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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:
 [1] Clifford A.H., Preston G.B.: The Algebraic Theory of Semigroups. Amer. Math. Soc., Providence I (1961). · Zbl 0111.03403 [2] Fabrici I.: Principal two-sided ideals in the direct product of two semigroups. Czechoslovak Math. J. 41 (1991), 411-421. · Zbl 0751.20043 [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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