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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 $\mathcal{T}_3$.

Shaun
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QueenLA
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  • What are your thoughts/attempts so far? The more information you can give us, the better we can tailor our answers to your needs. – Cameron Buie Oct 23 '13 at 06:34

1 Answers1

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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

J.-E. Pin
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  • Yes, its for semigroup T_n. I see this question in most of the books. However, I can't find the solution anywhere or do it myself. If anyone can tell me where I can find the solution for these proof, it will be really appreciated. – QueenLA Nov 06 '13 at 08:34