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In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero elements, often called the empty string and denoted by ε or λ, as the identity element.
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 ...
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 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 ...
The free monoid on a given set is the monoid whose elements are all the strings of zero or more elements from that set, ... The proof of the "only if" part is as follows.
The non-negative integers form a cancellative monoid under addition. Each of these is an example of a cancellative magma that is not a quasigroup. In fact, any free semigroup or monoid obeys the cancellative law, and in general, any semigroup or monoid embedding into a group (as the above examples clearly do) will obey the cancellative law.
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 =.
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.