A set of functions from a set to itself, which is closed under composition, is an example of the transformation type of these objects.
One of these entities is referred to as orthodox if its idempotents form a sub-one of these, and if every element of one of these objects is idempotent, it is referred to as a band. By the Krohn-Rhodes theorem, every  nite one of these entities is a divisor of a wreath product of certain other algebraic objects. A set of functions from a set to itself, which is closed under composition, is an example of the transformation type of these objects. These entities are distinguished from groupoids by their possession of the   associative property. For ten points, identify this algebraic object consisting of a set endowed with an associative binary operation.
One of these entities is referred to as orthodox if its idempotents form a sub-one of these, and if every element of one of these objects is idempotent, it is referred to as a band.
semigroups