operation ~, with ~x usually written as x−1. The axioms become: Associativity: . Identity element: ; formally: ∀x. e∗x=x=x∗e. Inverse element: formally: ∀x. x∗~x=e=~x∗x. To summarize, the usual definition has: a single binary operation (signature (2)) 1 equational law (associativity) 2 quantified laws (identity and inverse) while the universal algebra definition has: 3 operations: one binary, one unary, and one nullary (signature (2,1,0)) 3 equational laws (associativity, identity, and inverse) no quantified laws (except outermost universal quantifiers, which are allowed in varieties) A key point is that the extra operations do not add information, but follow uniquely from the usual definition of a group. Although the usual definition did not uniquely specify the identity element e, an easy exercise shows it is unique, as is each inverse element. The universal algebra point of view is well adapted to category theory. For example, when defining a group object in category theory, where the object in question may not be a set, one must use equational laws (which make sense in general categories), rather than quantified laws (which refer to individual elements). Further, the inverse and identity are specified as morphisms in the category. For example, in a topological group, the inverse must not only exist element-wise, but must give a continuous mapping (a morphism). Some authors also require the identity map to be a closed inclusion (a cofibration). Other examples Most algebraic structures are examples of universal algebras. Rings, semigroups, quasigroups, groupoids, magmas, loops, and others. Vector spaces over a fixed field and modules over a fixed ring are universal algebras. These have a binary addition and a family of unary scalar multiplication operators, one for each element of the field or ring. Examples of relational algebras include semilattices, lattices, and Boolean algebras. Basic constructions We assume that the type, , has been fixed. Then there are three basic constructions in universal algebra: homomorphic image, subalgebra, and product. A homomorphism between two algebras A and B is a function h: A → B from the set A to the set B such that, for every operation fA of A and corresponding fB of B (of arity, say, n), h(fA(x1,...,xn)) = fB(h(x1),...,h(xn)). (Sometimes the subscripts on f are taken off when it is clear from context which algebra the function is from.) For example, if e is a constant (nullary operation), then h(eA) = eB. If ~ is a unary operation, then