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Two free monoids or semigroups are isomorphic if and only if they have the same rank. In fact, every set of generators for a free monoid or semigroup S contains the free generators, since a free generator has word length 1 and hence can only be generated by itself. It follows that a free semigroup or monoid is finitely generated if and only if ...
The monoid is then presented as the quotient of the free monoid (or the free semigroup) by these relations. This is an analogue of a group presentation in group theory . As a mathematical structure, a monoid presentation is identical to a string rewriting system (also known as a semi-Thue system).
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.
A monoid object in the category of monoids (with the direct product of monoids) is just a commutative monoid. This follows easily from the Eckmann–Hilton argument. A monoid object in the category of complete join-semilattices Sup (with the monoidal structure induced by the Cartesian product) is a unital quantale.
The Schützenberger theorem relates the definition in terms of a multiplicative property to an additive property. [clarification needed] Let A ∗ be the free monoid on an alphabet A. Let X i be a sequence of subsets of A ∗ indexed by a totally ordered index set I. A factorisation of a word w in A ∗ is an expression
Let denote the free monoid on a set of generators , that is, the set of all strings written in the alphabet .The asterisk is a standard notation for the Kleene star.An independency relation on the alphabet then induces a symmetric binary relation on the set of strings : two strings , are related, , if and only if there exist ,, and a pair (,) such that = and =.
History monoids were first presented by M.W. Shields. [1] History monoids are isomorphic to trace monoids (free partially commutative monoids) and to the monoid of dependency graphs. As such, they are free objects and are universal. The history monoid is a type of semi-abelian categorical product in the category of monoids.
Generator may refer to: Signal generator , electronic devices that generate repeating or non-repeating electronic signals Electric generator , a device that converts mechanical energy to electrical energy.
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