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Keywords:
principal left ideal; direct product; direct product of two semigroups
principal left ideal; direct product; direct product of two semigroups
Summary:
A necessary and sufficient condition is given for a) a principal left ideal $L(s,t)$ in $S\times T$ to be equal to the direct product of the corresponding principal left ideals $L(s)\times L(t)$, b) an $\Cal L$-class $L_{(s,t)}$ to be equal to the direct product of the corresponding $\Cal L$-classes $L_s\times L_t$.
References:
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[3] Fabrici I.: On semiprime ideals in the direct product of semigroups. Mat. časopis 18 (1968), 201-203. MR 0237674
[4] Ivan J.: On the direct product of semigroups. Mat.-fyz. časopis (1953), 57-66. MR 0062733
[5] Petrich M.: Prime ideals in the cartesian product of two semigroups. Czechoslov. Math. J. 12 (1962), 150-152. MR 0140597
[6] Petrich M.: Introduction to semigroups. Charles E. Merrill Publishing CO. A Bell and Howell Company, Ohio. MR 0393206 | Zbl 0321.20037
[7] Plemmons R.: Maximal ideals in the direct product of two semigroups. Czechoslov. Math. J. 17 (1967), 257-260. MR 0214681 | Zbl 0189.02001
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