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The free partially commutative monoid, or trace monoid, is a generalization that encompasses both the free and free commutative monoids as instances. This generalization finds applications in combinatorics and in the study of parallelism in computer science .
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.
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.
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).
For example, the alphabet {a, b} with the rules { ab → ε, ba → ε }, where ε is the empty string, is a presentation of the free group on one generator. If instead the rules are just { ab → ε }, then we obtain a presentation of the bicyclic monoid. The importance of semi-Thue systems as presentation of monoids is made stronger by the ...
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 =.
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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