Examples of monoid

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But there, the languages are in another monoid, which we will introduce next.
In particular, when the set of actions is a singleton (containing a 'tic') we obtain the monoid of discrete time.
Additionally, we integrate into our setting a method for the annotation of hypergraphs with an algebraic structure (in our case, a commutative monoid).
We also consider interaction sets with a minimum of algebraic structure, namely that of a monoid.
Our basic example is the free monoid on a set of actions.
By the third clause, the prefix relation in a left divisibility monoid is a partial order relation.
A graded category is considered to be a simple extension of the usual definition of a graded monoid to the many-object case.
The additive structure, as well as the multiplicative one, is a commutative monoid.
As one of the referees remarked, these results could be stated for any saturated monoid of cancellable elements.
This function takes an argument which maps terms to a monoid, and combines the results.
When we use notions like units, divisional closures, and so on, we refer to the multiplicative monoid (when nothing else is stated).
We also show that, in the continuous case, the way-below relation on the commutative monoid is the quotient of the way-below on the monoid.
Therefore, all the requirements in the definition of a divisibility monoid are satisfied.
In other words, if you want to consider the additive and multiplicative monoid of a given ring, you have to define two distinct type classes.
From what we have shown so far, any divisibility monoid satisfies the conditions given here.
The monoid operations on v and e are extended to graphs and dynamic structures as follows.
We consider such a collection of resources, a preordered commutative monoid, from the relational semantics perspective.
When considering semirings, we use the multiplicative monoid for this definition.
This shows that it is very natural from the point of view of the multiplicative monoid.
In any monoid, the set of recognizable languages is closed under the usual set-theoretic operations, like complementation, intersection and difference.

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