Narrow Results

In the present paper, we classify varieties of algebraic systems of the type $((n),(m))$, for natural numbers $n$ and $m$, which are closed under particular derived algebraic systems. If we replace in an algebraic system the $n$-ary operation by an $n$-ary term operation and the $m$-ary relation by the $m$-ary relation generated by an $m$-ary formula, we obtain a new algebraic system of the same type, which we call derived algebraic system. We shall restrict the replacement to so-called “linear” terms and atomic “linear” formulas, respectively.

The set of linear terms, i.e. terms in which each variable occurs at most once, does not form a subsemigroup of the so-called diagonal semigroup. We consider the reduct of the diagonal semigroup to the linear terms, which is not a partial semigroup. We extend the set of linear terms by an expression “$\infty$”, that is formally a linear term, obtaining a semigroup. The algebraic structure of this semigroup will be studied in this paper. We characterize the Green’s relations and the regular elements as well as the idempotent elements. Moreover, we discuss the ideal structure.