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\begin{center}{\huge 
Order-Polynomially Complete Lattices Must Be Inaccessible}\end{center}
\begin{center}{\large Martin Goldstern}\end{center}
\begin{center}{\large Technical University of Vienna and 
Free University of Berlin}\end{center}

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\large

Let us call a lattice $(L, \wedge, \vee)$ `order-polynomially
complete' (o.p.c.) if every weakly monotone function from $L^n$ to $L$
is induced by a lattice-theoretic polynomial. 

The question whether infinite o.p.c.\ lattices exist is open.  
The existence of such lattices is not provable in ZFC, since  
the cardinality of an infinite o.p.c.\ lattice must be a
strongly inaccessible cardinal. 

Although the problem originates in algebra, the proof is purely
set-theoretical.  The main tools are partition and canonisation
theorems. 

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          *****************

This is a joint work with Saharon Shelah.  
Preprint available from 
        http://www.math.fu-berlin.de/~goldstrn/large.dvi
 or 
        http://math.rutgers.edu/~shelah/all/633.dvi


In TeX, that would be: 
        http://www.math.fu-berlin.de/\char`\~goldstrn/large.dvi
 or 
        http://math.rutgers.edu/\char`\~shelah/all/633.dvi