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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
or
. 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.
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
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
. 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 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''.
[
]
See also
* Symmetric group
Notes
References
*
*
*
Semigroup theory
Algebraic structures
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