Search results
Results from the WOW.Com Content Network
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]
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 ...
String diagrams (with generators from ) are arrows in the free monoidal category . [8] The interpretation in a monoidal category D {\displaystyle D} is a defined by a monoidal functor F : C Σ → D {\displaystyle F:C_{\Sigma }\to D} , which by freeness is uniquely determined by a morphism of monoidal signatures F : Σ → U ( D ...
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.
The monoid therefore is characterized by specification of the triple (S, • , e). Depending on the context, the symbol for the binary operation may be omitted, so that the operation is denoted by juxtaposition; for example, the monoid axioms may be written (ab)c = a(bc) and ea = ae = a. This notation does not imply that it is numbers being ...
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 ...
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 =.
Given a reliance alphabet (,,), a symmetric and irreflexive relation can be defined on the free monoid of all possible strings of finite length by: for all strings , and all independent symbols ,. The equivalence closure of ≐ {\displaystyle \doteq } is denoted ≡ {\displaystyle \equiv } or ≡ ( Σ , D , I ) {\displaystyle \equiv _{(\Sigma ...