# Free monoid

In abstract algebra, the **free monoid** on a set is the monoid whose elements are all the finite sequences (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*^{∗}. 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]}

As the name implies, free monoids and semigroups are those objects which satisfy the usual universal property defining free objects, in the respective 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.

The monoid (**N _{0}**,+) of natural numbers (including zero) under addition is a free monoid on a singleton free generator, in this case the natural number 1. According to the formal definition, this monoid consists of all sequences like "1", "1+1", "1+1+1", "1+1+1+1", and so on, including the empty sequence. Mapping each such sequence to its evaluation result

^{[4]}and the empty sequence to zero establishes an isomorphism from the set of such sequences to

**N**. This isomorphism is compatible with "+", that is, for any two sequences

_{0}*s*and

*t*, if

*s*is mapped (i.e. evaluated) to a number

*m*and

*t*to

*n*, then their concatenation

*s*+

*t*is mapped to the sum

*m*+

*n*.

In formal language theory, usually a finite set of "symbols" A (sometimes called the alphabet) is considered. A finite sequence of symbols is called a "word over *A*", and the free monoid *A*^{∗} is called the "Kleene star of *A*".
Thus, the abstract study of formal languages can be thought of as the study of subsets of finitely generated free monoids.

There are deep connections between the theory of semigroups and that of automata. For example, every formal language has a syntactic monoid that recognizes that language. For the case of a regular language, that monoid is isomorphic to the transition monoid associated to the semiautomaton of some deterministic finite automaton that recognizes that language. The regular languages over an alphabet A are the closure of the finite subsets of A*, the free monoid over A, under union, product, and generation of submonoid.^{[6]}

For the case of concurrent computation, that is, systems with locks, mutexes or thread joins, the computation can be described with history monoids and trace monoids. Roughly speaking, elements of the monoid can commute, (e.g. different threads can execute in any order), but only up to a lock or mutex, which prevent further commutation (e.g. serialize thread access to some object).

We define a pair of words in *A*^{∗} of the form *uv* and *vu* as **conjugate**: the conjugates of a word are thus its circular shifts.^{[7]} Two words are conjugate in this sense if they are conjugate in the sense of group theory as elements of the free group generated by *A*.^{[8]}

A free monoid is **equidivisible**: if the equation *mn* = *pq* holds, then there exists an *s* such that either *m* = *ps*, *sn* = *q* (example see image) or *ms* = *p*, *n* = *sq*.^{[9]} This result is also known as Levi's lemma.^{[10]}

The members of a set *A* are called the **free generators** for *A*^{∗} and *A*^{+}. The superscript * is then commonly understood to be the Kleene star. More generally, if *S* is an abstract free monoid (semigroup), then a set of elements which maps onto the set of single-letter words under an isomorphism to a semigroup *A ^{+}* (monoid

*A*

^{∗}) is called a

*set of free generators*for

*S*.

Each free semigroup (or monoid) *S* has exactly one set of free generators, the cardinality of which is called the *rank* of *S*.

Two free monoids or semigroups are isomorphic if and only if they have the same rank. In fact, *every* set of generators for a free semigroup or monoid *S* contains the free generators (see definition of generators in Monoid) since a free generator has word length 1 and hence can only be generated by itself. It follows that a free semigroup or monoid is finitely generated if and only if it has finite rank.

A set of free generators for a free monoid *P* is referred to as a **basis** for **P**: a set of words *C* is a **code** if *C** is a free monoid and *C* is a basis.^{[3]} A set *X* of words in *A*^{∗} is a **prefix**, or has the **prefix property**, if it does not contain a proper (string) prefix of any of its elements. Every prefix in *A*^{+} is a code, indeed a prefix code.^{[3]}^{[13]}

A submonoid *N* of *A*^{∗} is **right unitary** if *x*, *xy* in *N* implies *y* in *N*. A submonoid is generated by a prefix if and only if it is right unitary.^{[14]}

A factorization of a free monoid is a sequence of subsets of words with the property that every word in the free monoid can be written as a concatenation of elements drawn from the subsets. The Chen–Fox–Lyndon theorem states that the Lyndon words furnish a factorization. More generally, Hall words provide a factorization; the Lyndon words are a special case of the Hall words.

The intersection of free submonoids of a free monoid *A*^{∗} is again free.^{[15]}^{[16]} If *S* is a subset of a free monoid *A** then the intersection of all free submonoids of *A** containing *S* is well-defined, since *A** itself is free, and contains *S*; it is a free monoid and called the **free hull** of *S*. A basis for this intersection is a code.

The **defect theorem**^{[15]}^{[16]}^{[17]} states that if *X* is finite and *C* is the basis of the free hull of *X*, then either *X* is a code and *C* = *X*, or

A monoid morphism *f* from a free monoid *B*^{∗} to a monoid *M* is a map such that *f*(*xy*) = *f*(*x*)⋅*f*(*y*) for words *x*,*y* and *f*(ε) = ι, where ε and ι denotes the identity element of *B*^{∗} and *M*, respectively. The morphism *f* is determined by its values on the letters of *B* and conversely any map from *B* to *M* extends to a morphism. A morphism is **non-erasing**^{[18]} or **continuous**^{[19]} if no letter of *B* maps to ι and **trivial** if every letter of *B* maps to ι.^{[20]}

A morphism *f* from a free monoid *B*^{∗} to a free monoid *A*^{∗} is **simplifiable** if there is an alphabet *C* of cardinality less than that of *B* such the morphism *f* factors through *C*^{∗}, that is, it is the composition of a morphism from *B*^{∗} to *C*^{∗} and a morphism from that to *A*^{∗}; otherwise *f* is **elementary**. The morphism *f* is called a **code** if the image of the alphabet *B* under *f* is a code: every elementary morphism is a code.^{[25]}

For *L* a subset of *B*^{∗}, a finite subset *T* of *L* is a *test set* for *L* if morphisms *f* and *g* on *B*^{∗} agree on *L* if and only if they agree on *T*. The **Ehrenfeucht conjecture** is that any subset *L* has a test set:^{[26]} it has been proved^{[27]} independently by Albert and Lawrence; McNaughton; and Guba. The proofs rely on Hilbert's basis theorem.^{[28]}

The computational embodiment of a monoid morphism is a map followed by a fold. In this setting, the free monoid on a set *A* corresponds to lists of elements from *A* with concatenation as the binary operation. A monoid homomorphism from the free monoid to any other monoid (*M*,•) is a function *f* such that

where *e* is the identity on *M*. Computationally, every such homomorphism corresponds to a map operation applying *f* to all the elements of a list, followed by a fold operation which combines the results using the binary operator •. This computational paradigm (which can be generalized to non-associative binary operators) has inspired the MapReduce software framework.^{[citation needed]}

An **endomorphism** of *A*^{∗} is a morphism from *A*^{∗} to itself.^{[29]} The identity map *I* is an endomorphism of *A*^{∗}, and the endomorphisms form a monoid under composition of functions.

An endomorphism *f* is **prolongable** if there is a letter *a* such that *f*(*a*) = *as* for a non-empty string *s*.^{[30]}

The operation of string projection is an endomorphism. That is, given a letter *a* ∈ Σ and a string *s* ∈ Σ^{∗}, the string projection *p*_{a}(*s*) removes every occurrence of *a* from *s*; it is formally defined by

Note that string projection is well-defined even if the rank of the monoid is infinite, as the above recursive definition works for all strings of finite length. String projection is a morphism in the category of free monoids, so that

For free monoids of finite rank, this follows from the fact that free monoids of the same rank are isomorphic, as projection reduces the rank of the monoid by one.

for all strings *s*. Thus, projection is an idempotent, commutative operation, and so it forms a bounded semilattice or a commutative band.

Given a set *A*, the **free commutative monoid** on *A* is the set of all finite multisets with elements drawn from *A*, with the monoid operation being multiset sum and the monoid unit being the empty multiset.

The fundamental theorem of arithmetic states that the monoid of positive integers under multiplication is a free commutative monoid on an infinite set of generators, the prime numbers.

The **free commutative semigroup** is the subset of the free commutative monoid which contains all multisets with elements drawn from *A* except the empty multiset.

The free partially commutative monoid, or *trace monoid*, is a generalization that encompasses both the free and free commutative monoids as instances. This generalization finds applications in combinatorics and in the study of parallelism in computer science.