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A bounded [[Distributive lattice|distributive lattice]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o1100301.png" /> together with a dual lattice endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o1100302.png" />, i.e., a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o1100303.png" /> such that the [[de Morgan laws]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o1100304.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o1100305.png" /> hold for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o1100306.png" />. The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o1100307.png" /> of Ockham algebras is equational (i.e., is a variety; cf. also [[Algebraic systems, variety of|Algebraic systems, variety of]]). The Berman class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o1100308.png" /> is the subclass obtained by imposing on the dual endomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o1100309.png" /> the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003010.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003012.png" />). The Berman classes are related as follows:
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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/o/o110/o110030/o11003013.png" /></td> </tr></table>
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The smallest Berman class is therefore the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003014.png" /> described by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003015.png" /> and is the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003016.png" /> of de Morgan algebras. Perhaps the most important Berman class is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003017.png" />, described by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003018.png" />. This can be characterized as the class of Ockham algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003019.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003020.png" />. It contains also the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003023.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003024.png" />, and, in particular, the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003025.png" /> of Stone algebras (add the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003026.png" />).
+
A bounded [[Distributive lattice|distributive lattice]]  $  ( L; \wedge, \lor,0,1 ) $
 +
together with a dual lattice endomorphism  $  f $,
 +
i.e., a mapping  $  f : L \rightarrow L $
 +
such that the [[de Morgan laws]]  $  f ( x \lor y ) = f ( x ) \wedge f ( y ) $
 +
and  $  f ( x \lor y ) = f ( x ) \wedge f ( y ) $
 +
hold for all  $  x,y \in L $.  
 +
The class $  \mathbf O $
 +
of Ockham algebras is equational (i.e., is a variety; cf. also [[Algebraic systems, variety of|Algebraic systems, variety of]]). The Berman class  $  \mathbf K _ {p,q }  $
 +
is the subclass obtained by imposing on the dual endomorphism  $  f $
 +
the restriction  $  f ^ {2p+q } = f  ^ {q} $(
 +
$  p \geq  1 $,
 +
$  q \geq  0 $).  
 +
The Berman classes are related as follows:
  
An Ockham algebra congruence is an equivalence relation that has the substitution property for both the lattice operations and the [[unary operation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003027.png" />. A basic congruence is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003028.png" />, defined by
+
$$
 +
\mathbf K _ {p,q }  \subseteq \mathbf K _ {p  ^  \prime  ,q  ^  \prime  }  \iff  p \mid  p  ^  \prime  ,  q \leq  q  ^  \prime  .
 +
$$
  
<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/o/o110/o110030/o11003029.png" /></td> </tr></table>
+
The smallest Berman class is therefore the class  $  \mathbf K _ {1,0 }  $
 +
described by the equation  $  f  ^ {2} = { \mathop{\rm id} } $
 +
and is the class  $  \mathbf M $
 +
of de Morgan algebras. Perhaps the most important Berman class is  $  \mathbf K _ {1,1 }  $,
 +
described by  $  f  ^ {3} = f $.
 +
This can be characterized as the class of Ockham algebras  $  L $
 +
such that  $  ( f ( L ) ;f ) \in \mathbf M $.
 +
It contains also the class  $  \mathbf M \mathbf S $
 +
of  $  MS $-
 +
algebras  $  ( { \mathop{\rm id} } \leq  f  ^ {2} ) $,
 +
and, in particular, the class of  $  \mathbf S $
 +
of Stone algebras (add the relation  $  x \wedge f ( x ) = 0 $).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003030.png" />, then, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003033.png" /> indicates an isomorphism when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003034.png" /> is even and a dual isomorphism when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003035.png" /> is odd.
+
An Ockham algebra congruence is an equivalence relation that has the substitution property for both the lattice operations and the [[unary operation]]  $  f $.  
 +
A basic congruence is $  \Phi _ {n} $,
 +
defined by
  
An Ockham algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003036.png" /> is subdirectly irreducible if it has a smallest non-trivial congruence. Every Berman class contains only finitely many subdirectly irreducible algebras, each of which is finite.
+
$$
 +
( x,y ) \in \Phi _ {n}  \iff  f  ^ {n} ( x ) = f  ^ {n} ( y ) .
 +
$$
  
The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003037.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003038.png" /> is given by
+
If  $  ( L;f ) \in \mathbf K _ {p,q }  $,
 +
then, for  $  n \leq  q $,
 +
$  L/ \Phi _ {n} \sim f  ^ {n} ( L ) \in \mathbf K _ {p,q - n }  $,
 +
where  $  \sim $
 +
indicates an isomorphism when  $  n $
 +
is even and a dual isomorphism when  $  n $
 +
is odd.
  
<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/o/o110/o110030/o11003039.png" /></td> </tr></table>
+
An Ockham algebra  $  ( L;f ) $
 +
is subdirectly irreducible if it has a smallest non-trivial congruence. Every Berman class contains only finitely many subdirectly irreducible algebras, each of which is finite.
  
it is a locally finite generalized variety that contains all of the Berman classes. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003040.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003041.png" /> is subdirectly irreducible if and only if the lattice of congruences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003042.png" /> reduces to the chain
+
The class  $  \mathbf K _  \omega  $
 +
of $  \mathbf O $
 +
is given by
  
<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/o/o110/o110030/o11003043.png" /></td> </tr></table>
+
$$
 +
( L;f ) \in \mathbf K _  \omega  \iff  ( \forall x ) ( \exists m \neq 0,n )  f ^ {m + n } ( x ) = f  ^ {n} ( x ) ;
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003044.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003045.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003046.png" />.
+
it is a locally finite generalized variety that contains all of the Berman classes. If $  ( L;f ) \in \mathbf K _  \omega  $,  
 +
then $  L $
 +
is subdirectly irreducible if and only if the lattice of congruences of  $  L $
 +
reduces to the chain
  
Ockham algebras can also be obtained by topological duality. Recall that a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003047.png" /> in a [[Partially ordered set|partially ordered set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003048.png" /> is called a down-set if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003050.png" />, implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003051.png" />. Dually, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003052.png" /> is called an up-set if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003054.png" />, implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003055.png" />. An ordered [[Topological space|topological space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003056.png" /> (cf. also [[Order (on a set)|Order (on a set)]]) is said to be totally order-disconnected if, whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003057.png" />, there exists a closed-and-open down-set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003058.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003060.png" />. A Priestley space is a compact totally order-disconnected space. An Ockham space is a Priestley space endowed with a continuous order-reversing mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003061.png" />. The important connection with Ockham algebras was established by A. Urquhart and is as follows. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003062.png" /> is an Ockham space and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003063.png" /> denotes the family of closed-and-open down-sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003064.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003065.png" /> is an Ockham algebra, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003066.png" /> is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003067.png" />. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003068.png" /> is an Ockham algebra and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003069.png" /> denotes the set of prime ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003070.png" />, then, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003071.png" /> is equipped with the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003072.png" /> which has as base the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003074.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003076.png" /> is an Ockham space, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003077.png" />. Moreover, these constructions give a dual categorical equivalence. In the finite case the topology "evaporates" ; the dual space of a finite Ockham algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003078.png" /> consists of the ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003079.png" /> of join-irreducible elements together with the order-reversing mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003080.png" />.
+
$$
 +
\omega = \Phi _ {0} \prec \Phi _ {1} \prec \dots \prec \Phi _  \omega \prec \iota
 +
$$
  
Duality produces further classes of Ockham algebras. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003081.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003082.png" /> be the subclass of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003083.png" /> formed by the algebras whose dual space satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003084.png" />. Then every Berman class is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003085.png" />; more precisely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003086.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003087.png" /> is the dual space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003088.png" />, let, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003090.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003091.png" /> is finite, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003092.png" /> is subdirectly irreducible if and only if there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003093.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003094.png" />. The dual space of a subdirectly irreducible Ockham algebra in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003095.png" /> can therefore be represented as follows (here the order is ignored and the arrows indicate the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003096.png" />):
+
where  $  \Phi _  \omega  = \cup _ {i \geq  0 }  \Phi _ {i} $.
 +
If  $  L \in \mathbf K _ {p,q }  $,
 +
then  $  \Phi _  \omega  = \Phi _ {q} $.
 +
 
 +
Ockham algebras can also be obtained by topological duality. Recall that a set  $  D $
 +
in a [[Partially ordered set|partially ordered set]]  $  P $
 +
is called a down-set if  $  a \leq  b $,
 +
$  b \in D $,
 +
implies  $  a \in D $.  
 +
Dually,  $  U \subset  P $
 +
is called an up-set if  $  a \leq  b $,
 +
$  b \in U $,
 +
implies  $  b \in U $.  
 +
An ordered [[Topological space|topological space]]  $  ( X; \tau, \leq  ) $(
 +
cf. also [[Order (on a set)|Order (on a set)]]) is said to be totally order-disconnected if, whenever  $  x \Nle y $,  
 +
there exists a closed-and-open down-set  $  U $
 +
such that  $  y \in U $
 +
and  $  x \notin U $.
 +
A Priestley space is a compact totally order-disconnected space. An Ockham space is a Priestley space endowed with a continuous order-reversing mapping  $  g $.  
 +
The important connection with Ockham algebras was established by A. Urquhart and is as follows. If  $  ( X;g ) $
 +
is an Ockham space and if  $  {\mathcal O} ( X ) $
 +
denotes the family of closed-and-open down-sets of $  X $,
 +
then  $  ( {\mathcal O} ( X ) ;f ) $
 +
is an Ockham algebra, where  $  f $
 +
is given by  $  f ( A ) = X \setminus  g ^ {- 1 } ( A ) $.
 +
Conversely, if  $  ( L;f ) $
 +
is an Ockham algebra and if  $  I _ {p} ( L ) $
 +
denotes the set of prime ideals of  $  L $,
 +
then, if  $  I _ {p} ( L ) $
 +
is equipped with the topology  $  \tau $
 +
which has as base the sets  $  \{ {x \in I _ {p} ( L ) } : {x \ni a } \} $
 +
and  $  \{ {x \in I _ {p} ( L ) } : {x \Nso a } \} $
 +
for every  $  a \in L $,
 +
$  ( I _ {p} ( L ) ;g ) $
 +
is an Ockham space, where  $  g ( x ) = \{ {a \in L } : {f ( a ) \notin x } \} $.
 +
Moreover, these constructions give a dual categorical equivalence. In the finite case the topology  "evaporates" ; the dual space of a finite Ockham algebra  $  L $
 +
consists of the ordered set  $  I $
 +
of join-irreducible elements together with the order-reversing mapping  $  g $.
 +
 
 +
Duality produces further classes of Ockham algebras. For  $  m > n \geq  0 $,
 +
let  $  \mathbf P _ {m,n }  $
 +
be the subclass of  $  \mathbf O $
 +
formed by the algebras whose dual space satisfies $  g  ^ {m} = g  ^ {n} $.  
 +
Then every Berman class is a $  \mathbf P _ {m,n }  $;  
 +
more precisely, $  \mathbf K _ {p,q }  = \mathbf P _ {2p + q,q }  $.  
 +
If $  ( X;g ) $
 +
is the dual space of $  ( L;f ) $,  
 +
let, for every $  x \in X $,  
 +
$  g  ^  \omega  \{ x \} = \{ {g  ^ {n} ( x ) } : {n \in \mathbf N } \} $.  
 +
If $  ( L;f ) \in \mathbf O $
 +
is finite, then $  ( L;f ) $
 +
is subdirectly irreducible if and only if there exists an $  x \in X $
 +
such that $  g  ^  \omega  \{ x \} = X $.  
 +
The dual space of a subdirectly irreducible Ockham algebra in $  \mathbf P _ {m,n }  $
 +
can therefore be represented as follows (here the order is ignored and the arrows indicate the action of $  g $):
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/o110030a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/o110030a.gif" />
Line 31: Line 138:
 
Figure: o110030a
 
Figure: o110030a
  
The subdirectly irreducible Ockham algebra that corresponds to this discretely ordered space is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003097.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003098.png" /> is the algebra whose dual space is
+
The subdirectly irreducible Ockham algebra that corresponds to this discretely ordered space is denoted by $  L _ {m,n }  $.  
 +
In particular, $  ( L _ {3,1 }  ;f ) $
 +
is the algebra whose dual space is
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/o110030b.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/o110030b.gif" />
Line 43: Line 152:
 
Figure: o110030c
 
Figure: o110030c
  
The subdirectly irreducible algebras in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o11003099.png" /> are the nineteen subalgebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o110030100.png" />. Using a standard theorem of B.A. Davey from [[Universal algebra|universal algebra]], it is possible to describe completely the lattice of subvarieties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110030/o110030101.png" />.
+
The subdirectly irreducible algebras in $  \mathbf K _ {1,1 }  = \mathbf P _ {3,1 }  $
 +
are the nineteen subalgebras of $  ( L _ {3,1 }  ; f ) $.  
 +
Using a standard theorem of B.A. Davey from [[Universal algebra|universal algebra]], it is possible to describe completely the lattice of subvarieties of $  \mathbf K _ {1,1 }  $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.S. Blyth,  J.C. Varlet,  "Ockham algebras" , Oxford Univ. Press  (1994)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Berman,  "Distributive lattices with an additional unary operation"  ''Aequationes Math.'' , '''16'''  (1977)  pp. 165–171</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.A. Priestley,  "Ordered sets and duality for distributive lattices"  ''Ann. Discrete Math.'' , '''23'''  (1984)  pp. 39–60</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Urquhart,  "Lattices with a dual homomorphic operation"  ''Studia Logica'' , '''38'''  (1979)  pp. 201–209</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B.A. Davey,  "On the lattice of subvarieties"  ''Houston J. Math.'' , '''5'''  (1979)  pp. 183–192</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.S. Blyth,  J.C. Varlet,  "Ockham algebras" , Oxford Univ. Press  (1994)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Berman,  "Distributive lattices with an additional unary operation"  ''Aequationes Math.'' , '''16'''  (1977)  pp. 165–171</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.A. Priestley,  "Ordered sets and duality for distributive lattices"  ''Ann. Discrete Math.'' , '''23'''  (1984)  pp. 39–60</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Urquhart,  "Lattices with a dual homomorphic operation"  ''Studia Logica'' , '''38'''  (1979)  pp. 201–209</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B.A. Davey,  "On the lattice of subvarieties"  ''Houston J. Math.'' , '''5'''  (1979)  pp. 183–192</TD></TR></table>

Latest revision as of 08:03, 6 June 2020


A bounded distributive lattice $ ( L; \wedge, \lor,0,1 ) $ together with a dual lattice endomorphism $ f $, i.e., a mapping $ f : L \rightarrow L $ such that the de Morgan laws $ f ( x \lor y ) = f ( x ) \wedge f ( y ) $ and $ f ( x \lor y ) = f ( x ) \wedge f ( y ) $ hold for all $ x,y \in L $. The class $ \mathbf O $ of Ockham algebras is equational (i.e., is a variety; cf. also Algebraic systems, variety of). The Berman class $ \mathbf K _ {p,q } $ is the subclass obtained by imposing on the dual endomorphism $ f $ the restriction $ f ^ {2p+q } = f ^ {q} $( $ p \geq 1 $, $ q \geq 0 $). The Berman classes are related as follows:

$$ \mathbf K _ {p,q } \subseteq \mathbf K _ {p ^ \prime ,q ^ \prime } \iff p \mid p ^ \prime , q \leq q ^ \prime . $$

The smallest Berman class is therefore the class $ \mathbf K _ {1,0 } $ described by the equation $ f ^ {2} = { \mathop{\rm id} } $ and is the class $ \mathbf M $ of de Morgan algebras. Perhaps the most important Berman class is $ \mathbf K _ {1,1 } $, described by $ f ^ {3} = f $. This can be characterized as the class of Ockham algebras $ L $ such that $ ( f ( L ) ;f ) \in \mathbf M $. It contains also the class $ \mathbf M \mathbf S $ of $ MS $- algebras $ ( { \mathop{\rm id} } \leq f ^ {2} ) $, and, in particular, the class of $ \mathbf S $ of Stone algebras (add the relation $ x \wedge f ( x ) = 0 $).

An Ockham algebra congruence is an equivalence relation that has the substitution property for both the lattice operations and the unary operation $ f $. A basic congruence is $ \Phi _ {n} $, defined by

$$ ( x,y ) \in \Phi _ {n} \iff f ^ {n} ( x ) = f ^ {n} ( y ) . $$

If $ ( L;f ) \in \mathbf K _ {p,q } $, then, for $ n \leq q $, $ L/ \Phi _ {n} \sim f ^ {n} ( L ) \in \mathbf K _ {p,q - n } $, where $ \sim $ indicates an isomorphism when $ n $ is even and a dual isomorphism when $ n $ is odd.

An Ockham algebra $ ( L;f ) $ is subdirectly irreducible if it has a smallest non-trivial congruence. Every Berman class contains only finitely many subdirectly irreducible algebras, each of which is finite.

The class $ \mathbf K _ \omega $ of $ \mathbf O $ is given by

$$ ( L;f ) \in \mathbf K _ \omega \iff ( \forall x ) ( \exists m \neq 0,n ) f ^ {m + n } ( x ) = f ^ {n} ( x ) ; $$

it is a locally finite generalized variety that contains all of the Berman classes. If $ ( L;f ) \in \mathbf K _ \omega $, then $ L $ is subdirectly irreducible if and only if the lattice of congruences of $ L $ reduces to the chain

$$ \omega = \Phi _ {0} \prec \Phi _ {1} \prec \dots \prec \Phi _ \omega \prec \iota $$

where $ \Phi _ \omega = \cup _ {i \geq 0 } \Phi _ {i} $. If $ L \in \mathbf K _ {p,q } $, then $ \Phi _ \omega = \Phi _ {q} $.

Ockham algebras can also be obtained by topological duality. Recall that a set $ D $ in a partially ordered set $ P $ is called a down-set if $ a \leq b $, $ b \in D $, implies $ a \in D $. Dually, $ U \subset P $ is called an up-set if $ a \leq b $, $ b \in U $, implies $ b \in U $. An ordered topological space $ ( X; \tau, \leq ) $( cf. also Order (on a set)) is said to be totally order-disconnected if, whenever $ x \Nle y $, there exists a closed-and-open down-set $ U $ such that $ y \in U $ and $ x \notin U $. A Priestley space is a compact totally order-disconnected space. An Ockham space is a Priestley space endowed with a continuous order-reversing mapping $ g $. The important connection with Ockham algebras was established by A. Urquhart and is as follows. If $ ( X;g ) $ is an Ockham space and if $ {\mathcal O} ( X ) $ denotes the family of closed-and-open down-sets of $ X $, then $ ( {\mathcal O} ( X ) ;f ) $ is an Ockham algebra, where $ f $ is given by $ f ( A ) = X \setminus g ^ {- 1 } ( A ) $. Conversely, if $ ( L;f ) $ is an Ockham algebra and if $ I _ {p} ( L ) $ denotes the set of prime ideals of $ L $, then, if $ I _ {p} ( L ) $ is equipped with the topology $ \tau $ which has as base the sets $ \{ {x \in I _ {p} ( L ) } : {x \ni a } \} $ and $ \{ {x \in I _ {p} ( L ) } : {x \Nso a } \} $ for every $ a \in L $, $ ( I _ {p} ( L ) ;g ) $ is an Ockham space, where $ g ( x ) = \{ {a \in L } : {f ( a ) \notin x } \} $. Moreover, these constructions give a dual categorical equivalence. In the finite case the topology "evaporates" ; the dual space of a finite Ockham algebra $ L $ consists of the ordered set $ I $ of join-irreducible elements together with the order-reversing mapping $ g $.

Duality produces further classes of Ockham algebras. For $ m > n \geq 0 $, let $ \mathbf P _ {m,n } $ be the subclass of $ \mathbf O $ formed by the algebras whose dual space satisfies $ g ^ {m} = g ^ {n} $. Then every Berman class is a $ \mathbf P _ {m,n } $; more precisely, $ \mathbf K _ {p,q } = \mathbf P _ {2p + q,q } $. If $ ( X;g ) $ is the dual space of $ ( L;f ) $, let, for every $ x \in X $, $ g ^ \omega \{ x \} = \{ {g ^ {n} ( x ) } : {n \in \mathbf N } \} $. If $ ( L;f ) \in \mathbf O $ is finite, then $ ( L;f ) $ is subdirectly irreducible if and only if there exists an $ x \in X $ such that $ g ^ \omega \{ x \} = X $. The dual space of a subdirectly irreducible Ockham algebra in $ \mathbf P _ {m,n } $ can therefore be represented as follows (here the order is ignored and the arrows indicate the action of $ g $):

Figure: o110030a

The subdirectly irreducible Ockham algebra that corresponds to this discretely ordered space is denoted by $ L _ {m,n } $. In particular, $ ( L _ {3,1 } ;f ) $ is the algebra whose dual space is

Figure: o110030b

and is described as follows:

Figure: o110030c

The subdirectly irreducible algebras in $ \mathbf K _ {1,1 } = \mathbf P _ {3,1 } $ are the nineteen subalgebras of $ ( L _ {3,1 } ; f ) $. Using a standard theorem of B.A. Davey from universal algebra, it is possible to describe completely the lattice of subvarieties of $ \mathbf K _ {1,1 } $.

References

[a1] T.S. Blyth, J.C. Varlet, "Ockham algebras" , Oxford Univ. Press (1994)
[a2] J. Berman, "Distributive lattices with an additional unary operation" Aequationes Math. , 16 (1977) pp. 165–171
[a3] H.A. Priestley, "Ordered sets and duality for distributive lattices" Ann. Discrete Math. , 23 (1984) pp. 39–60
[a4] A. Urquhart, "Lattices with a dual homomorphic operation" Studia Logica , 38 (1979) pp. 201–209
[a5] B.A. Davey, "On the lattice of subvarieties" Houston J. Math. , 5 (1979) pp. 183–192
How to Cite This Entry:
Ockham algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ockham_algebra&oldid=48038
This article was adapted from an original article by T.S. Blyth (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article