Simple (abstract algebra)

In mathematics, the term simple is used to describe an algebraic structure which in some sense cannot be divided by a smaller structure of the same type. Put another way, an algebraic structure is simple if the kernel of every homomorphism is either the whole structure or a single element. Some examples are:
The general pattern is that the structure admits no non-trivial congruence relations.
The term is used differently in semigroup theory. A semigroup is said to be "simple" if it has no nontrivial
ideals, or equivalently, if Green's relation "J" is
the universal relation. Not every congruence on a semigroup is associated with an ideal, so a simple semigroup may
have nontrivial congruences. A semigroup with no nontrivial congruences is called "congruence simple".