The Structure of Locally Finite Varieties with Polynomially Many Models

In this paper, variety means a class of similar algebras—i.e., models of one firstorder language without relation symbols—which is defined by the satisfaction of some fixed sets of equations. A variety is locally finite if all of its finitely generated members are finite. Following J. Berman and P. Idziak [2], the G-spectrum, or generative complexity, of a locally finite variety C, is the function GC defined on positive integers so that GC(k) is the number of non-isomorphic (at most) k-generated members of C. The G-spectrum of C is a non-decreasing function from and to positive integers. The interest in this function as an invariant of a locally finite variety is tied to the fact, exhaustively demonstrated by J. Berman and P. Idziak [2], that the rate of growth of GV is closely related to numerous structural and algebraic properties of the members of V . Among other results, [2] contains a characterization of the finitely generated varieties V which omit type 1 and possess a G-spectrum GV(k) bounded by a singly exponential function 2 for some polynomial p(k). In this paper, we characterize the locally finite varieties which possess a Gspectrum bounded by a polynomial function. The literature contains two other papers dealing with these varieties. M. Bilski [4] characterizes the finitely generated varieties of semigroups with polynomially bounded G-spectrum. P. Idziak and R. McKenzie [6] prove that if a locally finite variety V omits type 1, then GV(k) ≤ k for some C > 0 and for all integers k > 1 iff V is polynomially equivalent to the variety of all unitary left R-modules for some finite ring R of finite representation type. This paper is a continuation of P. Idziak and R. McKenzie [6]. Here we extend their result to arbitrary locally finite varieties; and we prove that such a variety has a polynomially bounded G-spectrum if and only if it decomposes into a varietal product A ⊗ S1 ⊗ · · · ⊗ Sr, where A is polynomially equivalent to the variety of unitary left R-modules over a finite ring R of finite representation type, and where for 1 ≤ i ≤ r, Si is equivalent to a matrix power of the variety of Hi-sets with some constants, with Hi a finite group. The arguments in this paper show that if a locally finite variety V fails to decompose as V = A ⊗ S with A affine and S strongly Abelian, then GV(k) ≥ 2 C , for some positive C and for all k > 1. We do not know if this statement remains true when 2 C is replaced by 2 √ . Our arguments also demonstrate that if S is