he theory of semigroup varieties is one of the most important parts of the theory of semigroups. Many important results in this theory have been obtained in the last 15-20 years. This theory is well represented in the recent books on the theory of semigroups by Almeida and Pin.;Two of the most important classes of semigroup varieties are the class of finitely based varieties (that is varieties given by finitely many identities) and the class of finitely generated varieties (that is varieties generated by finite semigroups). There are two main questions concerning these classes: (1) When does a finite semigroup generate a finitely based variety? (2) When is a finitely based variety finitely generated? Both of these questions go back to the original work by Oates, Powell, Tarski and others on varieties of groups and universal algebras in general. In this thesis, we study both of these questions.;We give an algorithmic description of varieties of commutative semigroups generated by finite semigroups. As an application of our method, we describe all axiomatic and basis ladders of commutative semigroup varieties, answering a question by Jonsson, McNulty and Quackenbush. We also prove that the variety of idempotent semigroups is a minimal example of an inherently non-finitely generated variety of semigroups. This is the first example of this kind. We prove several results about the finite basis property of finite semigroups. With every finite language W we associate a finite monoid S(W) which is the syntactic monoid of the completion of W under taking subwords. We prove that the set of finite finitely based (infinitely based) monoids of the form S(W) is not closed either under direct products or under taking homomorphisms, or under taking subsemigroups. We also construct the first example of an infinite chain of finitely generated semigroup varieties where finitely based and infinitely based varieties alternate. Finally we give a complete description of all words w in a two-letter alphabet such that
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