
%  When you are satisfied with your abstract, please delete everything
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\titleofpresentation{Weakly Supplemented Lattices}

\presenter{S. Eylem Toksoy}  %  Name just one speaker, please; collaborators
            %  can be credited in the body of the abstract.

$L$ will mean a complete modular lattice with greatest element
1. An element $a$ in a lattice with zero is called an
\emph{atom} if $0\prec a$, i.e. there is no element $b\in L$ such
that $0< b <a$. A lattice $L$ is called \emph{semiatomic} if 1
is a join of atoms in $L$. An element $c$ of a complete
lattice $L$ is said to be \emph{compact} if for every subset
$X$ of $L$ with $c\leq \bigvee X$ there exists a finite
subset $F$ of $X$ such that $c\leq \bigvee F$. A lattice
$L$ is called a \emph{compact lattice} if 1 is compact and a
\emph{compactly generated lattice} if each of its elements is a join
of compact elements. We call an element $a$ of $L$ a
\emph{supplement} of an element $b$ if $a\vee b=1$ and $a$
is minimal with respect to this property. Minimality of $a$ is
equivalent to $a\wedge b\ll a/0$. Reducing this equivalent condition
to $a\wedge b\ll L$, we get the definition of \emph{weak
supplement}. $L$ is said to be \emph{supplemented
(\emph{respectively,} weakly supplemented)} if every element
$a$ of $L$ has a supplement (respectively, weak
supplement) in $L$. In a lattice $L$ the intersection of
all maximal ($\neq 1$) elements is called \emph{the radical} of
$L$, denoted by $r(L)$.


\begin{theorem}
If a compactly generated lattice $L$ is weakly supplemented
then $1/r(L)$ is semiatomic.
\end{theorem}

\begin{corollary}
  A compactly generated lattice $L$ with $r(L)=0$ is weakly
  supplemented if and only if it is semiatomic.
\end{corollary}

\begin{corollary}
Let $L$ be a compactly generated lattice with small radical. Then $L$ is
weakly supplemented if and only if $1/r(L)$
  is semiatomic.

\end{corollary}

\textbf{Joint work with:} Refail Alizade, \.{I}zmir Institute of
Technology

\email{rafailalizade@iyte.edu.tr}

\begin{ourbiblio}{99}  %  Delete this environment if you
               %  aren't giving a bibliography.

\bibitem{Toksoy1} Alizade, R., B\"{u}y\"{u}ka\c{s}{\i}k,
E.: \emph{Cofinitely supplemented modules}, Comm. Algebra, Vol. 31,
No. 11, pp. 5377-5390, (2003).

\bibitem{Toksoy2} B\"{u}y\"{u}ka\c{s}{\i}k, E.: \emph{Weakly cofinitely supplemented
modules}. Master's Thesis, DE\"{U} Fen Bilimleri Enstit\"{u}s\"{u},
\.{I}zmir, (2001).

\bibitem{Toksoy3} B\"{u}y\"{u}ka\c{s}{\i}k, E.: \emph{Weakly and cofinitely weak supplemented modules over
Dedekind rings}. Ph.D. Thesis, DE\"{U} Fen Bilimleri
Enstit\"{u}s\"{u}, \.{I}zmir, (2005).


\bibitem{Toksoy4} C\v{a}lug\v{a}reanu, G.: Lattice
Concepts of Module Theory, Kluwer Academic Publishers, Dordrecht,
Boston, London (2000).

\bibitem{Toksoy5} Clark, J., Lomp. C., Vanaja, N., Wisbauer, R.: Lifting Modules. Frontiers in
Mathematics, Birkh\"{a}user, (2006).

\bibitem{Toksoy6} Lomp, C.: \emph{On dual Goldie dimension}. Diplomarbeit (M.Sc.
Thesis), Mathematischen Institut der Heinrich-Heine Universit\"{a}t,
D\"{u}sseldorf. Revised version (2000).

\end{ourbiblio}

\presenteraddress{\.{I}zmir Institute of Technology}  % The name of a university is fine.

\email{eylemtoksoy@iyte.edu.tr}

\web{www.iyte.edu.tr/~eylemtoksoy}  % Those without homepages should delete this.

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