\titleofpresentation{Complete Semigroups of Binary Relations}

\presenter{Ya.~I. Diasamidze}

This is joint work with Sh.~I. Makharadze.

%2000 Mathematics Subject  Classification:  20M10, 20M10

Let $X$  be an arbitrary nonempty set. $D$ be complete
$X-$semilattice of unions, i.e. it is some nonempty set of subsets
from $X$  which is closed with respect to the set-theoretic union
of elements from $D$, and $f$ be arbitrary mapping of the set $X$
in the set $D$. To each mapping $f$ we assign a binary relation
$\alpha_{f}$ on the set $X$ which satisfies conditions
\[\alpha_{f}=\bigcup_{x\in X}(\{x\}\times f(x)).\]


 We denoted by $B_{X}(D)$  the set of all such $\alpha_{f}\,\, (f: X \longrightarrow D)$.
  It is well-known that $B_{X}(D)$ is subsemigroup of the semigroup $B_{X}$
   which is called a complete semigroup of binary relations, defined by an  $X-$semilattice of
unions  $D$.

The report is dedicated to semigroups $B_{X}(D)$. The issue on
substantiation of researched semigroups and some of their
properties is studied. Some early well-known classes of
semigroups that proved to be complete semigroups of binary
relations are pointed out. Divisibility conditions of binary
relations are found out. Describtion of idempotent and regular
elements; right units; exterior and irreducible elements; maximal
subgroup of semigroups  $B_{X}(D)$ is given. Some of the
properties of finite  $X-$semilattice unions are studied.


Further, semigroups $B_{X}(D)$ that are defined by semilattices,
generated by sets of nonchainwise paires; $X-$chaines; elementary
semilattices; nodal and generalized-elementary  semilattices are
studied.


\begin{ourbiblio}{99}

\bibitem{diasamidze1} Diasamidze Ya., ``Complete semigroups of binary relations", J.
of Math. Sciences, Plenum Publ. Cor., New York, 117,  No 4, 2003,
4271-4319.

\bibitem{diasamidze2} Diasamidze Ya., Makharadze Sh., ``Complete semigroups of
binary relations, defined by elementary and nodal $X-$semilattices
of unions", Journal of Mathematical Sciences, Plenum Publ. Cor.,
New York , V. 111,  No 1, 2002, 3171-3226.
\bibitem{diasamitze3} Diasamidze Ya., Sirabidze T., ``Complete semigroups of
binary relations, defined by $3$ elemented  chains", J. of Math.
Sciences, Plenum Publ. Cor., New York, 117,  No 4, 2003,
4320-4350.
\bibitem{diasamitze4} Diasamidze Ya., Makharadze Sh., Fartenadze G.,
Givradze O., ``On finite $X-$semilattices of unions", Bull. Georg.
Acad. Sci., 169, No 2. 2004, 263-266.

\bibitem{diasamitze5} Diasamidze Ya., Makharadze Sh., Fartenadze G., Givradze O.,
``Description of  $\Sigma(X,m)$ Class ($m-$finite number)", Bull.
Georg. Acad. Sci., 169, No 3. 2004, 463-465.
\end{ourbiblio}

\presenteraddress{Batumi State University}  % The name of a university is fine.

\email{diasamidze_ya@mail.ru }
