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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:

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

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|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" />

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

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/o110030b.gif" />

Figure: o110030b

and is described as follows:

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/o110030c.gif" />

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|universal algebra]], it is possible to describe completely the lattice of subvarieties of  $  \mathbf K _ {1,1 }  $.

====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>