In algebra, a transformation semigroup (or composition semigroup) is a collection of transformations (functions from a set to itself) that is closed under function composition. If it includes the identity function, it is a monoid, called a transformation (or composition) monoid. This is the semigroup analogue of a permutation group.
A transformation semigroup of a set has a tautological semigroup action on that set. Such actions are characterized by being faithful, i.e., if two elements of the semigroup have the same action, then they are equal.
An analogue of Cayley's theorem shows that any semigroup can be realized as a transformation semigroup of some set.
In automata theory, some authors use the term transformation semigroup to refer to a semigroup acting faithfully on a set of "states" different from the semigroup's base set.[1] There is a correspondence between the two notions.
^Dominique Perrin; Jean Eric Pin (2004). Infinite Words: Automata, Semigroups, Logic and Games. Academic Press. p. 448. ISBN 978-0-12-532111-2.
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In algebra, a transformationsemigroup (or composition semigroup) is a collection of transformations (functions from a set to itself) that is closed under...
Early results include a Cayley theorem for semigroups realizing any semigroup as a transformationsemigroup, in which arbitrary functions replace the role...
science, an action or act of a semigroup on a set is a rule which associates to each element of the semigroup a transformation of the set in such a way that...
way; their conjecture was proven in 2007 by Avraham Trahtman. A transformationsemigroup is synchronizing if it contains an element of rank 1, that is,...
alphabet Σ, or as the induced transformationsemigroup of Q. In older books like Clifford and Preston (1967) semigroup actions are called "operands"....
mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying...
the full transformationsemigroup or symmetric semigroup on X. (One can actually define two semigroups depending how one defines the semigroup operation...
In mathematics, the bicyclic semigroup is an algebraic object important for the structure theory of semigroups. Although it is in fact a monoid, it is...
semigroup is regular. Any full transformationsemigroup is regular. A Rees matrix semigroup is regular. The homomorphic image of a regular semigroup is...
set X (a.k.a. one-to-one partial transformations) forms an inverse semigroup, called the symmetric inverse semigroup (actually a monoid) on X. The conventional...
(partial transformations) on a given base set, X , {\displaystyle X,} forms a regular semigroup called the semigroup of all partial transformations (or the...
In group theory, an inverse semigroup (occasionally called an inversion semigroup) S is a semigroup in which every element x in S has a unique inverse...
The semigroup of all partial transformations of a set is a catholic semigroup. It follows that every semigroup is embeddable in a catholic semigroup. But...
with addition form a monoid, the identity element being 0. Monoids are semigroups with identity. Such algebraic structures occur in several branches of...
an I-semigroup and a *-semigroup. A class of semigroups important in semigroup theory are completely regular semigroups; these are I-semigroups in which...
mathematics, particularly in abstract algebra, a semigroup with involution or a *-semigroup is a semigroup equipped with an involutive anti-automorphism...
oscillator semigroup corresponds to a representation by contraction operators of the semigroup in SL(2,C) corresponding to Möbius transformations that take...
semigroups of measures and harmonic analysis on semigroups, transformationsemigroups, and applications of semigroup theory to other disciplines such as ring...
or occasionally as the full linear semigroup or general linear monoid. Notably, it constitutes a regular semigroup. If one removes the restriction of...
this monoid is known as the transition monoid, or sometimes the transformationsemigroup. The construction can also be reversed: given a δ ^ {\displaystyle...
January/February 1990, 15–17. Firstov, V. E. (2008). "A Special Matrix TransformationSemigroup of Primitive Pairs and the Genealogy of Pythagorean Triples". Mat...
of Michigan Engineering Summer (1968). Foundations of Information Systems Engineering. we see that an identity element of a semigroup is idempotent....
evolution for the full orbit: the monoid of the Picard sequence (cf. transformationsemigroup) has generalized to a full continuous group. This method (perturbative...
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