Geodesic
Complete semigroups of binary relations defined by $X$-semilattices of unions
Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 8, pp. 73-99.

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This paper is devoted to some results from the book “Complete Semigroups of Binary Relations”, which is not published yet. Conditions of divisibility of binary relations, idempotents, regular elements, one-side units, and one-side zeros are described. Finite $X$-semilattices and unions are characterized. Methods of finding formulas for counting the number of idempotents and regular elements are given. 
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Ya. Diasamidze; Sh. Makharadze. Complete semigroups of binary relations defined by $X$-semilattices of unions. Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 8, pp. 73-99. http://geodesic.mathdoc.fr/item/FPM_2008_14_8_a3/

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