As is known (see [5, 6]), idempotent elements of semigroups play an important role in the investigation of semigroups themselves. Moreover, it is known that any semigroup can be isomorphieally embedded into some semigroup of binary relations on some nonempty set X. This has generated great interest in the investigation of idempotent binary relations. Different authors used different approaches to study the construction of these elements and obtained different answers to the question under consideration (see, e.g., [1-3, 7, 8]). In this work, first, we characterize the complete X-semilattices of unions D for which there exist idempotent binary relations such that the set of all their cuts to the elements of D coincides with the given semilattice. Then we give a description of the structure of these idempotent binary relations in the language of limiting elements of some subset of the semilattice D (see Lemma 1 and Theorem 5) and show how we can find all idempotents of the complete group of binary relations B_X(D) defined by the complete X-semilattice of unions D (see Theorem 6 and Corollary 4). Moreover, with the help of chains of the semilattice D, we give a rule for constructing the semigroups of idempotent elements of the semigroup Bx(D) (see Theorem 7). In Theorem 9, we give a description of the structure of all regular elements of the semigroup B_X(D).
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