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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.
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 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 * .
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.
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 ...
Gen Alpha is defined as the group of people born between 2010 and 2024, succeeding Gen Z, who were born between the late 1990s and early 2010s, following millennials.
In category theory, a group with operators can be defined [3] as an object of a functor category Grp M where M is a monoid (i.e. a category with one object) and Grp denotes the category of groups. This definition is equivalent to the previous one, provided Ω {\displaystyle \Omega } is a monoid (if not, we may expand it to include the identity ...