A note on principal ideals and \(\mathcal J\)-classes in the direct product of two semigroups.

*(English)*Zbl 1038.20043Let \(S_1\) 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.

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.

Reviewer: Ján Jakubík (Košice)

##### MSC:

20M12 | Ideal theory for semigroups |

**OpenURL**

##### References:

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[3] | Fabrici I.: \(\mathcal J\)-classes in the direct product of two semigroups. Czechoslovak Math. J. 44 (1994), 325-335. · Zbl 0836.20086 |

[4] | Fabrici I.: One-sided principal ideals in the direct product of two semigroups. Math. Bohem. 4 (1993), 337-342. · Zbl 0810.20043 |

[5] | Tamura T.: Note on finite semigroups and determination of semigroups of order 4. J. Gakugei, Tokushima Univ. 5 (1954), 17-28. |

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