A characterization of Mal'cev's preiterative algebras

摘要

In universal algebras, propositional calculi, combinatorial switching circuits new operations are formed by composition. The process of composing operations is usually described by Menger algebras or by partial or graded algebras (clones). The inherent limitations are due to the preservation of arities. A.I. Mal'cev avoided this by introducing an algebra P@@@@ &equil; 0A,&zgr;,τ,Δ,* of type 1,1,1,2 on the set 0A of all finitary operations on A such that the subalgebras of P@@@@ (called preiterative algebras) agree with the sets of all polynomials of universal algebras on A. In this paper, we study the abstract structure of these concrete preiterative algebras; that is, we describe the preiterative algebras in terms of the properties of &zgr;, τ, Δ, and * only.