Principal ideals in the semigroup of binary relations on a finite set: What happens when one element is added to the set.

*(English)*Zbl 0763.20023As it is stated in the author’s abstract, the principal ideals in the semigroups of binary relations on a finite set \(X\) form a partially ordered set under inclusion. When the cardinality of the set \(X\) is increased by one, some of the covering relations present in the original set will be preserved and some will not. This article describes the difference. The main result in this paper discusses how the graph of the partially ordered set of ideals for an \(n\)-element set is embedded in the graph of the set of ideals for an \((n+1)\)-element set.

Reviewer: J.A.Hildebrant (Baton Rouge)

##### MSC:

20M20 | Semigroups of transformations, relations, partitions, etc. |

20M12 | Ideal theory for semigroups |

##### Keywords:

principal ideals; semigroups of binary relations; covering relations; partially ordered set of ideals##### References:

[1] | Kim, Ki Hang, ”Boolean Matrix Theory and Applications”, Marcel Dekker, New York, 1982. · Zbl 0495.15003 |

[2] | Plemmons, R. J. and M. T. West,On the Semigroup of Binary Relations, Pacific Journal of Mathematics35, no. 3, (1970), 743–753. · Zbl 0199.30602 |

[3] | Zaretskii, K. A.,The Semigroup of Binary Relations, Mat. Sbornik61, no. 3, (1963), 291–305 [in Russian]. |

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