  Let X be a finite set and let k be a commutative ring. We consider the
k-algebra of the monoid of all relations on X, modulo the ideal generated by
the relations factorizing through a set of cardinality strictly smaller than
Card(X), called inessential relations. This quotient is called the essential
algebra associated to X. We then define a suitable nilpotent ideal of the
essential algebra and describe completely the structure of the corresponding
quotient, a product of matrix algebras over suitable group algebras. In
particular, we obtain a description of the Jacobson radical and of all the
simple modules for the essential algebra.
