Geodesic
The Translational Hull of an Inverse Semigroup
Canadian journal of mathematics, Tome 26 (1974) no. 5, pp. 1050-1068

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Let S be a semigroup. A function λ(ρ) on S is a left (right) translation of S if, for all x, y ∊ S, λ(xy) = λ(x)y ((xy)ρ= x(yρ)). A left translation λ and a right translation ρ are said to be linked if x(λy) = (xρ)y, for all x,y ∊ S,and then the ordered pair (λ, ρ) is called a bitranslation. Clearly the set Λ(S) (P(S)) of all left (right) translations is a semigroup with respect to composition of functions. The set of bitranslations forms a subsemigroup of the direct product Λ(S) × P(S)which is called the translational hull, Ω(S), of S. A valuable survey of results relating to Ω(S) and its importance in relation to semigroup extensions will be found in Petrich's review [6], to which the reader is referred for basic results on translational hulls. 
Reilly, N. R. The Translational Hull of an Inverse Semigroup. Canadian journal of mathematics, Tome 26 (1974) no. 5, pp. 1050-1068. doi: 10.4153/CJM-1974-098-7
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