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A note on principal ideals and \(\mathcal J\)-classes in the direct product of two semigroups. (English) Zbl 1038.20043
Let \(S_1\) and \(S_2\) be semigroups, \(a\in S_1\), \(b\in S_2\). The author applies the following notation: \(J(a)\) is the principal two-sided ideal in \(S_1\) generated by \(a\), \(J(b)\) is the principal two-sided ideal in \(S_2\) generated by \(b\), \(J(a,b)\) is the principal two-sided ideal in \(S_1\times S_2\) generated by \((a,b)\). Further, \(J_a\) is the \(\mathcal J\)-class containing the element \(a\) in \(S_1\), \(J_b\) is the \(\mathcal J\)-class containing the element \(b\) in \(S_2\), \(J_{(a,b)}\) is the \(\mathcal J\)-class containing the element \((a,b)\) in \(S_1\times S_2\).
It is proved that the implication \(J(a,b)=J(a)\times J(b)\Rightarrow J_{(a,b)}=J_a\times J_b\) is valid and that the converse implication does not hold in general.

MSC:
20M12 Ideal theory for semigroups
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References:
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