = 2 n+2 ? 1 2 (n + 1)(n + 6) ; 2n+1 = 3 2 n+1 ? 1 2 (n 2 + 9n + 10) (1) for n 1. By the way, we extend the rule for the construction of linear archimedian involutory semigroups in L. Berg and W. Peters 4]. 1. Preliminaries According to 3], every commutative semigroup can be embedded into an involutory semi-group, where the operation is considered as addition and the neutral element is called the zero element. The nite linear involutory semigroups S can be characterized in the following two ways. Let n be the order of S, and we denote the elements of S by 1; 2; : : : ; n. For a xed i, let i + j = x j in the semigroup with i; j 2 f1; 2; : : : ; ng, and let d j = x j ? x j?1 in the natural sense with x 0 = 0. Note that x j also depends on i.