symmetric inverse semigroup (original) (raw)

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 compositionMathworldPlanetmath 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.