Finite semigroups as categories, ordered semigroups or compact semigroups
نویسنده
چکیده
The results presented in this section are a good illustration of the following quotation of Marshall Stone [34]: ’A cardinal principle of modern mathematical research may be stated as a maxim: “One must always topologize” ’. Varieties of finite semigroups are a good example where Stone’s principle was applied successfully. Recall that a variety of semigroups is a class of semigroups closed under taking subsemigroups, quotients and direct products. A variety of finite semigroups, or pseudovariety, is a class of finite semigroups closed under taking subsemigroups, quotients and finite direct products. It is a well known theorem of Birkhoff that varieties can be defined by identities. Formally, an identity is an equality of the form u = v between two elements of the free semigroup Σ on a countable number of generators Σ. A semigroup S satisfies an identity u = v if and only if uφ = vφ for every morphism φ : Σ → S. For instance, the identity xy = yx defines the variety of commutative semigroups and x = x defines the variety of bands. Birkhoff’s theorem can be summarized by saying that each variety is an equational class.
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