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A Lie algebra is a vector space g with a Lie bracket [X,Y], satisfying the Jacobi identity. Hence any element X gives a linear transformation given by ad(X)(Y)=[X,Y], (1) ...

A finite division algebra is a field.

A generalization of a Heyting algebra which replaces Boolean algebra in "intuitionistic" logic.

An operator A^~ such that A^~^2=A^~ or an element of an algebra x such that x^2=x.

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

A unar is an algebra A=(A,f), where f is a single unary operation.

A set considered together with the sigma-algebra on the set.

A set equipped with a sigma-algebra of subsets.

Every Lie algebra L is isomorphic to a subalgebra of some Lie algebra A^-, where the associative algebra A may be taken to be the linear operators over a vector space V.

A graded algebra over the integers Z. Cohomology of a space is a graded ring.