Abstract. If $A$ is a finite group or a finite lattice then the lattice of subvarieties of the variety generated by $A$ is finite (see Neumann [9] and Grätzer [3]). However, this is not the case for more general algebras (Dziobiah [2]) or for semigroups (Trakhtman [14]). It is an immediate consequence of the work of Kadourek [4] that there exist finite inverse semigroups generating varieties with an infinite number of subvarieties. Here we show that there exists an eight element inverse semigroup generating a variety with infinitely many subvarieties and which is the smallest such inverse semigroup. We also identify the smallest combinatorial inverse semigroups (two of size fourteen) with the same property.

AMS Subject Classification (1991): 20M07

Received August 16, 1991, and in revised from November 5, 1992. (Registered under 5530/2009.)