Algebras with minimal spectrum

Abstract

The spectrum of a class of algebras, ~ , is the set of cardinalities of the finite members of ~K; Spec (~K) denotes the spectrum of ~s If ~K is a variety (equational class), then Spec (~) is a multiplicative monoid of positive integers. Le t 92 be a finite non-trivial algebra and let ~s be the variety generated by 92; we say that 92 has minimal spectrum if Spec (~s = {1921 ~ [ n 0}; the aim of this paper is to characterize all algebras with minimal spectrum. Closely connected with this problem is the concept of the fine spectrum of ~ ; this was introduced and studied by Walter Taylor in [16]. The fine spectrum of ~K is the function fx such that fx(n) is the number of isomorphism classes of algebras in ~ of cardinality n. We will abbreviate fxr by f~; if 92 is finite, then fga(1) = 1 and f~(n) is finite. Note that if 92 has minimal spectrum, then fa(n)= 0 if n is not a power of [92 I. In [16] Taylor showed that there are exactly 9 2-e lement algebras with minimal spectrum (of course, up to polynomial equivalence). Moreover , he showed that for 7 of them, fga(2 "~) = 1 while for the other 2, fa(2 m) = 2 for m > 0. This property carries over to every algebra with minimal spectrum. Examples of the first kind are ~2, the 2-element boolean algebra, and @2, the 2-element group. An example of the second kind is ~2"=({0, 1};+, 1), the 2-element group in which the non-identity element is the value of a nuUary operation. The two isomorphism classes of cardinality 2 ~ are the one in which the added nullary operat ion is not the identity and the one in which it is the identity. For instance, if we factor ((~2") 2 by the congruence induced by the diagonal subgroup, then in the quotient the nullary is the identity. All 9 algebras are closely related to either ~2 or (~2. In particular, each example generates a congruence permutable variety. This property carries over to every algebra with minimal spectrum.