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2. Free monoid - Wikipedia

   en.wikipedia.org/wiki/Free_monoid

   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]

3. Monoid (category theory) - Wikipedia

   en.wikipedia.org/wiki/Monoid_(category_theory)

   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.

4. Monoid - Wikipedia

   en.wikipedia.org/wiki/Monoid

   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]

5. Monoidal category - Wikipedia

   en.wikipedia.org/wiki/Monoidal_category

   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 ...

6. Talk:Free monoid - Wikipedia

   en.wikipedia.org/wiki/Talk:Free_monoid

   If the first line is correct in it being all the finite sequences then the free monoid must be finite. How then can the natural numbers under addition be a free monoid as the set of natural numbers is not finite. — Preceding unsigned comment added by 80.229.45.38 10:18, 28 October 2020 (UTC) Each individual natural number is finite.

7. Semigroup - Wikipedia

   en.wikipedia.org/wiki/Semigroup

   The set of all finite strings over a fixed alphabet Σ with concatenation of strings as the semigroup operation – the so-called "free semigroup over Σ". With the empty string included, this semigroup becomes the free monoid over Σ. A probability distribution F together with all convolution powers of F, with convolution as the operation ...

8. Trace monoid - Wikipedia

   en.wikipedia.org/wiki/Trace_monoid

   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 =.

9. Numerical semigroup - Wikipedia

   en.wikipedia.org/wiki/Numerical_semigroup

   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 ...