Two Finitely Generated Varieties Having No Infinite Simple Members

Using a method of R. McKenzie, we construct a finitely generated semisimple variety of infinite type, and a finitely generated nonsemisimple variety of finite type, both having arbitrarily large finite but no infinite simple members. This amplifies M. Valeriote’s negative solution to Problem 11 from [1]. R. McKenzie [2] has constructed a finitely generated variety having arbitrarily large finite, but no infinite, subdirectly irreducible members; and a second finitely generated variety, this time of finite type, having a unique countably infinite subdirectly irreducible member but no uncountable subdirectly irreducible members. M. Valeriote [4] has subsequently shown that McKenzie’s second example can be modified to make the variety semisimple, i.e., so that every subdirectly irreducible member is simple. (In Valeriote’s example there are three, rather than one, countably infinite simple members.) In this note we (1) show that McKenzie’s first example can also be modified to make the variety semisimple; (2) modify McKenzie’s second example in a different way to obtain a finitely generated (nonsemisimple) variety of finite type having arbitrarily large finite, but no infinite, simple members. 1. McKenzie’s method The following is a summary of a method invented by McKenzie in [2, §6], as it is described in [5]. An M-algebra is any algebra A whose type includes ∧ (binary) and 0 (nullary) but no other nullary operation symbols, and which satisfies (1) The reduct 〈A,∧〉 is a height-1 meet semilattice with least element 0; (2) 0 is an absorbing element for each fundamental operation F of A; that is, if F is n-ary then 0 ∈ {a1, .., an} ⊆ A implies F (a1, ..., an) = 0. Suppose A is an M-algebra and U = A \ {0}. Consider an arbitrary subpower B ≤ A (I 6= ∅) with B(U) := B ∩ U I 6= ∅. On B(U) define the (reflexive) binary relation by f g if and only if F(h1, . . . , hn) = g for some fundamental operation symbol F and some hi ∈ B(U) such that f ∈ {h1, ..., hn}. Also let≫ be the transitive closure of . Now choose any p ∈ B(U) and let Bp = {f ∈ B(U) : f≫ p}. B(p) denotes the M-algebra, of the same type as A and B, whose universe is the disjoint union of Bp and {0}, and whose fundamental operations are defined as 1991 Mathematics Subject Classification. 08B26.