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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]
Many definitions and theorems about monoids can be generalised to small categories with more than one object. For example, a quotient of a category with one object is just a quotient monoid. Monoids, just like other algebraic structures, also form their own category, Mon, whose objects are monoids and whose morphisms are monoid homomorphisms. [8]
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
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 =.
A monoid object in the category of monoids (with the direct product of monoids) is just a commutative monoid. This follows easily from the Eckmann–Hilton argument. A monoid object in the category of complete join-semilattices Sup (with the monoidal structure induced by the Cartesian product) is a unital quantale.
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.
Numerical semigroups are commutative monoids and are also known as numerical monoids. [ 1 ] [ 2 ] The definition of numerical semigroup is intimately related to the problem of determining nonnegative integers that can be expressed in the form x 1 n 1 + x 2 n 2 + ... + x r n r for a given set { n 1 , n 2 , ..., n r } of positive integers and for ...
For every category C, the free strict monoidal category Σ(C) can be constructed as follows: its objects are lists (finite sequences) A 1, ..., A n of objects of C; there are arrows between two objects A 1, ..., A m and B 1, ..., B n only if m = n, and then the arrows are lists (finite sequences) of arrows f 1: A 1 → B 1, ..., f n: A n → B ...