Abstract
For several large classes of semigroups we provide a description of all semigroups which generate varieties with uncountably many subvarieties. These include the class of all Rees quotients of free monoids, the class of finite orthodox monoids, the class of monoids of index greater than two, and the class of finite inherently not finitely based semigroups. The first example of a finite, finitely based semigroup generating a variety with uncountably many subvarieties is presented and a number of related results are obtained. All varieties found with uncountably many subvarieties contain uncountable chains of subvarieties with the same ordering as the real numbers.
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