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\title{Constructing distributive lattices with a given link lattice.}
\author{Joanna Grygiel\\
Institute of Mathematics and Computer Science\\
Pedagogical University of Czestochowa, Poland}
\date{ }
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\begin{document}
\newcommand{\je}{{\bf 1}}
\newcommand{\ze}{{\bf 0}}
\newcommand{\bi}{{\cal B}}
\newtheorem{tw}{Theorem}
\newtheorem{lm}[tw]{Lemma}
\newtheorem{pr}{Example}
\newtheorem{wn}{Corollary}
\maketitle


 Let $\langle K, \leq \rangle$ be a lattice and suppose that
for each $x$ from $K$ there is a lattice $\langle K_x, \leq _x \rangle $ such that
\begin{enumerate}
\item if $y$ covers $x$ (i.e. $x \leq y$ and there is no $z$ such that $x < z < y$)
in $\cal K$ then $K_x \cap K_y \not = \emptyset$;
\item if $x \leq y$ and $K_x \cap K_y \not = \emptyset$ then $K_x \cap K_y$ is a filter of $K_x$
and an ideal of $K_y$. Moreover, the orderings $\leq _x$ and $\leq _y$ are
identical on $K_x \cap K_y$;
\item $K_x \cap K_y \subseteq K_{x \wedge y} \cap K_{x \vee y},$ for all $x, y$
from $K$.
\end{enumerate}

Herrmann proved [3] that if the above conditions hold then $\langle \bigcup _{x \in K} K_x, \leq
\rangle$, where $\leq$ is a transitive closure of the sum of $\leq _x$ for $x \in K$,
is a lattice, called $K$-sum of the family $\{ K_x \}_{x \in K}$. We shall call
the lattice $K$ the link lattice of the lattice $\bigcup _{x \in K} K_x$.
Moreover, every modular lattice not containing infinite chains is the $K$-sum
of its atomistic intervals. In the distributive case it means that a
finite distributive lattice $\cal D$ is the $K$-sum of its all maximal Boolean
fragments. Thus, if $\cal D$ has a scarce decomposition [2] then $\cal D$ is the
Wro/nski sum [7] and the Herrmann $K$-sum of the same components.

It was also prooved by Herrmann [3] that every finite lattice is a link  lattice
of some finite distributive lattice. However the problem of dimensions of the maximal
Boolean fragments of the finite distributive lattice with a given link lattice
is still open. We are going to discuss some theorems concerning this problem.

Moreover, we can prove the following theorem:
\begin{tw}
Let $K$ be a finite lattice. If ${\cal B}_0 \oplus {\cal B}_1$ is the Wro/nski sum
of Boolean lattices $\bi _0$ and $\bi _1$ and there are mappings
$f_0$ and $f_1$ such that $f_0$ is a $\vee$-homomorphism
from $K$ into $B_0$, $f_1$ is a $\wedge$-homomorphism from $K$ into $B_1$ and
\[f_0(0) = 0_0,\;f_0(1) = 0_1,\;f_1(0) = 1_0,\;f_1(1) = 1_1\]
then $f_0$ and $f_1$ determine some distributive lattice with the link lattice $K$.
Moreover, if the unit of $K$ is the join of atoms of $K$ and the zero of $K$ is 
the meet of  coatoms of $K$ then the above construction determines all finite
distributive lattice with the link lattice $K$.
\end{tw}

\vspace{.5cm}

\begin{center}
{\bf References}
\end{center}
\vspace{1cm}
$[1]$ Gr\"atzer G., General Lattice Theory, Birkhauser Verlag, 1978.\\
$[2]$ Grygiel J., Wojtylak P., The uniqueness of the decomposition of
distributive lattices into sums of Boolean lattices., Reports on Mathematical Logic,
31, 1997, 93-102.\\
$[3]$ Herrmann Ch., S-verklebte Summen von Verb\"anden, Math. Z. 130(1973), 255-274.\\
$[4]$ Ja/skowski S., Recherches sur le systeme de la logique intuitioniste,
Actes du Congres International de Philosophe Scientifique, VI Philosophie
des Mathematiques, Actualites Scientifiqies et Industrielles, 393 (1936), 58-61.\\
$[5]$ Kotas J., Wojtylak P., Finite distributive lattices as sums of Boolean
algebras, Reports on Mathematical Logic, 29(1995), 35-40.\\
$[6]$ Troelstra A.S., On intermediate propositional logic, Indagationes
Mathematicae, 27(1965), 141-152.\\
$[7]$ Wro/nski A., Remarks on intermediate logics with axioms containing
only one variable, Reports on Mathematical Logic, 2(1974), 63-76.\\
\vspace{1cm}


e-mail j.grygiel@wsp.czest.pl
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