Ockham algebra

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