Green's graphs of a semigroup

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>S</m:mi> </m:math> S be a semigroup. In this study, we first introduce the Green’s digraphs and Green’s graphs related to the Green’s relations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">L</m:mi> </m:math> {L} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">R</m:mi> </m:math> {R} , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">J</m:mi> </m:math> {J} of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>S</m:mi> </m:math> S . Further, the connectedness and completeness of the Green’s graphs are discussed. For a finite semigroup <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>S</m:mi> </m:math> S , we show that each of the Green’s graphs of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>S</m:mi> </m:math> S has a transitive orientation. Moreover, we obtain that these Green’s graphs are perfect. Finally, the structures of the Green’s graphs are characterized using the generalized lexicographic product.