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Given a subset B of a set A, the injection f:B->A defined by f(b)=b for all b in B is called the inclusion map.

If sets E and F are independent, then so are E and F^', where F^' is the complement of F (i.e., the set of all possible outcomes not contained in F). Let union denote "or" ...

A rational homomorphism phi:G->G^' defined over a field is called an isogeny when dimG=dimG^'. Two groups G and G^' are then called isogenous if there exists a third group ...

Every finite-dimensional Lie algebra of characteristic p!=0 has a faithful finite-dimensional representation.

In a lattice, any two elements a and b have a least upper bound. This least upper bound is often called the join of a and b, and is denoted by a v b. One can also speak of ...

A primitive subgroup of the symmetric group S_n is equal to either the alternating group A_n or S_n whenever it contains at least one permutation which is a q-cycle for some ...

A lattice embedding is a one-to-one lattice homomorphism.

A simple root of a Lie algebra is a positive root that is not the sum of two positive roots.

The infinitesimal algebraic object associated with a Lie groupoid. A Lie algebroid over a manifold B is a vector bundle A over B with a Lie algebra structure [,] (Lie ...

The commutation operation [a,b]=ab-ba corresponding to the Lie product.