symmetric inverse semigroup
Let X be a set. A partial map on X is an application defined from a subset of X into X. We denote by 𝔉(X) the set of partial map on X. Given α∈𝔉(X), we denote by dom(α) and ran(α) respectively the domain and the range of α, i.e.
dom(α),ranα⊆X,α:dom(α)→X,α(dom(α))=ran(α). |
We define the composition of two partial map α,β∈𝔉(X) as the partial map α∘β∈𝔉(X) with domain
dom(α∘β)=β-1(ran(β)∩dom(α))={x∈dom(β)|α(x)∈dom(β)} |
defined by the common rule
α∘β(x)=α(β(x)),∀x∈dom(α∘β). |
It is easily verified that the 𝔉(X) with the composition ∘ is a semigroup.
A partial map α∈𝔉(X) is said bijective when it is bijective as a map α:ran(α)→dom(α). It can be proved that the subset ℑ(X)⊆𝔉(X) of the partial bijective maps on X is an inverse semigroup (with the composition ∘), that is called symmetric inverse semigroup on X. Note that the symmetric group on X is a subgroup
of ℑ(X).
Title | symmetric inverse semigroup |
---|---|
Canonical name | SymmetricInverseSemigroup |
Date of creation | 2013-03-22 16:11:14 |
Last modified on | 2013-03-22 16:11:14 |
Owner | Mazzu (14365) |
Last modified by | Mazzu (14365) |
Numerical id | 6 |
Author | Mazzu (14365) |
Entry type | Definition |
Classification | msc 20M18 |
Defines | partial map |
Defines | composition of partial maps |
Defines | symmetric inverse semigroup |