Ockham algebras thomas blyth varlet

An Ockbam algebra is a bounded distributive lattice with a dual phism, the nomenclature being chosen since the notion of de Morgan nega-tion has been attributed to the logician William o

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An Ockbam algebra is a bounded distributive lattice with a dual phism, the nomenclature being chosen since the notion of de Morgan nega-tion has been attributed to the logician William of Ockham (cI290-c1349).The class of such algebras is vast, containing in particular the well-knownclasses of boolean algebras, de Morgan algebras, Kleene algebras, and Stonealgebras Pioneering work by Berman in 1977 has shown the importance ofOckham algebras in general, and has since stimulated much research in thisarea, notably by Urquhart, Goldberg, Adams, Priestley, and Davey.

endomor-Here our objective is to provide a reasonably self-contained and readableaccount of some of this research Our collaboration began in 1982 in the con-sideration of a common abstraction of de Morgan algebras and Stone algebras

which we called MS-algebras This class of Ockham algebras is characterised

by the fact that the dual endomorphism] satisfiesjO~]2,which implies that

] =]3 The subvariety M of de Morgan algebras is characterised by]O=]2

In general, it seems an impossible task to describe all the subvarieties ofOckham algebras The subvarieties of paramount importance are those inwhich]q=]2p+qfor somep,q;these are denoted byKp,q and are called the

Berman varieties. Of these, the most significant seems to be K1 1in whicheach algebra L is such that](L)EM Here we concentrate particularly on

K1 ,1, its subvarieties, subdirectly irreducibles, and congruences

No study of Ockham algebras can be considered complete without tion of the theory of duality, in which the work of Priestley is fundamental

men-We make full use of Priestley duality in considering the subvariety K1,1'Chapters 0-11 deal entirely with Ockham algebras whereas Chapters 12-

15 are devoted to a brief study of double algebras More precisely, we

con-sider algebras (L;0,+)for which (L;0)is an MS-algebra and (L;+)is a dual algebra with the unary operations0 and+ linked by the identities a O + =aOO

MS-and a+ o =.a++. Particular subvarieties of double MS-algebras are those ofdouble Stone algebras and trivalent Lukasiewicz algebras

As we have written this text with beginning graduate students in mind,

we have included many illustrative examples and diagrams, as well as severaluseful tabulations We would add that we make no claim that the list ofreferences is complete, nor that what is due to Cxsar has not been attributed

to Brutus

T.S.B.,].c.v.

O. Ordered sets, lattices, and universal algebra 1

1. Examples of Ockham algebras; the Berman classes 8

10. The dual space of a finite simple Ockham algebra 164

and universal algebra

Itis of course impossible to give a full account of ordered sets, lattices, anduniversal algebra in a few pages, so we refer the reader to the various bookscited in the bibliography Nevertheless, in order to make this monograph rea-sonably self-contained, we shall summarise in this introductory chapter thefundamental notions that we shall use throughout More specific conceptsthat we shall require will be defined as necessary As far as notation is con-cerned, we provide an index of the various symbols that are used throughout

The concept of order plays in mathematics a very prominent role,

proba-bly as important as that of size, though its importance has only rather recentlybeen recognised Itis probably the success of the work of George Boole inthe first half of the last century that has acted as a catalyst in producing a newarea of research, namely that of ordered sets and, more particularly, lattices

Anordered set (or partially ordered set or poset) is a set S on which there

is defined a binary relation R which is reflexive (aRa for all a ES), transitive (for all a, bE S the relations aRb and bRc imply aRc), and anti-symmetric (for all a, bE S the relations aRb and bRa imply a =b). Mathematics is

replete with examples of such order relations; for example, the relation of

magnitude on the set of real numbers, the relation <:: of inclusion on thepower set IP(E} of any setE, the relation I of divisibility on the set INa ofstrictly positive integers, etc Usually, an order relation is denoted by :::;;and its converse by ;:: Two elements x, y of an ordered set are said to

be comparable (in symbols, x My)ifx :::;; y or y :::;; x, and incomparable (in symbols, x IIy) if xif;y and y if;x The (order) dual Sop of an ordered set

S is the same set equipped with the converse order We writex (yifx:::;; y and {z Ix <z <y} =0 If x ( y then we say that x is covered by y, or that y covers x The relation j is clearly an order

There are several useful ways in which disjoint ordered setsP,Qcan be

combined to produce a third ordered set In particular, the disjoint union

PUQconsists of the setPUQwith the order defined by

x:::;;y -¢=} (X,YEPwithx:::;;yinP}or(x,YEQwithx:::;;yinQ) The linear sum P EElQconsists of PUQwith the order

x:::;;y -¢=} (x,y EP with x:::;;y inP) or (x,yEQwith x :::;;y inQ)

or(xEP andxEQ).

Finally, the vertical sum PEBQ is defined only whenP has a biggest element

a andQhas a smallest elementb,and is obtained from PEBQby identifying

aandb.

IfA, B are ordered sets then a mappingf : A t B is said to be preserving or isotone if it is such that

order-(Vx,y EA) x:(;y =; f(x):(;f(y),

andorder-reversing or antitone ifit is such that

(Vx,yEA) x:(;y =; f(x)~f(Y).

A andBare (order-) isomorphic if there is a surjectionf : A tB such that

and dually (order-) isomorphicif there is a surjectionf : A tB such that

An ordered set that is dually isomorphic to itself is said to be self dual. Amappingf : A t A such thatf2 =idA is called an involution (of periodtwo) An order-reversing involution is called apolarity.

Very often (and this is so for the three examples mentioned above) anypair of elementsx,y of an ordered set have agreatest lower bound(ormeet,

orinfimum)which is denoted byx I\y; and aleast upper bound(orjoin,or

supremum)which is denoted byxVy. Such ordered sets are calledlattices.

For example, (IR; :(;) is a lattice in which

x I\y=min{x,y}, xVy =max{x,y};

(IP(E);~) is a lattice in which

and (INo;I)is a lattice in which

m 1\ n =gcd{m, n}, mVn =lcm{m, n}.

Clearly, any finite subset of a latticeL has a meet and a join Ifevery subset

ofL has a meet (resp join) thenL is said to be meet-complete (resp complete). By a complete lattice we mean a lattice which is both meet-complete and join-complete

join-The concept of a lattice was introduced at the end of the last century by

C.S Peirce and E Schroder, but the study of lattices became really systematicwith G Birkhoffs first paper[30] in 1933 and his book [2], the first edition

of which appeared in 1940 and was for several decades the bible of latticetheoretists In recent years, lattice theory has grown considerably Lattices

(Vx,yEA)

(Vx,yEA)

x:(; y {==? f(x):(; f(y),

x:(; y {==? f(x) ~f(y).

apwear in all branches of mathematics: for any algebra the subalgebras, theeq~~~~erelations, the congruence relations form lattices; in a topologicalspace the open sets, the closed sets, the clopen (i.e closed and open) setsform lattices; the convex subsets of a vector space form a lattice; in classicallogic the propositions form a lattice in a way that we shall make precisebelow.

But what are lattices from the point of view of universal algebra? As is

well known, the aim of universal algebra is to highlight the properties that

various algebraic systems (e.g groups, rings, fields, modules, lattices, )have in common If we leave aside some early papers of Whitehead, wemight say that the first pioneer of universal algebra was also G Birkhoff

Fundamental to universal algebra is the notion of an operation If n is

a non-negative integer then an n -ary operation on a set A is a mapping

f : An -7A The integer n is called the arity of the operation We shall be

mainly concerned with the cases wheren =0,1,2which give respectively a

nullary operation (this simply picks out an element of A), a unary operation, and a binary operation. Analgebra oftype (nl, , na<) is a pair (A, F) where

A is a non-empty set and F is an a-tuple Ui, ,fa<) such that, for each i

with 1~i~a,}; is an nj-ary operation on A Thus, for example, a lattice is

an algebra of type(2,2), the two binary operations being meet and join, andsatisfying the folloWing identities :

If a lattice is bounded, i.e if it has a least element°and a greatest element

1, then it can be considered as an algebra of type (2,2,0,0)

If A and B are algebras of the same type (nl,' , na<) then a mapping

cP : A -7B is a morphism if, for each i such that 1~i ~a,

whenever (a1 , ,ani)EAni If, in addition, the mapping cp is surjective then cp is said to be an epimorphism with B an epimorphic image of A;

if it is injective then it is a monomorphism; and if it is both then it is an isomorphism A morphismf : A -7A is called an endomorphism on A; and

an isomorphism f : A -7A is called an automorphism on A.

Note thatifLandM are bounded lattices then any morphism cp : L -7M

has to satisfy CP(OL)=OM and cp(l L)=1M,

As we have mentioned above, lattice theory began properly with the work

of George Boole [4] in formal deductive logic with an attempt to codify thelaws of thought In fact, Boole considered very special (but also very im-portant) lattices in which, originally, the meet and the join were the binary

connectives called conjunction ('and') and disjunction ('or') respectively, with an additional unary operation called negation ('not') These so-called boolean lattices have turned out to be very useful in many areas of science

and mathematics: in electrical engineering, in computer science, in axiomatic

set theory, in model theory, and so on Precisely, a boolean lattice L has three

characteristics :

(1 ) it is bounded;

(2) it is distributive in the sense that

(Vx,Y,ZEL) x 1\(yVz)=(x I\y) V(x 1\ z).

Itis quite remarkable that this equality is equivalent to its dual

(Vx,y,ZEL) x V(yI\z) =(xvy)1\ (xVz).

(3) it is complemented in the sense that for every aEL there exists a'EL (called the complement of a) such that a 1\ a' = °and a va' =1; in other

words, the centre Z(L) of L, Le the set of complemented elements, is L itself.

The property of distributivity is shared by many lattices For instance,each of the three examples given above is distributive Since all the latticesthat we shall deal with will be distributive, we shall say nothing about thevarious forms of weak or restricted distributivity On the contrary, the notion

of complement is very strong and many weakened forms of it have beenconsidered

Note that in a boolean latticeLthe operationx I-tx'of complementation

is a polarity and satisfies the so-called de Morgan laws

(Vx,yEL) (x I\Y)'=x'Vy', (xVy)' =x' I\Y'.

As observed by H B Curry[8], 'the term is customary despite its historicalinaccuracy According to Bochenski, the formulas were known in the MiddleAges'

If, in a bounded distributive lattice, we can define a polarity that satisfies

the above de Morgan laws then we obtain what is called a de Morgan bra More precisely, this is an algebra (L; 1\,v,j,0,1) of type (2,2,1,0,0)where(L; 1\,V, 0, 1) is a bounded distributive lattice andf :L L is a unary

alge-operation that satisfies the identities

f(x I\Y) =f(x) Vf(y), f(x Vy)=f(x) I\f(y), f2(X) =X.

From these identities it follows that f(O) = 1 and f(1) = O De Morganalgebras were introduced by G.C.Moisil[75]and investigated by A Monteiro

[76] and his school A Kleene algebra is a de Morgan algebra satisfying the

inequalityx I\f(x):(;yVf(y).

Another way of generalising the notion of complementation is to retain

the identity a 1\ a' =0 and to drop the other In this manner we define a

semicomplementation A lattice L that is bounded below is said to be complementedifevery aELhas a semicomplement, Le a non-zero elementthat is disjoint froma. Here, of course, the second lattice operationV plays

semi-no part, so that the semi-notion of semicomplementation can be defined on a meetsemilattice Of considerable interest are those lattices (or meet semilattices)

in which, for any element a, the subset of elements disjoint from a has a

greatest element This is called the pseudocomplement of a and is denoted bya* Thus a* =max{xELI a 1\ x =O} Pseudocomplemented lattices arenecessarily bounded E~amplesof these are : the lattice of open subsets of

a topological space, the pseudocomplement of an open set being the rior of its complement; and the lattice of ideals of a distributive lattice that isbounded below

inte-Of course, if we require that a Va* =1 for every a EL, then a*

be-comes the complement ofaand, whenL is distributive,L is then a booleanlattice Guided by what occurs in many examples, Stone [92] suggested a

restriction of the identity aVa* to those elements a that are ments, Le that it would be fruitful to consider the identity a*Va** =1 forallaEL. Distributive pseudocomplemented lattices that satisfy this identity

pseudocomple-are therefore called Stone lattices When the unary operation a f -+a*is sidered as a fundamental operation of the algebraic system, we shall use the

con-term Stone algebra Note, therefore, that whereas a Stone lattice is of type

(2, 2, 0, 0), a Stone algebra is of type(2, 2,1,0, 0) This distinction is essentialnot only with regard to morphisms, but also with regard to subalgebras andcongruences

A subalgebra B of an algebra A is a non-empty subset of A which is closed under all the operations of A The operations on B are then those

of A restricted to B An algebra A and its subalgebras are of the same type.

For example, a subalgebra of a Stone lattice is just a sublattice, whereas a

subalgebra of a Stone algebra is a sublattice that is closed under a H a*.

A congruence relation on an algebra A of type (nl' ,nCl/) is an alence relation fJ on A which satisfies the substitution property ; for each

equiv-i E {1, ,a},

(a j ,b j )EfJ(j=1, ,nj) =?- (J;(al, ,a n /),};(b 1 , ,b ll ))EfJ.

For example, in a Stone latticeL an equivalence relationfJ is a congruenceif

(a,b)EfJ and (c,d) EfJ imply

If, on a bounded distributive latticeL,there is defined a unary operation

f which satisfies the de Morgan laws and is such thatf(l)= 0 andf(O)= 1(i.e if we drop the assumption thatf2 =idL,so thatf becomes a duallat-tice endomorphism, but not necessarily a dual lattice automorphism) then

we obtain what is called an Ockham algebra This idea goes back essentially

to 1977 in a short but very deep paper byJ. Berman[28]. Two years later,

A Urquhart[94] developed a topological duality theory for this type of

alge-bra, gave a logical motivation for his study, and introduced the name Ockham lattices with the justification : 'the term Ockham lattice was chosen because the so-called de Morgan laws are due (at least in the case of propositional logic) to William of Ockham' The name Ockham algebra has since become

classical and was used in the thorough doctoral thesis of M Goldberg [68]

and in a subsequent paper[69].

Since 1981 many papers have been published on Ockham algebras Theobjective of this book is therefore to develop the general properties of thisclass of algebras and to consider more particularly some important subclasseswhich are interesting not only in the framework of universal algebra but alsofor their significance in the algebra of logic At this point,itis not superfluous

to recall that de Morgan algebras arose in the researches on the algebraictreatment of constructive logic with strong negation The operationf that isinvolved in an Ockham algebra can also be interpreted as a negation (and forthisreasonf(a)is often written asrva),though this does not in general satisfythe law of double negation If we impose onf the restriction thatfn =id Lforsomen E IN\{I , 2} then we obtain a new logic whose interpretation, as far as

we know, has still to be made explicit In this connection, interesting workhas been done by D Schweigert and M Szymanska[88] on those Ockhamalgebras that belong to the class Pn,O (n odd) described in Chapter 4 The

class of algebras they deal with is shown to be the semantic for a propositionalcalculus called correlation logic.

The reader willseefrom the examples of Ockham algebras given in ter 1 that the study of Ockham algebras is far from being gratuitous

Chap-We close this brief introduction by observing that all the classes of bra that we have mentioned, and indeed all that we shall consider later, are

alge-equational,in the sense that they can be defined by a set of identities Itis acelebrated theorem of Birkhoff that the equational classes of algebras are pre-cisely those that are closed under the formation of subalgebras, epimorphicimages, and direct products, Le are varieties.

the Berman classes

Recall that a distributive Ockham algebra is an algebra (Lj /\,V,j,0, 1) oftype (2,2,1,0,0) in which(Li /\, V,0, 1) is a bounded distributive lattice and

x f-tf(x) is a unary operation such thatf(O)=1,f(l) = °and

(Vx,Y EL) f(x /\y) =f(x)Vf(y), f(x Vy)=f(x) /\f(y).

Without explicit mention to the contrary, all the Ockham algebras that weshall deal with will be distributive as lattices so we shall agree to drop the

adjective 'distributive' and talk of an Ockham algebra We shall often also

denote this by the simpler notation(L;f).

The class of Ockham algebras is equational (in other words, a variety), andwill be denoted by O As mentioned in Chapter 0, the concept of an Ockhamalgebra arose from successive attempts to generalise the notion of a booleanalgebra Important stepsinthis long history are the de Morgan algebras and the Stone algebras, these forming important subvarieties of the variety O In

1979, A. Urquhart [94] observed that 'an outstanding open problem is that

of determining all equational subclasses of the class of Ockham lattices' Tothis day, the problem remains unsolved However, very important subclasses

of 0 were introduced byJ. Berman[28] and we shall call them the Berman

classes. These are obtained by placing restrictions on the dual endomorphism

f. Precisely,ifwe letjO=id and definer recursively byr(x)=f[fn-l(X)] for n;:?:1, then forP,qEINwithP;:?:1 and q;:?:°we define the Berman class

Kp:q to be the subclass of0 obtained by adjoining the equation

The importance of the Berman classes is partly justified by the followingproperty

Proof Let(L;1) be a finite Ockham algebra, and consider the sets

{f,]3,]5, }, {f0,]2,]4,f6, }

of, respectively, dual endomorphisms and endomorphisms onL. Since both

of these must be finite, we have thatfq =f 2p + qfor somep,q. <>

Note that Theorem 1.1 is no longertrueifthe algebra in question is infinite,

as the following example shows

byai<a i fori <j; bi<b i fori <j; 0 <a i<b i <1for alli,j Define f: L -+L by

f(O)=l, f(l)=O, f(ai)=b_ il f(bi)=a-i-lo

Then (L; f) is an Ockham algebra We can depict the effect off as follows :

t

For every i EifandnE IN,we have the chains

<f2n(a i)< <f2(aJ <a i<f(a i)< <f2n+l(a i)< .

<f2n+l(b i)< <f(bJ <bi<f2(b i)< , <f2n(b i)< .

Itfollows thatL does not belong to any Berman class Note also that in thisexamplef is a bijection

The following three examples are by no means surprising.

op-eration as x ~ x. Every de Morgan algebra (L; -) belongs to the BermanclassK1,o, In fact, sincex =x for everyxEL, the operationx ~x is a dualautomorphism

op-eration as x ~ Xl. Every boolean algebra belongs to the subclass ofK1 0

obtained by adjoining the equationx /\J(x) =O In fact, from this itfollo~s

thatJ(x)Vj2(x) =1,Le.J(x)Vx=1,so thatJ(x)=Xl is the complement

ofx.

asx ~x*. Every Stone algebra(L;*)belongs to the subclass ofK1 1obtained

by adjoining the equationx /\ J(x) =O In fact, by the properties of thepseudocomplementation x ~ x* we have (x /\ y)* =x*Vy*, (x Vy)* = x* /\ y* with 0* =1 and 1*=0, so that (L;*) EO Since moreover x* =x*** it follows that (L;*)EK1,1' Finally, x /\ x* =0 by the definition of thepseudocomplement

Less trivial are the following examples

is bounded below, and letB be a distributive lattice that is bounded abovewith a dual isomorphism{j ; A - 7B. On the linear sumL=A EEl 5 EEl B define

a unary operationJby

{

{j(x) J(x)= x*

{j-1 (x)

ifxEA;

ifxE5;

ifxEB.

Then(L; J) is an Ockham algebra that belongs to K1,1

usual order, namely that given by

Moreover, this latticeF is distributive.

Now leta EIR+ be fixed and for everyp EFdefinef(P) by setting

(\IxEIR+) (t(P))(x) =1-p(x+a).

Clearly, we havef(O)=1 andf(1)=O Also, since

1 - rnin{p(x+a), q(x+an =max{l - p(x+a),1 - q(x+an

we seethatf(p I\q)=f(P)Vf(q), and likewisef(PVq)=f(P) I\f(q). Thus

(F;f) is an Ockham algebra Whena =0 we have (t(p))(x) =1-p(x)and

(t2(p))(X) =p(x). Itfollows that in this case (F;f)EK1,o, Moreover, since

rnin{l,l -p}~! ~max{q,1-q}

we havep I\f(P)~ qVf(q) and (F;f) is a Kleene algebra

in-terval1=[0,1]of real numbers under the usual order EveryxEI withxt:.1has a unique decimal representation

x =0 X1X2x3

where eachXi E{O, 1, , 9} For our convenience here, we shall write this

as x =(Xi )i;;'1' Leta be a fixed positive integer and for each such x define

f(x) =(9 - Xi+a)i;;.1,

withf(l)=O Then(I;f) is an Ockham algebra

In every Ockham algebra (L;f) the subset

S(L)={f(x) Ix EL}

is clearly a subalgebra ofLwhich we shall call theskeleton ofL. The skeleton

ofL is a de Morgan algebra precisely whenf3(x) =f(x) for everyx EL,

i.e precisely whenLbelongs to the Berman class K1,1' When this is the case,

we shall say thatL has a de Morgan skeleton. Note that in this case we alsohave

an idempotent{O, I}-lattice morphism such that the sublattice 1mip admits a polarity p Define a unary operation f : L 7L by the prescription

(VxEL) f(x) =p[ip(x)J.

Then (L;f) is an Ockhamalgebra with a de Morgan skeleton Moreover, every such Ockham algebra arises in this way.

f(x I\Y) =p[ip(x I\Y)] =p[ip(x) 1\ ip(y)]

=p[~(x)] Vp[ip(y)J

= f(x)Vf(y),

and similarlyf(x Vy)=f(x) I\f(y), so(L;f) is an Ockham algebra

Sinceipis idempotent we have thatip acts as the identity on Imip=Imp.

It follows from this thatipPip(x)=Pip(x) for everyx EL,so that

f3(x)=pippippip(x)=pipp2ip(X)=pip2(X)=Pip(x)=f(x).

Thus (L;f) has a de Morgan skeleton

Conversely, if(L;f) is an Ockham algebra with a de Morgan skeletonthen the mapping ip : X f + f2(x) describes a {O, 1}-lattice morphism on

L, and from f3 =f we deduce that ip2 = ip and that f is a polarity onImip={f2(X) Ix EL}. <>

As an application of Theorem 1.2, we shall obtain an affirmative answer

to the quite natural question of whether, given a bounded distributive lattice

L, it is always possible to makeL into an Ockhamalgebra with a de Morganskeleton For this purpose, we recall some definitions

A subsetQ of an ordered set P is said to be a down-set if it is decreasing,

in the sense thati EQ and j ~i imply that j EQ The down-set generated

by a subsetX ofP is defined by

XL ={yEPI(::JxEX)Y~x}.

In particular, whenX ={x} we write XL as xL An ideal of a lattice L is

a sublattice I ofL which is also a down-set; and an ideal of the form xL is called a principal ideal Dually, a subset Q of an ordered setP is said to be

an up-set if it is increasing, in the sense thati EQandj ;:;:: i imply thatj EQ.

The up-set generated by a subsetX ofP is defined by

XT={y EPI(::JxE X)Y ;:;:: x}.

In particular, when X ={x} we write XT as x T A filter of a lattice L is a sublattice F of L which is also an up-set; and a filter of the form xTis called

a principal filter Ideals and filters are convex, in the sense thatifa, bEl

Crespo F) anda~c~ b then cEI Crespo F). An ideal or a filter is said to

be be proper if it is not the whole lattice A proper ideal I of L is said to

be prime if a, bEL and a 1\ bEl imply that a EI or bEl The notion of a

prime filter is defined dually The set-theoretic complement of a prime ideal

is a prime filter In a distributive lattice L there are sufficiently many primeideals, so that any two elements can be 'separated' by a prime ideal, in thesense thatifa, bEL with a t-bthen there is a prime idealI of L that contains

one of these elements and not the other

Ock-ham algebra with a de Morgan skeleton.

chain of prime ideals

j(x)=p[ep(x)] for everyx EL,we see that (L;1)is an Ockham algebra with

a de Morgan skeleton.<:;

The preceding proof shows in particular that many non-isomorphic ham algebras can be defined on the same distributive lattice; for we canattribute different values to n,and even for a fixed value ofn there can existdifferent chains of prime ideals

0= ao <al < <an-l =1

Every antitone mapping ep : n t n such thatj(O) =1 andj(l) =0 termines an Ockham algebra Thus the number of non-isomorphic Ockhamalgebras definable on n is equal to the number of antitone mappings from

de-n - 2 to de-n, which is kde-nowde-n to be

Cl!n =(2n - n-23) For small values ofn,this number is given as follows:

n = 2 3 4 5 6 7

C={(x,Y,z) EIR3Ix,y,ZE[0,In.

With respect to the cartesian order, C is a bounded distributive lattice Themappingep :C tC given by

In this case we obtain

j'(x,y,z)=(1-y, 1-z,1-y),

which makes C into a different Ockham algebra with the same de Morganskeleton

a, bEBwithb<a and define ep : L- 7Lbyep(O)=1, ep(l)=0 and

(\-IxEB) ep(x)=(b Vx) 1\ a=bV(x 1\ a).

Clearly,epis an idempotent{O, 1}-lattice morphism We can define a polarity

p on Imep={O} U[b, a]U {I} byp(O)= 1,p(l)=0 and

(\-IxE [b, aD p(x)=ep(x')

where x' is the complement ofx inB. It is easy to verify thatp(x) is therelative complement ofx in[b, a]. Itfollows from Theorem1.2,withf(x)=p[ep(x)] for everyx EL, that(L;1) is an Ockham algebra with a de Morganskeleton Here we havef(O) =1,1(1)=0and, forxEB,

f(x) =p[ep(x)]=ep([ep(x)]') =ep[(b' 1\ x') va']=bV (a 1\ x')=ep(x').

p(a21l+1)=p'(a21l+d=a-21l-1;

p(ao)=ao, p(bo)=bo;

p'(ao)=bo, p'(bo)=ao·

a new smallest element0 and a new greatest element1 adjoined Considerthe mappingep ; L- 7L given by

{

X ifx E{O, 1,ao, bo}U{a21l+dIlEZl;

ep(x)= a21l-1 ifx = a21l (n'I0);

a 21l+1 ifx =b21l (n 'I0)

In the Hasse diagram opposite, the arrowheads

indi-cate the effect of ep. Itis readily seen that ep is an

idempotent {O,1}-lattice morphism There are two

polarities on Imep, namelyp,p' given by

These polarities give rise to different Ockham algebras with the same deMorgan skeleton

Example 1.12 LetE be a set and leta, bbe distinct elements ofE. Considerthe mappingip :IP(E)- 7IP(E)given by

Roughly speaking, ip adds a ifX already contains b, and removes a ifX

does not contain b Clearly, ip(0)= 0andip(E)=E. Itis readily seen thatip

is both a u-morphism and an n-morphism Moreover, ipis idempotent Now

ThenB~is a sublattice ofBn,with (0, , 0) as smallest element and (1 , , 1)

as greatest element Define ip :B~ ~ B~ by

ip(XI,X2,'" ,x n )=(XI,X n "" ,x n )·

Clearly, ip is an idempotent{O,l}-lattice morphism A polaritypon Imip is

We can therefore makeB~into an Ockham algebra with a de Morgan skeleton

by defining

We have seen above that if(L;/)EKI,1 then the mapping X f-7 j2(x)

is an idempotent lattice morphism We shall now investigate an importantspecial case of this, namely when the mappingXf-7j2(X) is a closure We

recall that a closure on an ordered set E is an isotone mapping j : E - 7E

such thatj=j2 ~idE' Thus Xf-7j2(X) is a closure precisely when

(Vx,YEL) x::;;Y ~ j2(X)::;;j2(y);

(VxEL) X ::;;j2(X);

(VxEL) j2(X)=j4(x).

Note that the first of these properties is satisfied by allLEO, and that thethird is satisfied by allLEKI,I'

Definition By a de Morgan-Stone algebra, or an MS-algebra, we mean

an algebra (L; 1\,V, 0,0, 1) of type (2,2,1,0,0) such that (L; 1\,V,0,1) is abounded distributive lattice andx r -t XO is a unary operation on L such that(MS1) 1°= 0;

(MS2) (V'x,y EL) (x l\y)O=XOVyO;

a dual endomorphism and x r -t x** is a closure So the notion of an algebra arises quite naturally by retaining the properties that are common tothese two classes of algebras

MS-The class MS of MS-algebras is equational; it is the subclass of KII tained by adjoining the equationx I\f2(X) =x In this connection, w~notethat M Ramalho and M Sequeira[82] have considered more generally thesubvarieties of 0 defined byx I\f21l(X)=x.

ob-The relation of MS-algebras to Ockham algebras is as follows

skeleton An Ockham algebra (L; f) is an MS-algebraifand onlyifx ~f2 (x) for every x EL.

0°=1°° ;::1 and so 0°=1 By (MS2), the mappingx r -tXO is antitone So

(xVy)O~XO 1\ yO~ (XO 1\ yO)OO=(x00 VyOO)O

Since clearlyxVy ~xOoVyOO, which implies that (xVy)O ;:: (x00VyOO)O, we

deduce that (xVy)o =XO I\ y o It follows that (L;0) is an Ockham algebra.Now by (MS3) we have x ~XOO and so xo;:: xOoo. But, again by (MS3),

XO ~XOOO Consequently, we have XO =XOOO and so (L;0) has a de Morganskeleton The second statement is immediate from the definitions.0

In an Ockham algebra(L; f) the biggest MS-subalgebra is

MS(L)={xELIX~f2(X)}.

In the case where (L;1) belongs to KI ,I and is obtained as in Theorem 1.2,

MS(L)={xELIx ~ep(x)}.

Thus, in Example 1.9 we have (for both of the algebras described)

°-preserving closure morphism such that the sublattice1m<p admits a polarity

p Define a unary operation 0 :L- 7L by the prescription

(VxEL) X O=p[<p(x)].

!ben (L;0) is an MS-algebra Moreover, every MS-algebra arises in this way.

we see that (L;0) belongs to KI I SinceX OO=<p(x)~x, it follows by rem 1.4 that (L;0) is an MS-algebra

Theo-Conversely, if(L;0) is an MS-algebra then the mapping <p : X f 7 X OO

describes a O-preserving closure morphism onL, andp : X f 7X Ois a polarity

on Im<p={X OO Ix EL} 0

+ :L- 7L will be called a dualMS-algebra if (LOP,+)is an MS-algebra, where

LOP denotes the order dual of the lattice1.

Itis clear that we can construct dual MS-algebras by using the dual ofTheorem 15, which involves a I-preserving dual closure morphism

{O} EBINoEB {oo} Letn = ITpfi be the decomposition into prime factors of

ThenC{Jn is a {O,00}-preselVing dual closure morphism with

Let (L;/)be an Ockham algebra Then an Ockham algebra congruence (or,briefly, acongruence)onLis an equivalence relation that has the substitutionproperty for both the lattice operations and for the unary operation j. It

follows that every congruence is in particular a lattice congruence and it isessential to distinguish these two types In order to do so, we shall use thesubscript 'lat' to denote a lattice congruence

Ifa, bEL anda:::;; bthen theprincipal congruence 19(a, b) generated by

a, b is defined by

19(a, b)=;\{<pEConLI(a, b)E<p}.

In other words, it is the smallest congruence that identifiesaandb Similarly,

theprincipal lattice congruence generated bya, bis

191at(a, b)= ;\{<pEConlatL I(a, b)E<p}.

Note that we then have

191at(a, b):::;; 19(a, b).

We recall that, in a distributive lattice,

and that the intersection of two principal lattice congruences is again a cipallattice congruence; in fact, ifa :::;;band c :::;;d then

prin-191at(a, b) 1\ 191at(c, d)=191at((aVc)1\ b 1\ d, b 1\ d).

A fundamental result concerning congruences that we shall require is that

ifL is an algebra and 19is a congruence onL then for any congruence<p of

L such that<p~19 the relation<p /19defined onL/19 by

([x]19,[y]19)E<pj19 ~ (x,y) E<p

is a congruence onL/19;and every congruence onL/19can be uniquely sented as<p /19for some congruence<p~19 Moreover, ConL/19is isomorphic

repre-to the filter[19,L] of ConL.

The following result, due toJ. Berman[28], is fundamental in the tigation of congruences

inves-Theorem 2.1 Let (L,f) be an Ockham algebra If a:::;; b in L then

19(a, b)= V 191at(r(a),fn(b)).

n;;'O

Proof Let cp = V19lat(rn(a),r(b)). We show as follows that cpEConL.

n;;,O

Suppose that (x,y)Ecp, so that there are integers i 1 , , i m and elements

ZO,Zl,'" ,Zm such that

X =Z0 == Z1 == Z2 == == Zm-l == Zm=y,where CPik =19lat(rik(a),jik (b)). Nowifi kis even we havejik(a)~fik(b)

and so

Zk-l /\lk(a)=Zk /\lk(a), Zk-l Vlk(b) =ZkVfik(b).

Applyingf to each of these equalities, we obtain

defini-Finally, since19(a, b) is a congruence we have that

(a, b)E19(a, b) =} (\In) vn(a),jn(b))E19(a, b),

from which we deduce that, for each n,

19(rn(a),r(b))~19(a, b)

and hence that

Thus cp=19(a, b) as asserted 0

exten-congruence 19 on A is the restriction of some exten-congruence cp on B (this being

denoted by CPIA =19) In fact, as was shown byA Day [66], in an tional class of algebras these properties are equivalent; indeed, they are eachequivalent to the condition

equa-for all subalgebrasA ofB and alla,bEA, 19A (a, b)=19 B (a, b)lA'

Using this fact,J. Berman[28] established the following result.

andf-L={)B(a, b). Then, by the above result of Day, it suffices to prove that

A=f-L IA' Now by Theorem 2.1 we have

we have An=f-LnlAo Now since A is a lattice congruence it has an extension

I E ConlatB such thatIIA =A Itfollows thatI;;::f-Ln for alln and so

A=IIA ;;:: ( Vf-Ln)IA;;:: A,

n;;>O

whence A=f-LIA' <>

Consider now, for anyaEL, the relation{)a defined by

(x,y)E{)a ¢==?- x 1\ a=y 1\ a and xV f(a) =y Vf(a).

(1) {)a isa principal lattice congruence; moreover, [a]{)a=[a, aVf (a)] and{)a=W* a ;;::f(a).

(2) Ifa~f2 (a) then {)aisa principal congruence.

(3) Ifa=j2(a) and a Ilf(a) then {)a and {)f(a) are non-trivial

congru-ences with {)a 1\ {)f(a) =w.

{)a={)lat(a, 1)1\{)lat(O,j(a») ={)lat(aI\f(a),j(a»).

That[a]{)a=[a, aVf (a)]follows immediately from the definition of {)a ;andthat{)a =W * a ;;::f(a)is immediate from the above

(2) Ifa~f2(a) and(x,y)E{)a then({(x),j(y») E{)a;for, fromx I\a=

y 1\ aweobtainf(x)Vf(a) =f(y) Vf(a) and fromx Vf(a)=y Vf(a) weobtainf(x) 1\j2(a)=f(y) I\f2(a), hencef(x) 1\ a=f(y) 1\ a.

(3) Ifa=f2(a) anda Ilf(a)then by (1) we have {)f(a) ={)lat(aI\f(a), a)

and consequently {)a1\ {)f(a) =w. It also follows by(1)that both {)aand {)f(a)

are non-trivial.<>

Theorem 2.4 Foran Ockham algebra (L;f),

(1) {a ELI a~fZ(a)} isa sublattice of L which contains 0 and 1;

(2)a~b =} 'I9a~1h;

(3)'I9al\'I9b='I9avb'

j2(a)Vj2(b).

(2) follows from the fact that x 1\ b=Y 1\ bimplies x 1\ a =y 1\ a, and

x Vf(b) =y Vf(b) implies x Vf(a) =y Vf(a).

(3) If (x,y)E'I9a 1\ 'I9bthen

xl\a=yl\a, xl\b=Yl\b, xVf(a)=yvf(a), xVf(b)=yvf(b),

which implies that

x 1\ (aVb)=Y 1\ (a Vb), XVf(a Vb)=y Vf(a Vb).

Consequently, '19 a 1\ '19 b~ '19 avb' The converse inequality follows from (2) <>

Theorems 2.3 and 2.4 show that in MS the meet of two principal

congru-ences each of the form'19ais again a principal congruence Unfortunately, thisproperty does not hold for arbitrary principal congruences, even when theOckham algebra belongs to a 'small' class such as that of de Morgan algebras.The interested reader may consult M E Adams [17].

For an Ockham algebra (L;f) consider now, for every n EIN,the relation

a practical description of which is as follows

some n Moreover, ifLisnon-trivial then <P w<L

Proof If (x,y)E<P w then there exist to, , t k and<Pi!,"" <Pik such that

x =to <Pi! t 1<Pi2 tz t k- 1<Pik t k =y.

Denote the greatest of these <Pi by <Pw Then we havex <PnY, Le fn(x) =fn(y). Conversely, ifr(x)=r(y) then clearly(x,y) E<P w'

Finally, ifLis non-trivial,r(O) t:fn(1) for all n gives <Pwt:L~

For every non-trivial Ockham algebra (L;1) it is clear that we have

w=<Po ( <PI (<P2 ( '" (<Pi(<PHI ( (<P w<~,

and, with ( meaning 'is a subalgebra of',

{a,I} ( '" (p+l(L) (fi(L) ( (f(L) (fO(L)=L.

Itis readily seen that [x]<p i f-tP(x) describes an Ockham algebra

iso-morphism when i is even and a dual isoiso-morphism when i is odd, a situation

which we shall denote by writing L/<Pi "-' P(L). The following result istherefore clear

Moreover, if(L;/) belongs to the Berman class Kp,q then each ofthe aboveis

equivalent to (L;/)EKp,o.

ProofThe equivalence of(1), (2),(3)is clear As for the final statement,ifforeveryx ELwehavej2p+q(x)= fq(x) then, by (3), wehavej2P(x)= x andthen (L;/)EKp,o, Conversely, suppose that (L;/) EKp,o and that <1>1 =f w.

Then there existx,yELsuch thatx =f y andf(x) =f(y). This implies that

f2P(x)= f2P(y)and hence the contradictionx =y. <>

Note that the hypothesis thatL belong to some Berman class is necessary

in the above As the following example shows, it is possible for f to beinjective withL¢}(,z,o for anyn.

f(x): 00 3 2°-2 -3 -00

Then (L; f) is an Ockham algebra withf injective But for m =f n we have

fm(x) =f r(x), sofm =frandLdoes not belong to any Berman class

In general, a <l>n-class can contain more than one element offn(L), oreven none, as the following example shows

x:Oabc1 f(x) :1 1 c a °

Then it is readily seen that(L;/)E0 and that the smallest Berman class that

it belongs to is K1,3' Nowf2(L) = {a,a,I}, and

[0]<1>2 ={O,a} with [0]<1>2nf2(L)={O,a};

[b]<I>2 ={b} with [b]<I>2 nf2(L)= 0.

Of some interest is the case where every <l>n-class ofL contains exactlyone element of1'l(L). Here we shall deal with the subvarieties of 0 defined

by x I\f2P(X) =x (see Chapter 1) and denoted by Ramalho and Sequeira

[82]byK~o. Clearly,

Kp,o CK~o CKp ,I'

Note that inK~o the mapping ep : Xf-7f2P(X) is a closure operator, since

ep(x) ~x; ep2(X)=ep(x); X~Y =? ep(x)~ep(y).

This closure is additivein the sense that ep(xV y)=ep(x) V ep(y), and also

multiplicativein the sense that ep(x 1\ y) =ep(x) 1\ ep(y). By Theorem 5 of

[97], it follows that rp(L) is a semiconvexsubalgebra ofL, Le ifx /\ y and

x Vy both belong to rp(L) then both x andy belong to rp(L). As a directconsequence, rp(L) containsZ(L), thecentre of L(Le the boolean sublattice

ofLformed by the complemented elements)

one element of fq (L) If LEKffo then every <l>zp -class has a greatest element, this being the only element ofthe class that belongs to fZP (L).

there-fore (tZP(x),f4P(x)) E<l>q and so (x,fZkp(X))E<l>q, Le (x,jq(x))E<l>q. It

follows that[x]<I>q contains exactly one element offq(L) since(x,y)E<l>q isequivalent tor(x)=fq(y).

Now letLE K~o' ThenLEKp,l ~Kp,zp. SincefZP is a closure operator,the greatest element of[x]<I>zp is the only one that belongs tofZP(L). ~

If an Ockham algebra (L; f) belongs to a Berman class then there is asmallest Berman class to which it belongs; we denote this byVB (L).

Theorem 2.9 For an Ockham algebra (L; f) the following statements are

equivalent:

(1)VB(L)=Kp,q;

(2) L:::>f(L) :::> :::>fq(L)=fq+l(L)= "', and VB (tq(L))=Kp,o,

fol-lows that fq(L) ~ fq+l(L), whence fq(L) = fq+l(L) = Suppose now,

by way of obtaining a contradiction, that for some n with n~q we have

fn-l(L) =r(L). Then clearlyfq-l(L) =fq(L). But, by(1), L¢Kp,q_l and sothere exists x ELwithfq-l(X) t j2p+q-l(X). Butfq-l(x) Efq-l(L) =fq(L)

and so fq-l(X) =fq(y) for some y EL. Hence fq(y)t fzp+q(y), and thiscontradicts the fact thatLEKp,q' This then establishes the chain

L:::>f(L) :::>fZ(L) :::> '" :::>fq(L) =r+1(L)=

Now sincefq =j2p+q we see thatj2P acts as the identity onfq(L). Hence

fq(L) EKp,o, ThatVB(tq(L)) =Kp,ofollows from the fact thatiffq(L)EKt,oC

Kp,0then necessarilytIPand, sincefZt is then the identity onr (L), we have

fq =fZt+qso thatLEKt,q,which contradictsVB(L)=Kp,q'

(2) =} (1) :IfVB (tq(L)) = Kp,o then clearlyfq = j2p+q and soL E Kp,q'

IfVB (L)=Kp/,q' then from (1)=} (2)it follows thatql = qandpi = p. ~

Corollary 2 ljVB(L)=Kp,q then

w=<1>0 <<1>1 <<1>2 < <<l>q = <l>q+l = ,

and Con L has length at least q+1 0

In a bounded distributive lattice every ideal I is the kernel of at least one congruence, i.e there is a lattice congruence {J such that [O]{J =I. We

also write I =Ker{J Dually, every filter F is the cokernel of a congruence,

i.e there is a lattice congruence 1f;such that [1]1f; =F. These properties donot carry over to Ockham algebras For instance, in Example 2.2 the principalideal c! is not the kernel of any congruence; for (0, c)E{Jimplies (a,1)E{J,

whence the contradiction (0,1)E{J. So it is of interest to characterise thoseideals of an Ockham algebra that are congruence kernels

LetI be an ideal of the Ockham algebra(L;/). For eachn ~0, define

Then clearlyI 2n is an ideal andpHI is a filter Finally, let

I00= V I 2n, 1 0 = V I 2n + 1 .

We shall now investigate, for a given ideal I, the smallest congruence

8 (1) on L that identifies the elements of I We recall that, in a distributive

lattice, 8 (1) is characterised as follows :

(x,y)E8(1) ~ (:liE!) xVi=yVi.

With the obvious subscript 'lat' to denote the corresponding smallest latticecongruence, we have the following result

8(1)=81at(JO)V81at(100)'

subset X ofLwe have 8 (X)= V{{J(a, b)Ia, bEX} 0

(x,y)E8(I) ~ (:liEIoo)(:l}EJO) (xVi)I\}=(yVi)I\}.

condition SinceL is distributive, 'II is a lattice congruence Suppose that

(x,y)E 'II Then there exist i E 1 00 and} EJO with (x V i) I\} =(yV i) I\},

whence

x I\} =x 1\ (xVi) I\} =x 1\(yVi) I\}

and hence (x, x 1\ (yVi)) E 9Iat(JO).

Now we also have

(x 1\ (yVi))Vi= (xV i)1\ (yV i)= (x I\y)V i

and so (x 1\ (y Vi), x I\Y)E9Iat(100)' Thus we see that

(x, x 1\ y)E9Iat(JO)V9Iat(100)'Similarly, we can show that

(y, x I\Y) E9Iat(r)V9Iat(100)

Thus, by Theorem 2.10, (x,y)E9(1)

Conversely, suppose that(x, y)E9 (1) Then, again by Theorem 2.10, wehave the finite sequence

x=PI=PZ="'=Pn=y

where each =is either9Iat(JO) or9Iat(100)' Suppose that we have

x 9lat(1oJPI 9Iat(r)pz 9Iat(100) 9Iat(r)y.

Then we have finitely many equalities

XViI =PI ViI, PIl\jl =pzl\h, pz Viz=P3 Viz, P31\jz =P41\jz,

where each ikE 1 00 and each j kE1 0

• Since1 00 is an ideal and1 0 is a filter,

we have VikE1 00 and!\hE1 0

• Moreover,(xVVik) 1\!\h =(PI VVik)l\!\h

=(PI 1\ 1\)k)V (V ik 1\ !\jk)

=(Pz1\ !\h)V (Vik 1\ !\h)

=(Pz VVik ) I\!\h

= (yVVik) 1\ !\h, from which we see that (x,y)E'P and hence that'P coincides with9(1).0The following observation is immediate

Lemma 2.1 Ifthe ideal I issuch that fZ(1) ~ I then,forevery n,

fZn(1) ~ I, fZn+I(1) ~ (t(1))i,

from which itfollows that

ker-nelifand onlyif

(O!) f2(1) ~ I;

({3) (\IxEL)(\ljE(t(1))i) Xl\jEI =* XEI.

(t2(i), 0) E 8, so thatj2(i) E I and we have property (O!) Ifj E L is such that

j ~f(i) for some i E I then (i, 0) E 8 gives (j, 1) E 8, whence (x I\j , x) E 8

for everyx EL. Property({3) is now immediate

Conversely, suppose that(O!) and({3) hold By Theorem 2.11, Lemma 2.1,and condition(O!),

(x,0)E8(1) ~ (3iEI)(3jE(t(I))i) (xVi)l\j=il\}.

We have i I\} E I and so, by ((3), x ViE I and hence x E I. Itfollows thatKer 8(1) ~ I. Since I = Ker 8Iat(1) ~ Ker 8(1), it follows that Ker 8(1)=

(necessarily unique) fixed point of f Then the congruence kernels of L are the proper ideals of a L that satisfy condition (O!) above, and L itself. <>

We now proceed with some considerations concerning the structure of

the congruence lattice Con L of an Ockham algebra (L;f) Since Con L is

a sublattice of ConlatL, which is known to be distributive, ConL is alsodistributive Itis also algebraic, in the sense that it is complete and compactly generated (every element of Con L is the supremum of a set of compact

elements, a compact element being an element1Jsuch that if1J ::;;:sup X forsomeX ~ConL then1J ::;;:supXl for some finiteX1 ~ ConL).

As far as the structure of ConLis concerned, it is important to observe thatevery lattice congruence that is contained in<1>1 is a congruence Itfollowsthat ifL is finite then the interval[w,<l>d of ConL is boolean We also pointout that<1>1is dually dense in Con L, in the sense that if 1JV<1>1 =~then1J=~.

In fact, for any1JE ConLwe have (0,1) E1JV<1>1 if and only if (0,1) E1J; for

°1J Xl <1>1 X2 1J X3 <1>1 <1>1 X n -2 1J X n -1 <1>1 1

Let A and B be algebras of the same type Then we say that A is an

ofB if every congruence on B has at most one extension to A, in which

case B is said to be a strongly large subalgebra of A Finally, A is a perfect

inwhich case B is a perfect subalgebra of A When this happens, we have

ConA ~ConB.

In [100] it is shown that, for any element a of a modular lattice L, the

This leads to the following property

ConL ~Con Ca'

Proof Clearly, Ca is a subalgebra ofL and this subalgebra is strongly large

In fact, since0 has the congruence extension property, Ca is a perfect algebra Hence we have the isomorphism stated <>

sub-The usefulness of sub-Theorem2.13lies in the fact that it enables us to workwith C a instead ofL, and we can benefit from this in two ways Firstly,the size ofCa can be conSiderably less than that ofL. Secondly, Ca will in

general belong to a subvariety of 0 that is smaller than that of L; for instance,

it is clear that (Ca;j) always satisfies the axiom x AJ(X)~y VJ(y) evenif

Then (L;f) is an Ockham algebra that belongs to K1,0 and has two fixedpoints, namely c ande. The cone Ce generated byeis a five-element chain.

As a subalgebra of (L;f) it also belongs to K1 0and satisfies the supplementaryrelationx I\f(x) ~ Y Vf(y), whichLitself'does not (consider the elements

eand c) It is easily seen that Con C e is a four-element boolean lattice andtherefore so also is ConL. Note also that by Theorem 2.12 the only ideals of

Lthat are congruence kernels are 01,a 1, b 1,and 11.

Example 2.4 (!be pineapple) Consider the ordered set L with Hasse

dia-gram

.0

and made into an Ockham algebra by definingf(xj ) =Xi+l for eachi Notethatf is injectiveso, by the Corollary of Theorem 2.7, W =<PI = =<P w'

We leave to the reader the task of verifying that ConL is the chain

W < -< 19(Xi+l ,Xi+z)-<19(Xi+l'Xj)-< <'P -< £

where 'P has classes {O},{I },L \ {O,1} We shall return to this particularexample later

Concerning the basic congruences in a general Ockham algebra (L;f) we

also have the following results

Theorem 2.14 If a, bEL with a~band f(a) =f(b) then 19(a, b) has a complement in [w,<Pd

Proof By Theorem 2.1 we have

'l9(a, b)= V 'l91 at (jn(a),r(b)).

11;;:0Sincef(a) =f(b) by hypothesis, it follows that'l9(a, b)='l91at(a, b) EConL

with, clearly, 19(a , b):s:;;<1>1' Write 19(a , b)=a and observe that since a is a

principal lattice congruence it has a complement in Conlat{L), namely

(3='l91~t(O,a)V 'l91at(b, 1)

Consider the lattice congruence a'=(3 /\ <1>1' Since every lattice congruence

contained in <1>1 is a congruence, we have a' E ConI Now

aVa'='l9(a, b)V((3/\ <1>1) =('191at(a,b)V(3) /\('191at(a,b)V<1>1)

=£ /\<1>1

=<1>1,and

a /\ a'='l91at(a,b) /\(3/\ <1>1 =w /\<1>1 =w.

Itfollows thata' is the complement ofa in [w,<l>d <>

Ifa is a congruence on an Ockham algebraL then an a-class [ala will

be calledlocallyfiniteif, wheneverx, y E[a]a withx :s:;;y,the interval[x, y]

is finite

The following two results are due to Jie Fang[67].

a-<b, (a,b)¢<I>n, (a,b)E<I>n+l' Then <l>n V'l9(a, b)isan atom of [<I>n,<l>n+d. Moreover, ifevery <l>n+l -classis

locallyfinite then every atom of [<I>n, <l>n+disobtained in this way.

'l9(j(a),j(b)):S:;; <l>n and consequently, by Theorem 2.1,

(1) <l>nV'l9(a, b)= <l>nV 'l91at(a,b)V'l9(j(a),j(b)) = <l>n V 'l91at(a,b).

Clearly, we have <l>n <<l>nV'l9(a, b):S:;; <l>n+l' Suppose that<pEConL is suchthat<l>n:S:;; <p <<l>nV'l9(a, b). Then we note that

Congruence relations 33

st-t then one of s,t must beaand the other must be b,whence (a, b)E<po

This gives the contradiction 19(a , b):::;; <po Hence we must have s =t, i.e

(xVa) 1\ b=(yVa) 1\ b. Butfrom(*)we havexva Vb=YVa Vband so,

by the distributivity ofL, x V a =y va. Again by(*)and the distributivity of

L,we obtainx =y and hence <p 1\191at(a,b)=W.

It now follows from (1) and (2) that

<p=<p 1\ (<pnV19(a, b))=<p 1\ (<pnV191at(a,b))=<p 1\<Pnand therefore<p :::;; <P n,whence <p =<Pw Hence <PnV19(a, b) is an atom of

[<p n , <P n +l].

Finally, let01be an atom of[<pn,<pn+d. Then there exista, bEL such that

a<b, (a, b)~<P n,and(a, b)E 01 ~<P n+ 1 If every <pn+l-class is locally finitethere existp, qE[a, b]such thatp -<q, (P, q) ~<P n, (P, q)E 01. For suchp, q

we have <Pn<<PnV19(p, q)~ 01, whence01=<PnV19(p, q).0

an atom ofConL.

hence an atom of ConL.0

non-trivial interval [<pn,<pn+d of ConL isa complete atomic boolean lattice.

ProofFor every<pEConL we have <p = V{19(a, b) I(a, b)E<p}. Thus, if

Note that [w,<l>d~ 24 and [<1>1, <1>2]~2.

The condition that every <l>w-class be locally finite cannot be removedfrom Theorem 2.16, as the following example shows

o <Xl <X2 < <a < <Y2 <Yl <1

made into an Ockham algebra by defining

f{O}=1, f{l} =0, {Vi} f{Xi} =f(Yi} =f{a} =a

Here we have <l>w=<1>1 with classes{O}, {I}, and C \ {O,I} The<l>w-class

C \ {O,I} is not locally finite Consider now the partition

{{O}, {xili~l}, {a}, {Yili~l}, {I}}

This defines a congruence in [w,<I>d which has no complement in [w,<I>d.

So in this case[w,<I>d is not boolean

The following interesting result was obtained byJ.Vaz de Carvalho[105].

length of Lism then ConL has at most m atoms, and hasprecisely m atoms

ifand onlyifL itselfisboolean.

that'l9EConL is such thatw~'l9 <'l9(a, b). By Theorem 2.1 we have

'l9(a, b)= V 'l9lat(tk(a)J (b)).

k=O

Since in this casef is a dual automorphism and a -< b, ifkis even we have

jk(a)-<fk(b), and ifk is odd thenfk(b) -< fk(a). In what follows we shall

suppose that k is even; a similar argument holds when k is odd We now

We have u, vE{fk(a)Jk(b}} and(u, v)E'l9 Suppose thatu =fv. Then one

of these elements isjk(a), the other isfk(b), and (tk(a)Jk(b))E'l9 Since'l9 is a congruence it follows that

(a, b)=(t2n-k[Jk(a)], f2n-k[Jk(b}]) E'l9,whence 'l9(a, b)~'l9, a contradiction Thus we have u =v. This, togetherwith condition (2) and the distributivity ofL, gives x Vjk(a) =y Vfk(a);

and this together with condition (1) gives likewisex=y.

Using this observation, we now have

ConL/ <l>q ~ [<I>q,~].

The result now follows by Theorem 2.16.<>

Then each summand of

in L Then we have

and soConL is a bounded distributive lattice whose greatest element is a join

of at mostm distinct atoms, whence it is boolean

If now L is boolean with m atoms then it is well known that ConL isalso boolean with m atoms Conversely, suppose thatLEKn,o is of length

m and that ConL is boolean with precisely m atoms Observe first that if

x EL \ {O,I} thenx Ilf(x). In fact, suppose that x <f(x) and consider amaximal chain passing through both, of the form

0= Zo -< -< x = Zi+l -< -< f(x)= Zt -<f(Zi) = Zt+l -< -<Zm = 1.Since ConL has precisely m atoms, all the congruences t9(zj,zj+d are dis-tinct But (j(Zi),f(X)) Et9(Zi' x) and t9(Zi, x) is an atom ofConL, so wehave the contradiction t9(Zi' x) =t9(j(X),f(Zi))' Similar arguments showthatx> f(x) andx =f(x) give contradictions

Since for every x EL we have f(x Vf(x)) ~x Vf(x), the above servation gives x Vf(x) =1; and from x I\f(x)~f(xI\f(x)) we obtain

ob-x I\f(ob-x)=O Thusx andf(x) are complementary, whenceLis boolean.<>