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A set of free generators for a free monoid P is referred to as a basis for P: a set of words C is a code if C* is a free monoid and C is a basis. [3] A set X of words in A ∗ is a prefix , or has the prefix property , if it does not contain a proper (string) prefix of any of its elements.
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.
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 ...
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 ...
In computer science, more precisely in automata theory, a recognizable set of a monoid is a subset that can be distinguished by some homomorphism to a finite monoid. Recognizable sets are useful in automata theory, formal languages and algebra. This notion is different from the notion of recognizable language.
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 =.
The free group F S with free generating set S can be constructed as follows. S is a set of symbols, and we suppose for every s in S there is a corresponding "inverse" symbol, s −1, in a set S −1. Let T = S ∪ S −1, and define a word in S to be any written product of elements of T. That is, a word in S is an element of the monoid ...