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The free monoid on a set A is usually denoted A ∗. The free semigroup on A is the subsemigroup of A ∗ containing all elements except the empty string. It is usually denoted A +. [1] [2] More generally, an abstract monoid (or semigroup) S is described as free if it is isomorphic to the free monoid (or semigroup) on some set. [3]
Every group is a monoid and every abelian group a commutative monoid. Any semigroup S may be turned into a monoid simply by adjoining an element e not in S and defining e • s = s = s • e for all s ∈ S. This conversion of any semigroup to the monoid is done by the free functor between the category of semigroups and the category of monoids. [3]
A Lyndon word over a totally ordered alphabet A is a word that is lexicographically less than all its rotations. [1] The Chen–Fox–Lyndon theorem states that every string may be formed in a unique way by concatenating a lexicographically non-increasing sequence of Lyndon words.
The Kleene star is defined for any monoid, not just strings. More precisely, let (M, ⋅) be a monoid, and S ⊆ M. Then S * is the smallest submonoid of M containing S; that is, S * contains the neutral element of M, the set S, and is such that if x,y ∈ S *, then x⋅y ∈ S *.
Let be an alphabet: the set of words over is a monoid, the free monoid on . The recognizable subsets of A ∗ {\displaystyle A^{*}} are precisely the regular languages . Indeed, such a language is recognized by the transition monoid of any automaton that recognizes the language.
A simpler example are the free monoids. The free monoid on a set X, is the monoid of all finite strings using X as alphabet, with operation concatenation of strings. The identity is the empty string. In essence, the free monoid is simply the set of all words, with no equivalence relations imposed.
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The monoid is then presented as the quotient of the free monoid (or the free semigroup) 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 a semi-Thue system). Every monoid may be presented by a semi-Thue ...