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A pseudo-complemented [[Distributive lattice|distributive lattice]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s0903501.png" /> (see [[Lattice with complements|Lattice with complements]]) in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s0903502.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s0903503.png" />. A pseudo-complemented distributive lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s0903504.png" /> is a Stone lattice if and only if the join of any two of its minimal prime ideals is the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s0903505.png" /> (the Grätzer–Schmidt theorem, [[#References|[3]]]).
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A Stone lattice, considered as a [[Universal algebra|universal algebra]] with the basic operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s0903506.png" />, is called a Stone algebra. Every Stone algebra is a [[Subdirect product|subdirect product]] of two-element and three-element chains. In a pseudo-complemented lattice, an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s0903507.png" /> is said to be dense if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s0903508.png" />. The centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s0903509.png" /> of a Stone lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s09035010.png" /> (cf. [[Centre of a partially ordered set|Centre of a partially ordered set]]) is a [[Boolean algebra|Boolean algebra]], while the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s09035011.png" /> of all its dense elements is a distributive lattice with a unit. Moreover, there is a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s09035012.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s09035013.png" /> into the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s09035014.png" /> of filters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s09035015.png" />, defined by
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s09035016.png" /></td> </tr></table>
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A pseudo-complemented [[Distributive lattice|distributive lattice]]  $  L $(
 +
see [[Lattice with complements|Lattice with complements]]) in which  $  a  ^  \star  + a  ^ {\star\star} = 1 $
 +
for all  $  a \in L $.
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A pseudo-complemented distributive lattice  $  L $
 +
is a Stone lattice if and only if the join of any two of its minimal prime ideals is the whole of  $  L $(
 +
the Grätzer–Schmidt theorem, [[#References|[3]]]).
 +
 
 +
A Stone lattice, considered as a [[Universal algebra|universal algebra]] with the basic operations  $  \langle \lor , \wedge , {}  ^  \star  , 0, 1\rangle $,
 +
is called a Stone algebra. Every Stone algebra is a [[Subdirect product|subdirect product]] of two-element and three-element chains. In a pseudo-complemented lattice, an element  $  x $
 +
is said to be dense if  $  x  ^  \star  = 0 $.  
 +
The centre  $  C( L) $
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of a Stone lattice  $  L $(
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cf. [[Centre of a partially ordered set|Centre of a partially ordered set]]) is a [[Boolean algebra|Boolean algebra]], while the set  $  D( L) $
 +
of all its dense elements is a distributive lattice with a unit. Moreover, there is a homomorphism  $  \phi  ^ {L} $
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from  $  C( L) $
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into the lattice  $  F( D( L)) $
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of filters of  $  D( L) $,
 +
defined by
 +
 
 +
$$
 +
a \phi  ^ {L}  = \
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\{ {x } : {x \in D( L), x \geq  a  ^  \star  } \}
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,
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$$
  
 
which preserves 0 and 1.
 
which preserves 0 and 1.
  
The triplet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s09035017.png" /> is said to be associated with the Stone algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s09035018.png" />. Homomorphisms and isomorphisms of triplets are defined naturally. Any triplet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s09035019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s09035020.png" /> is a Boolean algebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s09035021.png" /> is a distributive lattice with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s09035022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s09035023.png" /> is a homomorphism preserving 0 and 1, is isomorphic to the triplet associated with some Stone algebra. Stone algebras are isomorphic if and only if their associated triplets are isomorphic (the Chen–Grätzer theorem, [[#References|[2]]]).
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The triplet $  \langle  C( L), D( L), \phi  ^ {L} \rangle $
 +
is said to be associated with the Stone algebra $  L $.  
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Homomorphisms and isomorphisms of triplets are defined naturally. Any triplet $  \langle  C, D, \phi \rangle $,  
 +
where $  C $
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is a Boolean algebra, $  D $
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is a distributive lattice with a $  1 $
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and $  \phi : C \rightarrow F( D) $
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is a homomorphism preserving 0 and 1, is isomorphic to the triplet associated with some Stone algebra. Stone algebras are isomorphic if and only if their associated triplets are isomorphic (the Chen–Grätzer theorem, [[#References|[2]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.C. Chen,  G. Grätzer,  "Stone lattices I-II"  ''Canad. J. Math.'' , '''21''' :  4  (1969)  pp. 884–903</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Grätzer,  E.T. Schmidt,  "On a problem of M.H. Stone"  ''Acta Math. Acad. Sci. Hung.'' , '''8''' :  3–4  (1957)  pp. 455–460</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  T.S. Fofanova,  "General theory of lattices" , ''Ordered sets and lattices'' , '''3''' , Saratov  (1975)  pp. 22–40  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.C. Chen,  G. Grätzer,  "Stone lattices I-II"  ''Canad. J. Math.'' , '''21''' :  4  (1969)  pp. 884–903</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Grätzer,  E.T. Schmidt,  "On a problem of M.H. Stone"  ''Acta Math. Acad. Sci. Hung.'' , '''8''' :  3–4  (1957)  pp. 455–460</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  T.S. Fofanova,  "General theory of lattices" , ''Ordered sets and lattices'' , '''3''' , Saratov  (1975)  pp. 22–40  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Stone lattices occur, in particular, as the open-set lattices of extremally-disconnected spaces (see [[Extremally-disconnected space|Extremally-disconnected space]]), and are so named in honour of M.H. Stone's investigation of such spaces [[#References|[a1]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s09035024.png" /> is the lattice of all open sets of a compact extremally-disconnected space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s09035025.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s09035026.png" /> is a complete Boolean algebra, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s09035027.png" /> is its [[Stone space|Stone space]]; thus, in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s09035028.png" /> is entirely determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090350/s09035029.png" />.
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Stone lattices occur, in particular, as the open-set lattices of extremally-disconnected spaces (see [[Extremally-disconnected space|Extremally-disconnected space]]), and are so named in honour of M.H. Stone's investigation of such spaces [[#References|[a1]]]. If $  L $
 +
is the lattice of all open sets of a compact extremally-disconnected space $  X $,  
 +
then $  C( L) $
 +
is a complete Boolean algebra, and $  X $
 +
is its [[Stone space|Stone space]]; thus, in this case $  L $
 +
is entirely determined by $  C( L) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.H. Stone,  "Algebraic characterization of special Boolean rings"  ''Fund. Math.'' , '''29'''  (1937)  pp. 223–303</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Grätzer,  "General lattice theory" , Birkhäuser  (1978)  (Original: Lattice theory. First concepts and distributive lattices. Freeman, 1978)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.H. Stone,  "Algebraic characterization of special Boolean rings"  ''Fund. Math.'' , '''29'''  (1937)  pp. 223–303</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Grätzer,  "General lattice theory" , Birkhäuser  (1978)  (Original: Lattice theory. First concepts and distributive lattices. Freeman, 1978)</TD></TR></table>

Latest revision as of 08:23, 6 June 2020


A pseudo-complemented distributive lattice $ L $( see Lattice with complements) in which $ a ^ \star + a ^ {\star\star} = 1 $ for all $ a \in L $. A pseudo-complemented distributive lattice $ L $ is a Stone lattice if and only if the join of any two of its minimal prime ideals is the whole of $ L $( the Grätzer–Schmidt theorem, [3]).

A Stone lattice, considered as a universal algebra with the basic operations $ \langle \lor , \wedge , {} ^ \star , 0, 1\rangle $, is called a Stone algebra. Every Stone algebra is a subdirect product of two-element and three-element chains. In a pseudo-complemented lattice, an element $ x $ is said to be dense if $ x ^ \star = 0 $. The centre $ C( L) $ of a Stone lattice $ L $( cf. Centre of a partially ordered set) is a Boolean algebra, while the set $ D( L) $ of all its dense elements is a distributive lattice with a unit. Moreover, there is a homomorphism $ \phi ^ {L} $ from $ C( L) $ into the lattice $ F( D( L)) $ of filters of $ D( L) $, defined by

$$ a \phi ^ {L} = \ \{ {x } : {x \in D( L), x \geq a ^ \star } \} , $$

which preserves 0 and 1.

The triplet $ \langle C( L), D( L), \phi ^ {L} \rangle $ is said to be associated with the Stone algebra $ L $. Homomorphisms and isomorphisms of triplets are defined naturally. Any triplet $ \langle C, D, \phi \rangle $, where $ C $ is a Boolean algebra, $ D $ is a distributive lattice with a $ 1 $ and $ \phi : C \rightarrow F( D) $ is a homomorphism preserving 0 and 1, is isomorphic to the triplet associated with some Stone algebra. Stone algebras are isomorphic if and only if their associated triplets are isomorphic (the Chen–Grätzer theorem, [2]).

References

[1] G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973)
[2] C.C. Chen, G. Grätzer, "Stone lattices I-II" Canad. J. Math. , 21 : 4 (1969) pp. 884–903
[3] G. Grätzer, E.T. Schmidt, "On a problem of M.H. Stone" Acta Math. Acad. Sci. Hung. , 8 : 3–4 (1957) pp. 455–460
[4] T.S. Fofanova, "General theory of lattices" , Ordered sets and lattices , 3 , Saratov (1975) pp. 22–40 (In Russian)

Comments

Stone lattices occur, in particular, as the open-set lattices of extremally-disconnected spaces (see Extremally-disconnected space), and are so named in honour of M.H. Stone's investigation of such spaces [a1]. If $ L $ is the lattice of all open sets of a compact extremally-disconnected space $ X $, then $ C( L) $ is a complete Boolean algebra, and $ X $ is its Stone space; thus, in this case $ L $ is entirely determined by $ C( L) $.

References

[a1] M.H. Stone, "Algebraic characterization of special Boolean rings" Fund. Math. , 29 (1937) pp. 223–303
[a2] G. Grätzer, "General lattice theory" , Birkhäuser (1978) (Original: Lattice theory. First concepts and distributive lattices. Freeman, 1978)
How to Cite This Entry:
Stone lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stone_lattice&oldid=48866
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article