Proof of Green's Relation

Question

Can anyone tell me where I can get the proofs for the following Green's relations?

1. $a\mathcal{L}b$ iff $\operatorname{Im}(a) = \operatorname{Im}(b)$,

2. $a\mathcal{R}b$ iff $\operatorname{ker}(a) = \operatorname{ker}(b)$,

3. $a\mathcal{D}b$ iff $\operatorname{rank}(a) = \operatorname{rank}(b)$,

where $a,b$ are transformations in $\mathcal{T}_3$.

Answer 1

I suppose you are working in the semigroup $\cal{T}_n$ of all transformations on $\{1, ..., n\}$. Note that the results you mention only hold for regular $\cal D$-classes if you are working in a subsemigroup of $\cal{T}_n$. That being said, a good reference for your question is the book

G. Lallement, Semigroups and combinatorial applications, Wiley, 1979