Geodesic
Irreducible generating sets of complete semigroups of unions $B_X(S)$ defined by semilattices of the class $\Sigma_1(X,4)$
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, Tome 177 (2020), pp. 69-73
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In complete semigroups of unions $B_X(D)$ defined by semilattices of the class $\Sigma_1(X,4)$, we describe the set of all external elements and show that it is a generating (and, therefore, irreducible) set of the semigroup $B_X(D)$. For a finite semigroup $B_X(D)$, we give a formula for calculating the number of elements of the generating set.
Keywords: semilattice of unions, complete semigroup of binary relations, generating set, quasinormal representation of binary relations. 
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     author = {O. Givradze},
     title = {Irreducible generating sets of complete semigroups of unions $B_X(S)$ defined by semilattices of the class $\Sigma_1(X,4)$},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {69--73},
     year = {2020},
     volume = {177},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2020_177_a6/}
}
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O. Givradze. Irreducible generating sets of complete semigroups of unions $B_X(S)$ defined by semilattices of the class $\Sigma_1(X,4)$. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, Tome 177 (2020), pp. 69-73. http://geodesic.mathdoc.fr/item/INTO_2020_177_a6/

[1] Givradze O., “Irreducible generating sets of complete semigroups of unions $B_X(D)$ defined by semilattices of the class $\Sigma_2(X,4)$”, J. Math. Sci. (N.Y.)., 186:5 (2012), 745–750 | DOI | MR | Zbl

[2] Givradze O., “The number of equivalences on a finite set”, Proc. A. Razmadze Math. Inst., 131 (2003), 121–122 | Zbl