Examples
The class of all semigroups forms a variety of algebras of signature (2). A sufficient defining equation is the associative law:
It satisfies the HSP closure requirement, since any homomorphic image, any subset closed under multiplication and any direct product of semigroups is also a semigroup.
The class of groups forms a class of algebras of signature (2,1,0), the three operations being respectively multiplication, inversion and identity. Any subset of a group closed under multiplication, under inversion and under identity (i.e. containing the identity) forms a subgroup. Likewise, the collection of groups is closed under homomorphic image and under direct product. Applying Birkhoff's theorem, this is sufficient to tell us that the groups form a variety, and so it should be defined by a collection of identities. In fact, the familiar axioms of associativity, inverse and identity form one suitable set of identities:
A subvariety of a variety V is a subclass of V that has the same signature as V and is itself a variety. Notice that although every group becomes a semigroup when the identity as a constant is omitted (and/or the inverse operation is omitted), the class of groups does not form a subvariety of the variety of semigroups because the signatures are different. On the other hand the class of abelian groups is a subvariety of the variety of groups because it consists of those groups satisfying with no change of signature. Viewing a variety V and its homomorphisms as a category, a subclass U of V that is itself a variety is a subvariety of V implies that U is a full subcategory of V, meaning that for any objects a, b in U, the homomorphisms from a to b in U are exactly those from a to b in V. On the other hand there is a sense in which Boolean algebras and Boolean rings can be viewed as subvarieties of each other even though they have different signatures, because of the translation between them allowing every Boolean algebra to be understood as a Boolean ring and conversely; in this sort of situation the homomorphisms between corresponding structures are the same.
Read more about this topic: Variety (universal Algebra)
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