Search results
Results from the WOW.Com Content Network
The free monoid on a set A is usually denoted A ∗. The free semigroup on A is the subsemigroup of A ∗ containing all elements except the empty string. It is usually denoted A +. [1] [2] More generally, an abstract monoid (or semigroup) S is described as free if it is isomorphic to the free monoid (or semigroup) on some set. [3]
In mathematics, a factorisation of a free monoid is a sequence of subsets of words with the property that every word in the free monoid can be written as a concatenation of elements drawn from the subsets.
In algebra, an action of a monoidal category S on a category X is a functor ⋅ : S × X → X {\displaystyle \cdot :S\times X\to X} such that there are natural isomorphisms s ⋅ ( t ⋅ x ) ≃ ( s ⋅ t ) ⋅ x {\displaystyle s\cdot (t\cdot x)\simeq (s\cdot t)\cdot x} and e ⋅ x ≃ x {\displaystyle e\cdot x\simeq x} and those natural ...
The Kleene star is defined for any monoid, not just strings. More precisely, let (M, ⋅) be a monoid, and S ⊆ M. Then S * is the smallest submonoid of M containing S; that is, S * contains the neutral element of M, the set S, and is such that if x,y ∈ S *, then x⋅y ∈ S *.
Let be an alphabet: the set of words over is a monoid, the free monoid on . The recognizable subsets of A ∗ {\displaystyle A^{*}} are precisely the regular languages . Indeed, such a language is recognized by the transition monoid of any automaton that recognizes the language.
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.
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
Every group is a monoid and every abelian group a commutative monoid. Any semigroup S may be turned into a monoid simply by adjoining an element e not in S and defining e • s = s = s • e for all s ∈ S. This conversion of any semigroup to the monoid is done by the free functor between the category of semigroups and the category of monoids. [3]