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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 ...
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 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 ...
Associated with any semiautomaton is a monoid called the characteristic monoid, input monoid, transition monoid or transition system of the semiautomaton, which acts on the set of states Q. This may be viewed either as an action of the free monoid of strings in the input alphabet Σ, or as the induced transformation semigroup of Q .
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 ...
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.
Beta decay: beta particle is emitted from an atomic nucleus Compton scattering: scattering of a photon by a charged particle Neutrino-less double beta decay: If neutrinos are Majorana fermions (that is, their own antiparticle), Neutrino-less double beta decay is possible. Several experiments are searching for this. Pair production and annihilation