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...

characters is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide...

the syntactic monoid is unique up to unique isomorphism. An alphabet is a finite set. The free monoid on a given alphabet is the monoid whose elements...

a group are imposed. 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...

the set of all finite sequences (or strings) of members of X forms a free monoid under the operation of concatenation of sequences, with sequence reversal...

the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading...

non-commutative polynomial ring may be identified with the monoid ring over R of the free monoid of all finite words in the Xi. Since the words over the...

Associated with any semiautomaton is a monoid called the characteristic monoid, input monoid, transition monoid or transition system of the semiautomaton...

on the free monoid Σ∗ (or the free semigroup Σ+) generated by Σ. The monoid is then presented as the quotient of the free monoid (or the free semigroup)...

complete equivalence under all reorderings. The trace monoid or free partially commutative monoid is a monoid of traces. Traces were introduced by Pierre Cartier...

as groups or rings. If A is a set, the free monoid over A (denoted A*, also called Kleene star of A) is a monoid containing all the finite sequences (or...

operation – the so-called "free semigroup over Σ". With the empty string included, this semigroup becomes the free monoid over Σ. A probability distribution...

monoids were first presented by M.W. Shields. History monoids are isomorphic to trace monoids (free partially commutative monoids) and to the monoid of...

factor monoid M R = Σ ∗ / ↔ R ∗ {\displaystyle {\mathcal {M}}_{R}=\Sigma ^{*}/{\overset {*}{\underset {R}{\leftrightarrow }}}} of the free monoid Σ ∗ {\displaystyle...

regular expressions. In mathematics, it is more commonly known as the free monoid construction. Given a set V {\displaystyle V} , define V 0 = { ε } {\displaystyle...

beyond the monad laws themselves. Just as a free monoid concatenates elements without evaluation, a free monad allows chaining computations with markers...

operation. A monoid homomorphism is a map between monoids that preserves the monoid operation and maps the identity element of the first monoid to that of...

mathematics, a factorisation 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...

an algebraic definition of the free partially commutative monoid or trace monoid, or equivalently, the history monoid, which provides a concrete algebraic...

category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category's objects...

If it includes the identity function, it is a monoid, called a transformation (or composition) monoid. This is the semigroup analogue of a permutation...

separate sorting algorithm. The monoid of words over an alphabet A is the free monoid over A. That is, the elements of the monoid are the finite sequences (words)...

important special case is a monoid action or act, in which the semigroup is a monoid and the identity element of the monoid acts as the identity transformation...

then the free category on Q has only one object, and corresponds to the free monoid on the edges of Q. The category of small categories Cat has a forgetful...

concatenation operation form a monoid, the free monoid generated by Σ. In addition, the length function defines a monoid homomorphism from Σ* to the non-negative...

addition of integers; in modern mathematics, this system is called a free monoid. In 1956 Alonzo Church wrote: "Like any branch of mathematics, theoretical...

{*}{\underset {R}{\leftrightarrow }}}} of the free monoid Σ ∗ {\displaystyle \Sigma ^{*}} by the Thue congruence. If a monoid M {\displaystyle {\mathcal {M}}} is...

concatenation operation, form an associative algebraic structure called a free monoid. The identity element is the null string. Sets of strings with concatenation...

(\mathbb {N} ,+)} is a commutative monoid with identity element 0. It is a free monoid on one generator. This commutative monoid satisfies the cancellation property...

provide a unique monoid factorisation of the free monoid. They are also totally ordered, and thus provide a total order on the monoid. This is analogous...