Top Qs
Timeline
Chat
Perspective

Symmetric inverse semigroup

From Wikipedia, the free encyclopedia

Remove ads
Remove ads

In abstract algebra, the set of all partial bijections on a set X (a.k.a. one-to-one partial transformations) forms an inverse semigroup, called the symmetric inverse semigroup[1] (actually a monoid) on X. The conventional notation for the symmetric inverse semigroup on a set X is [2] or .[3] In general is not commutative.

Details about the origin of the symmetric inverse semigroup are available in the discussion on the origins of the inverse semigroup.

Remove ads

Finite symmetric inverse semigroups

When X is a finite set {1, ..., n}, the inverse semigroup of one-to-one partial transformations is denoted by Cn and its elements are called charts or partial symmetries.[4] 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.[5]

The cycle notation of classical, group-based permutations generalizes to symmetric inverse semigroups by the addition of a notion called a path, which (unlike a cycle) ends when it reaches the "undefined" element; the notation thus extended is called path notation.[5]

Remove ads

See also

Notes

References

Loading content...
Loading related searches...

Wikiwand - on

Seamless Wikipedia browsing. On steroids.

Remove ads