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...

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...

synchronization points or thread joins. The trace monoid or free partially commutative monoid is a monoid of traces. In a nutshell, it is constructed as...

from linear algebra is the multiplicative monoid of real square matrices of order n (called the full linear monoid). The map which sends a matrix to its transpose...

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...

that recognizes the language L {\displaystyle L} . The free monoid on a given set is the monoid whose elements are all the strings of zero or more elements...

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)...

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

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...

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...

symbols or characters. In mathematics, it is more commonly known as the free monoid construction. The application of the Kleene star to a set V {\displaystyle...

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

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

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...

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)...

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

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...

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...

variables with coefficients in the ring R is the monoid ring R[N], where the monoid N is the free monoid on n letters, also known as the set of all strings...

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...

defining relations on X. Let X* be the set of all words in X (i.e. the free monoid generated by X). Since the relations R generate an equivalence relation...

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

In abstract algebra, a monoid ring is a ring constructed from a ring and a monoid, just as a group ring is constructed from a ring and a group. Let R be...

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...

equations is finite. A free monoid on a finite alphabet is compact. A free monoid on a countable alphabet is compact. A finitely generated free group is compact...

In mathematics, a rational monoid is a monoid, an algebraic structure, for which each element can be represented in a "normal form" that can be computed...

syntactic monoid M, meaning it is the preimage {w ∈ Σ* | f(w) ∈ S} of a subset S of a finite monoid M under a monoid homomorphism f: Σ* → M from the free monoid...

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...