Editing Free monoid (section)

{{Short description|Concept in mathematics}}
In [[abstract algebra]], the '''free monoid''' on a [[Set (mathematics)|set]] is the [[monoid]] whose elements are all the [[finite sequence]]s (or strings) of zero or more elements from that set, with [[string concatenation]] as the monoid operation and with the unique sequence of zero elements, often called the [[empty string]] and denoted by ε or λ, as the [[identity element]]. The free monoid on a set ''A'' is usually denoted ''A''<sup>&lowast;</sup>. The '''free semigroup''' on ''A'' is the sub[[semigroup]] of ''A''<sup>&lowast;</sup> containing all elements except the empty string. It is usually denoted ''A''<sup>+</sup>.<ref name="Lot23">{{harvtxt|Lothaire|1997|pp=2–3}}, [https://books.google.com/books?id=eATLTZzwW-sC&pg=PA2]</ref><ref name=PF2>{{harvtxt|Pytheas Fogg|2002|p=2}}</ref>

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.<ref name=Lot5>{{harvtxt|Lothaire|1997|p=5}}</ref>

As the name implies, free monoids and semigroups are those objects which satisfy the usual [[universal property]] defining [[free object]]s, in the respective [[category (mathematics)|categories]] of monoids and semigroups. It follows that every monoid (or semigroup) arises as a homomorphic image of a free monoid (or semigroup). The study of semigroups as images of free semigroups is called combinatorial semigroup theory.

Free monoids (and monoids in general) are [[associative]], by definition; that is, they are written without any parenthesis to show grouping or order of operation. The non-associative equivalent is the [[free magma]].