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A free monoid is equidivisible: if the equation mn = pq holds, then there exists an s such that either m = ps, sn = q (example see image) or ms = p, n = sq. [9] This result is also known as Levi's lemma. [10] A monoid is free if and only if it is graded (in the strong sense that only the identity has gradation 0) and equidivisible. [9]
A simpler example are the free monoids. The free monoid on a set X, is the monoid of all finite strings using X as alphabet, with operation concatenation of strings. The identity is the empty string. In essence, the free monoid is simply the set of all words, with no equivalence relations imposed.
In mathematics, it is more commonly known as the free monoid construction. The application of the Kleene star to a set V {\\displaystyle V} is written as V ∗ {\\displaystyle V^{*}} . It is widely used for regular expressions , which is the context in which it was introduced by Stephen Kleene to characterize certain automata , where it means ...
This monoid is denoted Σ ∗ and is called the free monoid over Σ. It is not commutative if Σ has at least two elements. Given any monoid M, the opposite monoid M op has the same carrier set and identity element as M, and its operation is defined by x • op y = y • x. Any commutative monoid is the opposite monoid of itself.
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). Every monoid may be presented by a semi-Thue ...
Conversely, the set of isomorphism classes (if such a thing makes sense) of a monoidal category is a monoid w.r.t. the tensor product. Any commutative monoid (,,) can be realized as a monoidal category with a single object. Recall that a category with a single object is the same thing as an ordinary monoid.
The bicyclic semigroup is the quotient of the free monoid on two generators p and q by the congruence generated by the relation p q = 1. Thus, each semigroup element is a string of those two letters, with the proviso that the subsequence "p q" does not appear. The semigroup operation is concatenation of strings, which is clearly associative.
A monoid object in the category of complete join-semilattices Sup (with the monoidal structure induced by the Cartesian product) is a unital quantale. A monoid object in (Ab, ⊗ Z, Z), the category of abelian groups, is a ring. For a commutative ring R, a monoid object in (R-Mod, ⊗ R, R), the category of modules over R, is a R-algebra.