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''Fortuin–Kasteleyn–Ginibre inequality''

An inequality [[#References|[a3]]] that began a series of [[correlation inequalities]] for finite partially ordered sets. Let  $\Gamma$
be a finite [[partially ordered set]] ordered by  $\prec$ (irreflexive, transitive) with  $  ( \Gamma, \prec ) $
a [[distributive lattice]]:  $  a \lor b =  \min  \{ {z \in \Gamma } : {a \preceq z, b \preceq z } \} $, 
$  a \wedge b =  \max  \{ {z \in \Gamma } : {z \preceq a, z \preceq b } \} $, 
and  $  a \wedge ( b \lor c ) = ( a \wedge b ) \lor ( a \wedge c ) $
for all  $  a,b, c \in \Gamma $. 

Suppose  $  \mu : \Gamma \rightarrow {[ 0, \infty ) } $
is log supermodular:

$$ 
\mu ( a ) \mu ( b ) \leq  \mu ( a \lor b ) \mu ( a \wedge b )  \textrm{ for  all  }  a, b \in \Gamma,
$$

and that  $  f : \Gamma \rightarrow \mathbf R $
and  $  g : \Gamma \rightarrow \mathbf R $
are non-decreasing:

$$ 
a \prec b \Rightarrow \{ f ( a ) \leq  f ( b ) , g ( a ) \leq  g ( b ) \}  \textrm{ for  all  }  a,b \in \Gamma .
$$

The FKG inequality is:

$$ 
\left [ \sum _ {a \in \Gamma } \mu ( a ) f ( a ) \right ] \left [ \sum _ {a \in \Gamma } \mu ( a ) g ( a ) \right ] \leq  
\left [ \sum _ {a \in \Gamma } \mu ( a ) \right ] \left [ \sum _ {a \in \Gamma } \mu ( a ) f ( a ) g ( a ) \right ] .
$$

If  $  \Gamma $
is a [[Boolean algebra]] and  $  \mu $
is a [[probability measure]] on  $  \Gamma $, 
the inequality is  $  {\mathsf E} _  \mu  ( f ) {\mathsf E} _  \mu  ( g ) \leq  {\mathsf E} _  \mu  ( fg ) $, 
where  $  {\mathsf E} _  \mu  $
denotes [[mathematical expectation]].

Related inequalities are discussed in [[#References|[a1]]], [[#References|[a2]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a9]]].

See also [[Ahlswede–Daykin inequality]]; [[Fishburn–Shepp inequality|Fishburn–Shepp inequality]]; [[Holley inequality|Holley inequality]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Bollobás, "Combinatorics" , Cambridge Univ. Press  (1986)</TD></TR><TR>
<TD valign="top">[a2]</TD> <TD valign="top">  P.C. Fishburn, "Correlation in partially ordered sets"  ''Discrete Appl. Math.'' , '''39'''  (1992)  pp. 173–191</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  C.M. Fortuin, P.N. Kasteleyn, J. Ginibre, "Correlation inequalities for some partially ordered sets"  ''Comm. Math. Phys.'' , '''22'''  (1971)  pp. 89–103</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  R.L. Graham,   "Linear extensions of partial orders and the FKG inequality"  I. Rival (ed.) , ''Ordered sets'' , Reidel  (1982)  pp. 213–236</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  R.L. Graham,   "Applications of the FKG inequality and its relatives" , ''Proc. 12th Internat. Symp. Math. Programming'' , Springer  (1983)  pp. 115–131</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  R. Holley,   "Remarks on the FKG inequalities"  ''Comm. Math. Phys.'' , '''36'''  (1974)  pp. 227–231</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  K. Joag-Dev,   L.A. Shepp,   R.A. Vitale,   "Remarks and open problems in the area of the FKG inequality" , ''Inequalities Stat. Probab.'' , ''IMS Lecture Notes'' , '''5'''  (1984)  pp. 121–126</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L.A. Shepp,   "The XYZ conjecture and the FKG inequality"  ''Ann. of Probab.'' , '''10'''  (1982)  pp. 824–827</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  P M. Winkler,   "Correlation and order"  ''Contemp. Math.'' , '''57'''  (1986)  pp. 151–174</TD></TR>
</table>