The axiomatization of algebras of functions or partial functions under various operations has been studied by several authors. Menger [2] deals with the problem of axiomatizing the algebraic properties of 1-ary functions from the reals to the reals. He considers algebras with one of three binary operations corresponding to addition, multiplication or composition. This study is furthered by Schweizer and Sklar in [3]–[6]. In addition to the operations mentioned above they also introduce a partial ordering which corresponds to restriction of functions. In the second of these papers they give a set of axioms such that any system satisfying these axioms is order isomorphic to a concrete system of partial functions under composition and ordered by restriction. In [7] Schweizer and Sklar extend their work to the algebra of multiplace vector-valued functions. In [8] Whitlock studies abstract multiplaced function systems given by super-associative laws and shows that these systems are isomorphically embeddable in a concrete system of multiplaced functions where the operation on the functions is substitution of an m-ary function in an n-ary function.

In this paper we consider the set of functions from Uα, the set of α-sequences, to U where a is an infinite ordinal. As opposed to the composition operations studied in the above works, we consider the composition operations *κ, for κ < α, in a narrow sense—the substitution of one function in the κth place of another. The algebras studied have the form ‹A, *κ, Vκ› κ<α where Vκ are the projection (selector) functions on the κth place. In particular, the polynomials over an algebra form such an algebra called a polynomial substitution algebra. In §6 we show that a first-order axiom system and a condition of local finiteness, given by Pinter in a talk at Berkeley in 1972 to characterize term substitution abgebras, also characterize these polynomial substitution algebras.