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An extension F of a field K is said to be algebraic if every element of F is algebraic over K (i.e., is the root of a nonzero polynomial with coefficients in K).

A set of class representatives is a subset of X which contains exactly one element from each equivalence class.

Let A be a unital C^*-algebra, then an element u in A is called co-isometry if uu^*=1.

If A is a unital Banach algebra where every nonzero element is invertible, then A is the algebra of complex numbers.

An addition chain for which there is a subset H of members such that each member of the chain uses the largest element of H which is less than the member.

The map which assigns every member of a set A to the same element id_A. It is identical to the identity function.

A relation R on a set S is irreflexive provided that no element is related to itself; in other words, xRx for no x in S.

An algebra in which the associator (x,x,x)=0. The subalgebra generated by one element is associative.

A group G is quasi-unipotent if every element of G of order p is unipotent for all primes p such that G has p-group rank >=3.

A p-element x of a group G is semisimple if E(C_G(x))!=1, where E(H) is the commuting product of all components of H and C_G(x) is the centralizer of G.