Abstract
In this paper we give a full description of idempotent elements of the semigroup BX (D), which are defined by semilattices of the class ∑1 (X, 10). For the case where X is a finite set we derive formulas by means of which we can calculate the numbers of idempotent elements of the respective semigroup.
Highlights
Let X be an arbitrary nonempty set, D be an X-semilattice of unions, i.e. such a nonempty set of subsets of the set X that is closed with respect to the set-theoretic operations of unification of elements from D, f be an arbitrary mapping of the set X in the set D
It is easy to prove that BX ( D) is a semigroup with respect to the operation of multiplication of binary relations, which is called a complete semigroup of binary relations defined by an X-semilattice of unions D
How to cite this paper: Makharadze, S., Aydın, N. and Erdoğan, A. (2015) Complete Semigroups of Binary Relations Defined by Semilattices of the Class Σ1 ( X,10)
Summary
Let X be an arbitrary nonempty set, D be an X-semilattice of unions, i.e. such a nonempty set of subsets of the set X that is closed with respect to the set-theoretic operations of unification of elements from D, f be an arbitrary mapping of the set X in the set D To each such a mapping f we put into correspondence a binary relation α f on the set X that satisfies the condition. A binary relation α having a quasinormal ( ) repres= entation α T∈V (D,α ) YTα ×T is an idempotent element of this semigroup iff a) V ( D,α ) is complete XI-semilattice of unions;.
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