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...

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...

and to trees (see tree automaton). Rational set generalizes the notion (of regular/rational language) to monoids that are not necessarily free. Likewise...

that Z {\displaystyle \mathbb {Z} } under multiplication is a commutative monoid. However, not every integer has a multiplicative inverse (as is the case...

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...

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...

imaginary numbers such as the square roots of −1. The real numbers include the rational numbers, such as the integer −5 and the fraction 4 / 3. The rest of the...

finite index in G. In contrast, H is rational if and only if H is finitely generated. Rational set Rational monoid John Meakin (2007). "Groups and semigroups:...