HOME

TheInfoList



OR:

__NOTOC__ In
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
, the set of all
partial bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
s on a set ''X'' ( one-to-one partial transformations) forms an
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 semig ...
, called the symmetric inverse semigroup (actually a
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids a ...
) 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 In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
. 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 pr ...
. A (famous) example of (sets of) charts are the hypomorphic mapping sets from the
reconstruction conjecture Informally, the reconstruction conjecture in graph theory says that graphs are determined uniquely by their subgraphs. It is due to KellyKelly, P. J.A congruence theorem for trees ''Pacific J. Math.'' 7 (1957), 961–968. and Ulam.Ulam, S. M ...
in
graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
. The
cycle notation 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 pro ...
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 In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...


Notes


References

* * * Semigroup theory Algebraic structures {{Abstract-algebra-stub