Birkhoff [3] introduced the concept of a variety of algebras, a class of algebras all of the same type, closed under the taking of homomorphic images, subalgebras of the same type and direct products, Birkhoff showed that such classes were defined by sets of identities (for example, within the class of all semigroups, the collection of all commutative semigroups is a variety defined by the identity xy = yx). Conversely, any class of algebras, all of the same type, all of which satisfy the same set of identities, is a variety. Thus began the study of universal algebra. Over the years, numerous classes of algebras were studied using the new tools of universal algebra: groups, rings, lattices and, importantly for us, semigroups.
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