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Inverse Semigroup
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'' ''y'' in ''S'' in the sense that ''x = xyx'' and ''y = yxy'', i.e. a regular semigroup in which every element has a unique inverse. Inverse semigroups appear in a range of contexts; for example, they can be employed in the study of partial symmetries. (The convention followed in this article will be that of writing a function on the right of its argument, e.g. ''x f'' rather than ''f(x)'', and composing functions from left to right—a convention often observed in semigroup theory.) Origins Inverse semigroups were introduced independently by Viktor Vladimirovich Wagner in the Soviet Union in 1952, and by Gordon Preston in the United Kingdom in 1954. Both authors arrived at inverse semigroups via the study of partial bijections of a set: a partial transformation ''α'' of a set ''X'' is a function from ''A'' to '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Inverse Semigroup
__NOTOC__ In abstract algebra, the set of all partial bijections on a set ''X'' ( one-to-one partial transformations) forms an inverse semigroup, called the symmetric inverse semigroup (actually a monoid) on ''X''. The conventional notation for the symmetric inverse semigroup on a set ''X'' is \mathcal_X or \mathcal_X. In general \mathcal_X is not commutative. Details about the origin of the symmetric inverse semigroup are available in the discussion on the origins of the inverse semigroup. Finite symmetric inverse semigroups When ''X'' is a finite set , the inverse semigroup of one-to-one partial transformations is denoted by ''C''''n'' and its elements are called charts or partial symmetries. The notion of chart generalizes the notion of permutation. A (famous) example of (sets of) charts are the hypomorphic mapping sets from the reconstruction conjecture in graph theory. The cycle notation of classical, group-based permutations generalizes to symmetric inverse semigroups by t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Symmetries
__NOTOC__ In abstract algebra, the set of all partial bijections on a set ''X'' ( one-to-one partial transformations) forms an inverse semigroup, called the symmetric inverse semigroup (actually a monoid) on ''X''. The conventional notation for the symmetric inverse semigroup on a set ''X'' is \mathcal_X or \mathcal_X. In general \mathcal_X is not commutative. Details about the origin of the symmetric inverse semigroup are available in the discussion on the origins of the inverse semigroup. Finite symmetric inverse semigroups When ''X'' is a finite set , the inverse semigroup of one-to-one partial transformations is denoted by ''C''''n'' and its elements are called charts or partial symmetries. The notion of chart generalizes the notion of permutation. A (famous) example of (sets of) charts are the hypomorphic mapping sets from the reconstruction conjecture in graph theory. The cycle notation of classical, group-based permutations generalizes to symmetric inverse semigroups by th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set and an Binary operation, operation that combines any two Element (mathematics), elements of the set to produce a third element of the set, in such a way that the operation is Associative property, associative, an identity element exists and every element has an Inverse element, inverse. These three axioms hold for Number#Main classification, number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
