Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In algebra, a presentation of a monoid (or semigroup) is a description of a monoid (or semigroup) in terms of a set of generators and a set of relations on the free monoid * (or free semigroup +) generated by . The monoid is then presented as the quotient of the free monoid by these relations. This is an analogue of a group presentation in group theory. As a mathematical structure, a monoid presentation is identical to a string rewriting system (also known as semi-Thue system). Every monoid may be presented by a semi-Thue system (possibly over an infinite alphabet). The relations are given as a (finite) binary relation R on *. To form the quotient monoid, these relations are extended to monoid congruences as follows. First one takes the symmetric closure R R–1 of R. This is then extended to a symmetric relation E * x * by defining x ~E y if and only if x = sut and y = svt for some strings u, v, s, t * with (u,v) R R–1. Finally, one takes the reflexive and transitive closure of E, which is then a monoid congruence.

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