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 by a finite transducer: multiplication in such a monoid is "easy", in the sense that it can be described by a rational function.
In mathematics, a rationalmonoid is a monoid, an algebraic structure, for which each element can be represented in a "normal form" that can be computed...
is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide a foundation...
precisely in automata theory, a rational set of a monoid is an element of the minimal class of subsets of this monoid that contains all finite subsets...
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...
that Z {\displaystyle \mathbb {Z} } under multiplication is a commutative monoid. However, not every integer has a multiplicative inverse (as is the case...
finite index in G. In contrast, H is rational if and only if H is finitely generated. Rational set Rationalmonoid John Meakin (2007). "Groups and semigroups:...
a monoid, one can still use the notion of a generating set S {\displaystyle S} of G {\displaystyle G} . S {\displaystyle S} is a semigroup/monoid generating...
and to trees (see tree automaton). Rational set generalizes the notion (of regular/rational language) to monoids that are not necessarily free. Likewise...
successor function. Formally, one has a injective homomorphism of ordered monoids from the natural numbers N {\displaystyle \mathbb {N} } to the integers...
In abstract algebra, a branch of mathematics, an affine monoid is a commutative monoid that is finitely generated, and is isomorphic to a submonoid of...
Archimedean monoids, linearly ordered monoids that obey the Archimedean property. Examples include the natural numbers, the non-negative rational numbers...
R\langle A\rangle } . A formal series is a R-valued function c, on the free monoid A*, which may be written as ∑ w ∈ A ∗ c ( w ) w . {\displaystyle \sum _{w\in...
arises as the function composition of endomorphisms over any commutative monoid. The theory of (associative) algebras over commutative rings can be generalized...
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...
numbers can be replaced by the more general notion of a monoid. Let M {\displaystyle M} be a monoid with identity element 1 ∈ M , {\displaystyle 1\in M,}...
mathematics, the Grothendieck group, or group of differences, of a commutative monoid M is a certain abelian group. This abelian group is constructed from M in...
group is the dyadic monoid, which is the monoid of all strings of the form STkSTmSTn... for positive integers k, m, n,.... This monoid occurs naturally in...
multiplicative monoids called the structure sheaf. An affine monoid scheme is a monoidal space that is isomorphic to the spectrum of a monoid, and a monoid scheme...
multiplicatively, and a multiplicative identity denoted by 1 is a monoid. In such a monoid, exponentiation of an element x is defined inductively by x 0 =...
monoid M is then the monoid of all such finite-length left-right moves. Writing γ ∈ M {\displaystyle \gamma \in M} as a general element of the monoid...
order. A monoid is a set with an associative operation that has an identity element. The invertible elements in a monoid form a group under monoid operation...
(\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...
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...
natural numbers as an ordered monoid. The embedding of the rationals then gives a way of speaking about the rationals, integers, and natural numbers...
we say that M is a normal monoid. For example, the monoid Nn consisting of n-tuples of natural numbers is a normal monoid, with the Grothendieck group...
are given by the monoid that describes the symmetries of the infinite binary tree or Cantor space. This so-called period-doubling monoid is a subset of...
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