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Rational Monoid
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. Definition Consider a monoid ''M''. Consider a pair (''A'',''L'') where ''A'' is a finite subset of ''M'' that generates ''M'' as a monoid, and ''L'' is a language on ''A'' (that is, a subset of the set of all strings ''A''∗). Let ''φ'' be the map from the free monoid ''A''∗ to ''M'' given by evaluating a string as a product in ''M''. We say that ''L'' is a ''rational cross-section'' if ''φ'' induces a bijection between ''L'' and ''M''. We say that (''A'',''L'') is a ''rational structure'' for ''M'' if in addition the kernel of ''φ'', viewed as a subset of the product monoid ''A''∗×''A''∗ is a rational set. A quasi-rational monoid is one for which ''L'' is a rational relation: a rationa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoid
In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being . Monoids are semigroups with identity. Such algebraic structures occur in several branches of mathematics. The functions from a set into itself form a monoid with respect to function composition. More generally, in category theory, the morphisms of an object to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object. In computer science and computer programming, the set of strings built from a given set of characters is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide a foundation for process calculi and concurrent computing. In theoretical computer science, the study of monoids is fundamental for automata theory (Krohn–Rhodes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
