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In algebra and theoretical computer 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 the product of two elements of the semigroup (using the semigroup operation) is associated with the composite of the two corresponding transformations. The terminology conveys the idea that the elements of the semigroup are acting as transformations of the set. From an algebraic perspective, a semigroup action is a generalization of the notion of a group action in group theory. From the computer science point of view, semigroup actions are closely related to automata: the set models the state of the automaton and the action models transformations of that state in response to inputs.
An important special case is a monoid action or act, in which the semigroup is a monoid and the identity element of the monoid acts as the identity transformation of a set. From a category theoretic point of view, a monoid is a category with one object, and an act is a functor from that category to the category of sets. This immediately provides a generalization to monoid acts on objects in categories other than the category of sets.
Another important special case is a transformation semigroup. This is a semigroup of transformations of a set, and hence it has a tautological action on that set. This concept is linked to the more general notion of a semigroup by an analogue of Cayley's theorem.
(A note on terminology: the terminology used in this area varies, sometimes significantly, from one author to another. See the article for details.)
theoretical computer 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...
the semigroup analogue of a permutation group. A transformation semigroup of a set has a tautological semigroupaction on that set. Such actions are characterized...
group actionSemigroupaction Ring Action (mathematics) Action (firearms), the mechanism that manipulates cartridges and/or seals the breech Action! (programming...
maps and equivalence relations however. See semigroupaction. Instead of actions on sets, we can define actions of groups and monoids on objects of an arbitrary...
Preston (1967) semigroupactions are called "operands". In category theory, semiautomata essentially are functors. A transformation semigroup or transformation...
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...
representation leads to a semigroup of contraction operators, introduced as the oscillator semigroup by Roger Howe in 1988. The semigroup had previously been...
definitions also apply to semigroups. In ring theory, the centralizer of a subset of a ring is defined with respect to the semigroup (multiplication) operation...
In mathematics, a topological semigroup is a semigroup that is simultaneously a topological space, and whose semigroup operation is continuous. Every topological...
with addition form a monoid, the identity element being 0. Monoids are semigroups with identity. Such algebraic structures occur in several branches of...
or occasionally as the full linear semigroup or general linear monoid. Notably, it constitutes a regular semigroup. If one removes the restriction of...
semi-algebraic systems in computer algebra Regular semigroup, related to the previous sense *-regular semigroup Borel regular measure Cauchy-regular function...
transformation semigroup or symmetric semigroup on X. (One can actually define two semigroups depending how one defines the semigroup operation as the...
notion generalizes to semigroups and, as such, is a central construction in the Krohn–Rhodes structure theory of finite semigroups. Let A {\displaystyle...
easy to see that this generates a semigroup in some sense—it is not absolutely integrable and so cannot define a semigroup in the above strong sense. Many...
is a topological space with continuous group action Topological module Topological semigroup – semigroup with continuous operationPages displaying wikidata...
their conjecture was proven in 2007 by Avraham Trahtman. A transformation semigroup is synchronizing if it contains an element of rank 1, that is, an element...
Thus, in such lossy systems, the renormalization group is, in fact, a semigroup, as lossiness implies that there is no unique inverse for each element...
group action Topological ring – ring where ring operations are continuousPages displaying wikidata descriptions as a fallback Topological semigroup – semigroup...
maximal cone. A similar decomposition already occurs in the semigroup. The oscillator semigroup of Roger Howe concerns the special case of this theory for...
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