## Congrucences and Green’s relations on regular semigroups.(English)Zbl 0257.20057

Let $$S$$ be a regular semigroup and $$\rho$$ a congruence relation on $$S$$. The main problem treated in this paper is the relationship between Green’s relations on $$S$$ and those on $$S/\rho$$. Let $$\mathcal K$$ be any of the Green’s relations $$\mathcal H$$, $$\mathcal L$$, $$\mathcal R$$, $$\mathcal D$$ and $$\mathcal J$$. The following problem is formulated.
Question H: If $$A$$ and $$B$$ are elements of $$S/\rho$$ that are $$\mathcal K$$-related in $$S/\rho$$, then are there elements $$a\in A$$, $$b\in B$$ such that $$a$$ and $$b$$ are $$\mathcal K$$-related in $$S$$? The following is proved.
Theorem A: Let $$A_1,\ldots, A_n$$ be any elements $$\in S/\rho$$ such that $$A_1\mathcal L A_2\mathcal L \ldots \mathcal L A_n$$ in $$S\rho$$. Then there exist elements $$a_i\in A_i$$ such that 1) $$a_1\mathcal L a_2\ldots \mathcal L a_n$$ in $$S$$; 2) $$a_1$$ is an idempotent if $$A_1$$ is an idempotent of $$S/\rho$$.
An analogous result holds for the $$\mathcal D$$-relation.
Examples are given which show that the question H may have a negative answer if $$\mathcal K=\mathcal H$$ or $$\mathcal K = \mathcal J$$. There are some further results concerning non-necessarily regular semigroups, e.g.,
Theorem B: Let $$\mathcal K$$ be any of the Green’s relations $$\mathcal H$$, $$\mathcal L$$, $$\mathcal R$$, $$\mathcal J$$. Let $$\rho$$ be a congruence contained in on a semigroup $$S$$. Then $$\mathcal K(S/\rho) =\mathcal K(S)/\rho$$.
The paper contains many other results which lead to new proofs of known results about congruences on regular semigroups.

### MSC:

 20M17 Regular semigroups
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### References:

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