homomorphism, subalgebra, and product. One direction of the equivalence mentioned above, namely that a class of algebras satisfying some set of identities must be closed under the HSP operations, follows immediately from the definitions. Proving the converse—classes of algebras closed under the HSP operations must be equational—is more difficult. Using the easy direction of Birkhoff's theorem, we can for example verify the claim made above, that the field axioms are not expressible by any possible set of identities: the product of fields is not a field, so fields do not form a variety. Subvarieties A subvariety of a variety of algebras V is a subclass of V that has the same signature as V and is itself a variety, i.e., is defined by a set of identities. 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. Similarly, the class of semigroups which are groups is not a subvariety of the variety of semigroups. The class of monoids which are groups contains and does not contain its subalgebra (more precisely, submonoid) . However, 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. The finitely generated abelian groups do not form a subvariety, since by Birkhoff's theorem they don't form a variety, as an arbitrary product of finitely generated abelian groups is not finitely generated. Viewing a variety V and its homomorphisms as a category, a subvariety U of V 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. Free objects Suppose V is a non-trivial variety of algebras, i.e. V contains algebras with more than one element. One can show that for every set S, the variety V contains a free algebra FS on S. This means that there is an injective set map i : S → FS which satisfies the following universal property: given any algebra A in V and any map k : S → A, there exists a unique V-homomorphism f : FS → A such that . This generalizes the notions of free group, free