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.


20M17 Regular semigroups
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