Trace monoid

In computer science, a trace is a set of strings, wherein certain letters in the string are allowed to commute, but others are not. It generalizes the concept of a string, by not forcing the letters to always be in a fixed order, but allowing certain reshufflings to take place. Traces were introduced by Cartier and Foata in 1969 to give a combinatorial proof of MacMahon's Master theorem. Traces are used in theories of concurrent computation, where commuting letters stand for portions of a job that can execute independently of one another, while non-commuting letters stand for locks, synchronization points or thread joins.

The trace monoid or free partially commutative monoid is a monoid of traces. In a nutshell, it is constructed as follows: sets of commuting letters are given by an independency relation. These induce an equivalence relation of equivalent strings; the elements of the equivalence classes are the traces. The equivalence relation then partitions up the free monoid (the set of all strings of finite length) into a set of equivalence classes; the result is still a monoid; it is a quotient monoid and is called the trace monoid. The trace monoid is universal, in that all homomorphic monoids are in fact isomorphic.

Trace monoids are commonly used to model concurrent computation, forming the foundation for process calculi. They are the object of study in trace theory. The utility of trace monoids comes from the fact that they are isomorphic to the monoid of dependency graphs; thus allowing algebraic techniques to be applied to graphs, and vice versa. They are also isomorphic to history monoids, which model the history of computation of individual processes in the context of all scheduled processes on one or more computers.

Let ${\displaystyle \Sigma ^{*}}$ denote the free monoid, that is, the set of all strings written in the alphabet ${\displaystyle \Sigma }$. Here, the asterisk denotes, as usual, the Kleene star. An independency relation ${\displaystyle I}$ on ${\displaystyle \Sigma }$ then induces a binary relation ${\displaystyle \sim }$ on ${\displaystyle \Sigma ^{*}}$, where ${\displaystyle u\sim v\,}$ if and only if there exist ${\displaystyle x,y\in \Sigma ^{*}}$, and a pair ${\displaystyle (a,b)\in I}$ such that ${\displaystyle u=xaby}$ and ${\displaystyle v=xbay}$. Here, ${\displaystyle u,v,x}$ and ${\displaystyle y}$ are understood to be strings (elements of ${\displaystyle \Sigma ^{*}}$), while ${\displaystyle a}$ and ${\displaystyle b}$ are letters (elements of ${\displaystyle \Sigma }$).

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