Decomp osition of Domains

y

Achim Jung Leonid Libkin Hermann Puhlmann

Abstract

The problem of decomp osing domains into sensible factors is addressed and

solved for the case of dIdomains A decomp osition theorem is proved which allows

the represention of a large sub class of dIdomains in a pro duct of at domains

Direct pro duct decomp ositions of Scottdomains are studied separately

Intro duction

This work was initiated by Peter Bunemans interest in generalizing relational databases

see He quite radically dismissed the idea that a database should b e forced into

the format of an nary relation Instead he allowed it to b e an arbitrary antichain in a

Scottdomain The reason for this was that advanced concepts in database theory such

as null values nested relations and complex ob jects force one to augment relations

and values with a notion of information order Following Bunemans general approach

the question arises how to dene basic database theoretic concepts such as functional

dep endency for antichains in Scottdomains For this one needs a way to sp eak ab out

relational schemes which are nothing but factors of the pro duct of which the relation is

a Buneman successfully dened a notion of scheme for Scottdomains and it is

that denition which at the heart of this work We show that his generalized schemes

b ehave almost like factors of a pro duct decomp osition Consequently we cho ose the word

semifactor for them In the light of our results Peter Bunemans theory of generalized

databases b ecomes less miraculous a large class of domains can b e understo o d as sets of

tuples

Bunemans denition of scheme was discussed in and an alternative denition was

prop osed The idea of b oth denitions is that the elements of a domain are treated as

ob jects and pro jecting an element into a scheme corresp onds to losing some information

ab out this ob ject The denition of is based on the assumption that the same piece

of information is lost for every ob ject For example if ob jects are records it means that

we lose information ab out some attributes values The idea of is that every scheme

has a sort of complement and if we pro ject one ob ject to a scheme and the other to its

App eared in Proceedings of the Conference on Mathematical Foundations of Programming Semantics

Springer Lecture Notes in Computer Science 598

y

Addresses Achim Jung and Hermann Puhlmann Fachb ereich Mathematik Technische Ho chschule

Darmstadt Schlogartenstrae D Darmstadt Germany Leonid Libkin Department of Computer

and Information Sciences University of Pennsylvania Philadelphia PA USA LLibkin was

supp orted in part by NSF Grants IRI and CCR and ONR Grant NOOOK

complement then there exists a join of two pro jections ie every ob ject consists of two

indep endent pieces of information Intuitively it means that the domain itself could b e

decomp osed into two corresp onding domains

The denition of is stronger than the denition of It is the rst denition that

is used in our decomp osition theory while the second denition serves as a to ol to describ e

direct pro duct decomp ositions of domains Combining the decomp osition theorems we

will prove a formal statement that claries the informal reasonings from the previous

paragraph

There is also a more philosophical or p edagogical motivation for this work A feature

that novices to frequently nd unsettling is the profusion of dierent def

initions it oers Often these denitions are laid out at the b eginning and the relation to

the semantics of programming languages is established only later In particular useful

closure prop erties of the resp ective categories are derived In his Pisa Lecture Notes

Gordon Plotkin chose a rather more gentle approach The domains he considers are very

primitive at the b eginning just sets and step by step new constructs and prop erties are

added to them a b ottom element transforms sets into at domains and thus the in

formation order is intro duced next come slightly more complicated orders created by

forming nite pro ducts of at domains function spaces call for the denition of dcp o

and Scottcontinuous function and via bilimits and p owerdomains he nally arrives at

binite domains Furthermore along the way he develops a syntax which allows to denote

most of the elements of the domains making them available for computation the pro d

uct app ears as a set of arrays the function space as a set of terms etc This asp ect

is also describ ed elegantly and comprehensively in In this way Plotkin creates the

impression that all binite domains are built up from at domains using various domain

constructors This may b e reassuring for the novice but of course it is not explicitly con

rmed in the text Plotkin is just very carefully expanding his denitions and motivating

each new concept But we may still ask to what extent this rst impression could b e

transformed into a theorem To b e more precise we may ask Is it true that every

binite domain can b e derived from at domains using only lifting pro duct coalesced

sum function space and convex p owerdomain as constructors A similar question was

in fact asked and found dicult by Carl Gunter for the universal binite domain

How would one attack such a problem We think the natural way to do it is to work

backwards and to try to decompose domains into pieces that decomp ose no further If we

can show that the only irreducible domains are the at domains then we are done

At this p oint the informed reader may already have b ecome nervous b ecause he may

know small nite counterexamples to the ab ove question But there are many variations

of it which are equally interesting We can restrict or augment the numb er of allowed

constructions we can change the class of domains we want to analyze we can allow

more or fewer primitive ie irreducible building blo cks The choice we have made

for this pap er is to consider Scott and dIdomains cf and a single alb eit rather

general constructor and instead of prescribing the irreducible factors we are curious

what they will turn out to b e The advantage of a decomp osition theorem of this kind

is apparent instead of proving a prop erty for general domains we can prove that it

holds for the irreducible factors and that it is preserved under the constructions We

allow ourselves to compare this endeavor with the similar and only recently completed

pro ject of decomp osing nite groups into nite simple groups although the comparison

is somewhat attering we cannot exp ect to nd so much mathematically intriguing

structure in domains

What are the practical implications of our decomp osition theorem Well in our

particular setting we derive a very concrete representation of dIdomains as a set of

tuples which should simplify the implementation of dIdomains as abstract data typ es

Of course there is a welldevelop ed theory of eective representations see

where one enumerates the set of compact elements and represents a subset of the

innite elements by recursively enumerable sets of compact approximations However

this is more theoretical work and no one exp ects that we really ever use domains as

data typ es represented this way Instead our representation is much more concrete

To give an example consider a domain which is the pro duct of two at domains The

traditional eective domain theory simply enumerates all elements and if enumerations

of the elements of the two factors are already given then these are combined with the

help of pairing functions We work rather in the opp osite direction For a given domain

we seek to decomp ose it as far as p ossible and we will only enumerate the bases of the

irreducible factors in the traditional way The representation of the original domain is

then put together as a set of tuples

The pap er is organized as follows In the next section we shall quickly review some

basic denitions from domain theory mostly to x notation and to remind the reader

of a few less common concepts In Section we intro duce semifactors and prove basic

prop erties of them We apply these ideas and get a rst decomp osition theorem This

representation still contains a lot of redundancy and in Section we show how to factor

away this redundancy The resulting decomp osition theorem yields a representation of

dIdomains which is very tight These sections rep ort work by the rst and the third

author

A direct pro duct decomp osition is a particular and interesting instance of our general

goal and deserves more detailed study In Sections and which were written by the sec

ond author this is done by establishing a relationship b etween these decomp ositions and

particular instances of congruence relations and neutral ideals The idea to describ e direct

pro duct decomp ositions via neutral ideals is b orrowed from theory where neutral

ideals describ e decomp ositions of b ounded lattices For domains we will obtain a more

general kind of decomp osition including direct pro duct and coalesced sum as limit cases

These decomp ositions are given by families of of a domain such that every element

of the domain has a unique representation as the join of suitably chosen representatives

of these sets Pairs of p ermutable complemented congruences also describ e direct pro duct

decomp ositions as well as they describ e decomp ositions of Having proved char

acterizations of decomp ositions we establish the result showing the relationship b etween

the two notions of scheme

Denitions

We are using the standard denitions such as they can b e found in and in In

particular dcpos are directedcomplete partial orders and they have suprema for all di

rected sets Most of the time they have a least element which we denote by Compact

elements in a domain are such that they cannot b e b elow a supremum of a

without b eing b elow some element of that set already and if there are enough compact

elements such that every element is the supremum of a directed collection of them we

call the dcp o algebraic More suprema than just those of directed sets can exist if every

b ounded set has a join then we call the dcp o boundedcomplete if every set has a join

then we have a In case a b oundedcomplete dcp o is also algebraic we

call it a Scottdomain The expression algebraic complete lattice is shortened to alge

braic lattice We will mostly study distributive Scottdomains for which it is sucient to

require the distributive law to hold in the principal ideals The standard textb o ok on

distributive lattices is Even more restrictive is the denition of dIdomains cf

they are distributive Scottdomains in which every principal generated by a compact

element is nite Because of this strong niteness prop erty we can usually derive theo

rems ab out dIdomains very quickly from the same theorems stated for nite distributive

Scottdomains

All our functions are Scottcontinuous which means they carry the supremum of a

directed set to the supremum of the image of the set We do not make much use of

them in this generality but mostly consider projections which are in addition idemp otent

and b elow the identity Recall that pro jections always preserve existing inma and are

completely determined by their image Even the order b etween pro jections can b e read

o their image it is simply inclusion For more detailed information we refer to

An element x in a lattice is join meet irreducible if from the equation y z x

y z x we can deduce that x equals y or z In the presence of distributivity this

is equivalent to the stronger prop erty of join meet primeness but we will not make

much use of this

Domain theory also includes the concept of ideal which is a directed and downward

closed subset This is a generalization of ideal as it is known in lattice theory where

these are sets which are downward closed and closed under nite suprema We need a

generalization which go es in a dierent direction

Denition A stable subdomain in a Scottdomain D is a downward closed subset which

is closed under all existing joins

The same concept is dened in and in where such subsets of Scottdomains

are called strong ideal and complete ideal resp ectively We nd either expression rather

misleading as we are not dealing with a sp ecial kind of domain theoretic ideal but with

a completely dierent concept Instead we take the viewp oint that such subsets are

sp ecial substructures ie sp ecial sub domains As it happ ens they corresp ond onetoone

to images of pro jections p for which y x implies py y px An even stronger

prop erty holds see Prop osition ii b elow In domain theory such functions are known

as stable projections hence our terminology

Factors of pro ducts of dcp os with b ottom have the prop erty that there is always a

canonical pro jection onto them This is also true for stable sub domains in Scottdomains

Lemma Let A be a stable subdomain of the Scottdomain D Then p D D dened

A

by

p x x A

A

is a projection on D with image A

Our rst decomp osition has the form of a general categorical limit A concrete de

scription is given in terms of certain elements of the pro duct of the dcp os involved

Denition Let D b e a set of dcp os and let F b e a set of Scottcontinuous functions

b etween elements of D in the language of category theory a diagram in DCPO Fur

thermore let x x b e an element a tuple of the cartesian pro duct of all elements

D D D

of D We say that x is commuting if the equation x f x holds for all functions

E D

f D E f F and all elements D E in D Similarly it is called hypercommuting if

the inequality x f x holds

E D

The set of all commuting tuples forms the categorical limit of the diagram D F and

we denote it by lim D The set of hyp ercommuting tuples we call the hyperlimit and

F

we reserve the notation hyp erlim D for it The latter construction is a sp ecial case of a

F

more general concept develop ed in the theory of categories namely lax limits It is easy

to see that DCPO is closed under limits and this kind of lax limit Whether any of the

other prop erties generally asso ciated with domains is preserved dep ends on the structure

of the diagram For more detailed information consult

Stable sub domains semifactors and the First De

comp osition Theorem

We b egin by recalling from and some of the prop erties of stable sub domains

Prop osition Let D be a Scottdomain Then the fol lowing hold

i f g and D are stable subdomains of D

D

ii If x is an element of a stable subdomain A of D and if p y is less than x then

A

p y x y

A

iii If D is distributive then p preserves existing suprema

A

iv The set Q of al l stable subdomains of D ordered by inclusion is an algebraic lattice

D

v If D is distributive then Q is distributive

D

vi In Q the nite meet of stable subdomains is given by their intersection and p

D AB

p p p p

A B B A

vii If D is distributive then arbitrary suprema in Q can be calculated pointwise and

D

W

W

p x for A Q x D p x

A D

AA

A

Pro ofs can b e found in

The concept of stable sub domain is still to o general to serve as a denition of distin

guished piece of a domain For example every element x of a domain generates a stable

sub domain x but in general such a principal ideal cannot b e hop ed to lead to a sensible

decomp osition In a more restrictive denition is intro duced that of a scheme and it

is motivated by the database applications we had in mind there Here we can give a new

motivation based on the desired decomp osition result Consider the following theorem

Theorem Let D be a nite distributive Scottdomain and let A be a set of stable

subdomains the supremum of which equals D Let F be the set of projections

Q

D

p j B A where A B are two elements of A Furthermore let D consist of those

A B

commuting tuples x x for which the set fx j A Ag is bounded in D Then D

A AA A

is isomorphic to D with the isomorphisms

D D x p x

A AA

D D x fx j A Ag

A

The pro of of this theorem is straightforward one only has to b ear in mind that suprema

in Q are calculated p ointwise The theorem is unsatisfying however b ecause in order

D

to represent D through a set of stable sub domains we need to include information that

can only b e gained by lo oking at D itself the b oundedness of the co ordinates of x We

shall now give a denition of a semifactor such that b oundedness comes for free if only

the tuple commutes

Denition A stable sub domain A of a Scottdomain D is called semifactor if p x a

A

implies that x and a are b ounded for all x D and a A

In and in it is shown that this denition works well in the test case of direct

pro duct decomp ositions the semifactors of a direct pro duct D E are in corresp on

dence with pro ducts of semifactors of D and E In particular D f g and f g E

E D

are semifactors in D E

We collect the basic prop erties of semifactors in a fashion similar to that for stable

sub domains

Prop osition Let D be a distributive Scottdomain Then the fol lowing hold

i f g and D are semifactors of D

D

ii The set S of al l semifactors of D ordered by inclusion is a distributive complete

D

lattice

iii If S and T are semifactors of D then so are S T and S T where again the

join is taken pointwise The latter also holds for arbitrary joins

iv S is a sublattice of Q

D D

For the pro ofs see

The following lemma states that our denition yields the desired extension prop erty

Lemma Let S be a family of semifactors of a nite distributive Scottdomain D and

let F consist of al l connecting projections as in Theorem above Let S be such that with

S T S we also have S T S If x x is a commuting tuple then the set

S S S

fx j S S g is bounded in D S

Pro of We rst show this for the case in which S consists of just three semifactors S T

and S T Let x b e a commuting tuple in S T S T

x p x p id

T S T T S T D

x x is commuting

S T

p x ditto

S T S

p p x vi iii

T S S

p x p j id

T S S T T

By the dening prop erty of semifactors fx x g is b ounded in D

S T

W

n

S and T S By the induction The general pro of is by induction Set S

i n+1

i=1

W

n

g exists and we may set x x The tuple x hyp othesis the join of fx

S S S S

n

i=1 1

i

x x p x is commuting for the three semifactors S T and S T b ecause pro

S T S T S

W

n

x jections preserve suprema by iii p x p p x p x p

S S T S T S S T S T

i=1

i

W W W

n n n

p x x p x p x p p x p x So we can

T S S T S T S T S T T S T T

i=1 i i=1 i i=1 i

apply the result for the three element case for the induction step 2

In our decomp osition theorem we want to use as few semifactors as p ossible which

in turn should b e as primitive as p ossible As a rst approximation we cho ose the set

J S of semifactors which are joinirreducible in Q This set has two prop erties which

D D

make it attractive every semifactor is a join of irreducibles in the nite case but it will

generalize to dIdomains and a joinirreducible cannot b e reached by a join of strictly

smaller semifactors so it is in a sense unavoidable But in order to apply the previous

lemma we need a set closed under nite intersections and in general J Q will not do

D

us this favor We need another preparatory lemma

Lemma Let D be a nite distributive Scottdomain and let J S be the set of join

D

irreducible semifactors of D Let x x be a commuting tuple for J S and

S D

S J (S )

D

the connecting projections F Let F be the appropriately extended set of connecting

projections for al l of S Then x can be extended uniquely to a commuting element x for

D

S F

D

Pro of We rst show that for two joinirreducible semifactors U and V we have the

following commutation rule p x p x Indeed if U V is the join of the

U V V U

joinirreducible semifactors U U then we can calculate p x p p x

1 n U V U V V

W W W

n n n

p x p x p p x p x x p x p x

U U U V U V U U V U U U V U V V

i=1 i=1 i=1

i i i

We extend the tuple x to all of S by setting

D

x fx j S U J S g

S U D

We have to show that x x is commuting so let S T b e two semifactors of D

S S S

D

Then we have

p x p fx j T U J S g by def

S T S U D

fp x j T U J S g iii

S U D

fp x j T U J S S V J S g vii

V U D D

fp x j T U J S S V J S g as shown b efore

U V D D

fp x j S V J S g vii

T U D

fx j S V J S g V S T

V D

x by def 2

S

We can now state

Theorem The First Decomp osition Theorem Let J S be the set of al l join

D

irreducible semifactors of the nite distributive Scottdomain D and let F be the set

of connecting projections Then D is isomorphic to the limit of J S over F The

D

isomorphisms are given by

D lim J S

D

F

x p x

S S J (S )

D

and

lim J S D

D

F

x x

S S J (S ) S

D

S J (S )

D

The pro of of this theorem is contained completely in the previous lemma where we

showed how to extend a commuting tuple to all of S in particular to D S itself 2

D D

We illustrate the First Decomp osition Theorem for three nite domains

Example D M M a nite set ie D is a at domain We nd that D p ossesses

only the trivial semifactors fg and D the latter b eing joinirreducible in S Hence we

D

conclude

Observation Flat domains are indecomp osable

Example D the fourelement Bo olean Since D is a lattice it is

isomorphic to its lattice of semifactors The joinirreducibles are and and

the decomp osition yields D where This is

not a coincidence

Observation Direct pro duct structure is recognized

Example

c

c c

c c c c

c d

c c c c

a b

D c S c

D

We nd that D is joinirreducible in S and hence must b e contained in any decom

D

p osition based on the First Decomp osition Theorem This is obviously not satisfactory

and we shall derive a b etter decomp osition theory b elow Before doing so let us study

the situation for innite domains Here we have to deal with the following complication

the intersection of an innite family of semifactors is not necessarily a semifactor again

We therefore do not know whether S is algebraic in general We view this as the ma jor

D

op en problem in our decomp osition theory In the case of dIdomains we are ne

op

Prop osition Let D be a dIdomain Then S is algebraic and coalgebraic ie S

D

D

is algebraic

Pro of We only give an outline b ecause we dont have the space to intro duce the details

of the theory of approximation via compact elements in domains in general and in our

decomp osition theory in particular

One rst observes that stable sub domains and semifactors are completely determined

by the set of compact elements they contain Also the canonical pro jection onto a stable

sub domain can b e seen as mapping each element onto the supremum of those compact

W

elements of the sub domain which are b elow it p x x K D A Furthermore

A

the canonical pro jection as a Scottcontinuous map is completely determined by its

b ehavior on compact elements Since it is also sucient to state the extension prop erty

of semifactors for compact elements only we have reduced the whole theory to K D

the set of compact elements in D With this in mind it is now easy to see that the

arbitrary intersection of semifactors is again a semifactor b elow a in

a dIdomain there are only nitely many elements at all and so for a particular compact

element the intersection b ehaves as if it were over a nite index set

Similarly it is easy to see that the directed union of semifactors yields a semifactor

again Together this shows that the set S of semifactors forms an inductive hull system

D

on D which implies algebraicity A semifactor is compact in S if and only if it is

D

generated by a nite set of compact elements of D

The cocompact elements are found as follows supp ose a semifactor S do es not

contain a certain element x of D By algebraicity of D it follows that there is a compact

element c of D which S do es not contain Furthermore b ecause c is a nite distributive

lattice there is a joinirreducible k b elow c which again do es not b elong to S On the

other hand if k is joinirreducible hence prime in K D then the join of all semi

factors which do not contain k will again not contain this element From this it follows

along standard lines that any nite set of joinirreducible elements of K D denes a

cocompact semifactor and that there are enough cocompact semifactors to generate

the whole lattice S So it is coalgebraic as well 2

D

From we recall that algebraic lattices have an infbasis of meetirreducible ele

ments and so for a dIdomain D the distributive lattice S has b oth a supbasis of

D

joinirreducibles and an infbasis of meetirreducibles We can therefore state

Corollary The First Decomposition Theorem holds for dIdomains

Factoring by stable sub domains and the Second

Decomp osition Theorem

In group theory and in ring theory we are familiar with the following technique For a given

strong substructure normal subgroup ideal resp ectively one studies the equivalence

relation which identies those elements which dier only by an amount contained in the

substructure A similar notion works for ideals in distributive lattices If A is an ideal

in L then we can set x y if there is an a A such that x a y a for details

see Since domains lack arbitrary suprema we have to rework this denition a little

bit

Denition Let A b e a stable sub domain in a distributive Scottdomain D On D dene

a binary relation by setting x y if there is a A such that y x a Let b e the

A A A

symmetric and transitive hull of that is the smallest equivalence relation containing

A

S

1

n

can b e describ ed concretely as

A A

nN

A

This denition proves to b e extremely fruitful We list the following prop erties

Prop osition Let D be a nite distributive Scottdomain and let A be a stable subdo

main in D Then the fol lowing hold

i x y x y

A

ii x y y x p y and for al l a Aif y x a then a p y

A A A

iii

A A A

iv x y z D z x z y and z x z y Provided the suprema exist

A A A

v is a congruence relation on D with respect to nite inma and existing suprema

A

vi Each equivalence class of contains a least element

A

vii

A A

viii Each equivalence class of is order convex

A

1

ix

A A

A

x p is injective on every equivalence class of

A A

We denote the function which maps each element onto the smallest element in its

equivalence class by q With this notation we can add the following clauses

A

xi q is a projection on D

A

xii q preserves existing suprema

A

Pro of i is trivial for ii recall that p is joinpreserving by iii

A

iii x y z y x a and z y a x a a and with a and a

A A 1 2 1 2 1 2

elements of A their join is again in A

iv x y y x a z y z x a z x z a and with a A

A

the element z a is again in A For suprema x y y x a z y z x a

A

1

v It is immediate from the denition of as a union of pro ducts of and

A A

A

that iv also holds for Now if x y and x y then x x y x y y

A A A A A

and analogously for suprema

vi follows from v by taking the inmum of the equivalence class

vii Supp ose x y and x y Then by denition there is a chain x x x

A 1 2 n

1 1

x x x y By taking the x x of elements such that x x

4 n1 A x 2 A 3 1

A A

supremum of each element of this sequence with x and then the inmum with y we derive

a new sequence which is completely contained in the interval x y x is then necessarily

2

1

equal to x We further shorten the sequence as follows x x x x x x

2 2 4 A 3 4

A

1

x x x x Since x x is b elow x and in relation it is actually equal

4 4 4 A 5 3 4 4

A

to x so the sequence now reduces to x x x Applying iii we nd that x is in

4 A 4 A 5

relation to x already Continuing in this fashion will reduce the sequence eventually

A 5

to x y which is what we want

A

viii Assume x y and x z y By vii we have x y which implies y x a

A A

for some a A But then z z y x z x a x z a which gives us x z

A

The relation z y follows directly from y x a

A

ix Combining vi and vii we nd that the least element of an equivalence class is

in relation to each memb er

A

x A pro jection always preserves inma and so if p maps two elements x and y to

A

the same image a it will map x y to a as well and if x y then x y y and

A A

by vii x y y So consider wlog x y and p x p y We directly get

A A A A

y x p y x p x x

A A

xi We only have to show that q is monotone So supp ose x y By vii we have

A

q y y which yields with iv x q y x y x But q x is the smallest

A A A A A

element in the equivalence class of x Hence q x x q y q y follows

A A A

xii From z x y q x x q y y we conclude by iv that q x q y

A A A A A A A

x y z Since q is monotone it follows that q x q y must b e equal to q z 2

A A A A

Given a representation of a p oset P as the cartesian pro duct of two p osets R and S

we can understand P as follows it consists of jR j many copies S of S and if x y in R

x

then each element of S is b elow the corresp onding element of S A semifactor S in a

x y

nite domain leads to a similar representation for each element x in the image R of q

S

we take the principal lter F p x in S instead of the whole semifactor These

x S

lters are connected as b efore that is if x y in R then each element of F is b elow the

x

corresp onding element of F However there may b e elements of F for which there is no

y x

corresp onding element in F This is the content of the following prop osition A picture

y

illustrating this representation is given in Figure

Knowing S im q and the action of p on the image of q we can reconstruct the

S S S

domain

Prop osition Let D be a nite distributive domain and let S be a semifactor in D

i The image of an equivalence class of under p is upward closed in S

S S

ii D is isomorphic to the set D fx s im q S j p x sg ordered pointwise

S S

The isomorphism is given by q p D D and by the supremum function for the

S S

other direction

iii If x D p q x then S is a direct factor of D

S S D

Pro of i If s is ab ove p x in S then by the extension prop erty of semifactors s x

S

1

exists and is in relation to x Also p s x p s p x s p x s

S S S S S

@

@

@

b b

b b

@ @

@ @ b b

b b

@ @

b b

b b

@ @

b b

@ @ b b

b

b

@

b

@

b

b

@

b

b

@

b

@

@ S

Figure Decomp osition of a domain by a semifactor S

ii For x D we have q x x and therefore p q x p x So the pair

S S S S

q x p x b elongs to D The mapping q p is injective by Prop osition x We

S S S S

claim that the inverse is given by the supremum function First of all the supremum

exists for the pairs in D b ecause S is a semifactor It is clearly monotonic and it inverts

q p b ecause q p x s q x s p x s q x q s p x p s

S S S S S S S S S S

x p x s x s and for the other comp osition q x p x x b ecause

S S S

q x x by denition

S S

iii This follows b ecause for p q the condition in the denition of

S S

D D

D is always satised 2

This prop osition works with elements of the domain But there is also a way of

lo oking at this situation using congruence relations Recall that every homomorphism

f D E induces a canonical congruence relation on D called the kernel of f ker f

which identies exactly those elements of D which are mapp ed to the same element

Obviously ker q Let C onD b e the complete lattice of all congruences with

A A

resp ect to nite inma and existing suprema on D

Prop osition Let D be a nite distributive Scottdomain and let A be a stable subdo

main in D Then the fol lowing is true

i ker p is a congruence with respect to arbitrary inma and arbitrary existing

A

suprema

ii ker p

A A D D C on(D )

iii ker p D D

A A

C on(D )

Pro of i holds b ecause p is a pro jection on a distributive domain ii restates x

A

and nally iii follows b ecause every x D is related to in the following way

x q x ker p p q x 2

A A A A A A

The results of this section extend to dIdomains

Prop osition Proposition and Proposition hold for dIdomains in particular

equivalence classes of and ker p are closed under directed suprema and q is Scott

A A A

continuous

Pro of The main technical diculty is to prove that equivalence classes of have a

A

least element For details we refer the reader to 2

We use factorization to improve on our First Decomp osition Theorem We observed

that it pro duces representations which are redundant namely if two comparable semi

factors S T are joinirreducible in S then b oth take part in the representation T re

D

p eating the information given by S We shall now factor out this rep eated information

Given a collection S of semifactors we dene for each element S S its lower S cover

W

S by S fT S j T S g Also if S T S let S stand for imq j j With this

D T T S

notation we are now ready to formulate

Theorem The Second Decomp osition Theorem Let D be a nite distributive

Scottdomain a dIdomain and let J S be the set of al l joinirreducible semifactors

D

of D Dene

0

j S J S g RJ S fS

D D S

and

0

p j S T J S g F fq

S T D S

0

T

Then D is isomorphic to the hyperlimit of RJ S over F with the isomorphisms

D

D hyp erlim RJ S

D

F

0

p x x q

S S

S J (S )

D

and

hyp erlim RJ S D

D

F

x x

S S

S J (S )

D

S J (S )

D

The pro of of this should b e clear from the First Decomp osition Theorem and Prop o

sition

We illustrate the representation of domains provided by the Second Decomp osition

Theorem with Example from the last section The three joinirreducible semifactors

are a b and D itself By factoring D through the join of a and b we can replace it

by the three element domain f c dg

Decomp osition into at domains is particularly satisfying and one may wonder whether

it is achievable for all distributive Scottdomains or for all dIdomains The answer is no

a counterexample is given in Figure

However it turns out that the category F of those distributive Scottdomains which

are representable as hyp erlimits of at domains is cartesian closed and contains strictly

all concrete domains cf Indeed the connection to concrete domains seems

to b e very strong Recent work by Geva and Bro okes see their contribution to this

volume suggests that every domain in F can b e represented as a generalized concrete

data structure

c c c

c c c

c

Figure A nonat indecomp osable dIdomain

Characterization of direct pro duct decomp ositions

There exist several nice characterizations of the direct pro duct decomp ositions of arbitrary

algebras see In this section we will nd the analogues to two of them for domains

There are several reasons to study the direct decomp ositions of domains The rst rea

son is of course purely theoretical However knowing domain decomp ositions may b e

imp ortant from the practical p oint of view In the intro duction we briey describ ed the

idea of generalizing relational databases that denes a relation as a nite antichain in a

Scott domain There are several advantages of this approach Firstly it gives a formal

framework for having attributes of arbitrary typ es p erhaps admitting null values Sec

ondly it allows constructions more general than simply relations matrices for example

record and variant constructors can b e applied to form very complex generalized records

see Supp ose that we have a Scottdomain whose nite antichains are considered as

relations A question arises how far can this domain b e seen as the direct pro duct of its

sub domains In the other words how far can our relations b e seen as usual relations

that is sets of tuples and what are the attributes of these tuples To answer this question

we need a characterization of the direct decomp ositions of Scottdomains

We will characterize direct pro duct decomp ositions via complemented p ermutable con

gruences and neutral complemented ideals Surprisingly the characterization we will ob

tain is based on one of the concepts related to the domain approach to databases In fact

all factors in the direct decomp ositions will b e schemes as they were intro duced in

Schemes generalize semifactors and serve as the main to ol to intro duce relational algebra

and dep endencies in the domain mo del of databases

The decomp osition theorems also enable us to prove a result which explains the other

notion of scheme prop osed in In fact the denition of schemes of here we call

them semifactors works for domains which are similar to domains of at records Ac

cording to the denition of a scheme is a stable sub domain such that the asso ciated

pro jection maps the maximal elements of the domain to the maximal elements of the

stable sub domain This denition emphasizes the fact that schemes are signicant parts

of domains It is more general than the original denition of semifactor which in turn is

based on prop erties that schemes of domains of at records should satisfy We will prove

that in a certain class of domains the condition that every scheme is a semifactor implies

that the domain is very similar to domain of at records Thus our decomp osition results

are helpful in understanding the domaintheoretic generalization of databases

This section contains ve subsections In the rst we recall two wellknown results from

The second demonstrates how domains can b e treated as partial alge

bras with op erations of innite arity and intro duces the concepts of ideal and congruence

for them the rst one b eing identical to the stable sub domains In the third subsection we

give our basic lemma which reduces direct pro duct decomp ositions of domains to those of

principal ideals generated by maximal elements Neutral ideals are studied in the fourth

subsection They give rise to socalled general decompositions which have coalesced sum

and direct pro duct decomp ositions as two limit cases The fth subsection deals with

characterizations of direct pro duct and general decomp ositions via congruences

From now on all domains are Scottdomains and we will write domain instead of

Scottdomain We will denote the sets of maximal elements of a domain D and of a stable

max max

sub domain A by D and A resp ectively

We will characterize only decomp ositions into a nite numb er of factors Notice that

in fact we do not need to consider more than two factors Therefore in all the results

b elow only decomp ositions into two factors are characterized

For the sake of brevity only sketches of the pro ofs will b e given and the technical

details will b e omitted

Algebraic preliminaries

Let us recall two techniques that are used in universal algebra and lattice theory to

describ e direct decomp ositions

Given an algebra hA i a congruence on it is an equivalence relation on A such

that for any of arity n and for any x y A i n i n x y

i i i i

implies x x y y Congruences considered as binary relations form

1 n 1 n

an algebraic lattice C onhA i in which the b ottom element is the equality and the

top element is the total equivalence relation Then direct pro duct decomp ositions of

hA i ie decomp ositions hA i hA i hA i are in onetoone corresp ondence

1 2

with pairs such that is a complement of in C onhA i and are

1 2 2 1 1 2

p ermutable ie see In fact hA i hA i and hA i

1 2 2 1 1 1 2

hA i or vice versa

2

Let L b e a lattice An element a L is called neutral if for every x y L the

sublattice hx y ai generated by x y a is distributive If L is b ounded then every direct

a where a is pro duct decomp osition L L L can b e represented as L a

1 2

a is its complement see Both ideals a and a neutral complemented element and

a are neutral elements of the ideallattice ie socalled neutral ideals It is also well

known that there is a onetoone corresp ondence b etween direct pro duct decomp ositions

L L L and pairs I I where I I are neutral ideals and I is a complement of I

1 2 1 2 1 2 2 1

in the ideallattice In fact L I and L I or vice versa

1 1 2 2

Domains as algebras Congruences and ideals

In order to transfer the previous characterizations to domains we have to intro duce

algebraic structure on domains Let D b e a domain We consider it as a partial algebra

containing the op erations of innite arity namely inma and existing suprema for all

p ossible subsets X D Thus D b ecomes a partial algebra whose op erations may b e of

arbitrary arity It is wellknown that the previous results ab out decomp ositions are not

true for algebras with partial op erations of innite arity

A subalgebra of this partial algebra could b e called subdomain but in semantics this

notion has no generally accepted meaning In lattice theory an ideal is a downward closed

set which is closed under nite joins In our algebraic interpretation ideals are downward

closed subsets of a domain which are closed under all existing joins ie they are stable

subdomains A congruence is an equivalence relation such that for any x y D i I I

i i

V V

an arbitrary set of indices x y for all i I implies fx i I g fy i I g and if

i i i i

W W

b oth x fx i I g and y fy i I g exist then xy

i i

If D is a lattice our denition of congruence coincides with the denition of complete

congruence of a lattice intro duced recently in These congruences form a complete

lattice denoted C onD However in contrast to the case of op erations of nite arity this

lattice may fail to b e algebraic It was proved in that for every complete lattice L

there exists a lattice M such that L is isomorphic to the lattice of complete congruences

of M Moreover M can b e chosen among mo dular algebraic lattices Therefore we

cannot guarantee algebraicity of C onD

x

If x D then is the restriction of to x that is x x The lattices

of congruences of x will b e denoted by C onx if D is understo o d x denotes the

equivalence class of x ie x fy xy g If is a congruence then is an ideal

that is a stable sub domain

We will also also need a concept of scheme It was intro duced in in order to

generalize the analogous concept of known in this pap er as semifactor A stable

sub domain A D is called a scheme if p x is a maximal element of A for every

A

max

x D Any semifactor is a scheme Although all the factors we will decomp ose

domains into will b e semifactors it is enough to require that they b e schemes

The main lemma

The following lemma generalizes the result of which describ es direct pro duct decom

p ositions of b ounded lattices Theorem of says that the direct pro duct decom

p ositions of a b ounded lattice L into two factors are in onetoone corresp ondence with

a neutral complemented elements a L In fact L a a or equivalently L a

a and it is In the lemma b elow a pair of complemented schemes plays the role of a

required that the pro jections of any maximal element x into these schemes b e neutral and

complementary in x

Notice also that according to direct decomp ositions can b e equivalently describ ed

a and we by pairs of neutral complementary ideals in fact these ideals are just a and

can view the pair of schemes as an analogy of these ideals rather than neutral elements

However in contrast to the lattice case neutral complementary elements of the lattice of

ideals stable sub domains of a domain do not characterize direct decomp ositions as we

will see in the next subsection

Lemma The direct product decompositions of a domain D are given by pairs of

max max

schemes A A such that A A fg and x A y A x y ex

1 2 1 2

1 2

max

ists x y D and x is a neutral element in the principal ideal x y In fact

D A A Moreover A and A are neutral complementary elements of the lattice

1 2 1 2

Q of stable subdomains

D

Pro of Let D D D We can think of b oth D and D as b eing subsets of D ie

1 2 1 2

we can asso ciate D with fx x D g and D with f y y D g Denote

1 1 2 2

these subsets by A and A Obviously b oth A and A are stable sub domains We

1 2 1 2

max max

have A A f g If x A and y A then x hx i y h y i where

1 2

1 2

D

max max

max

x D y D Thus x y hx y i D Obviously x and y complement

1 2

each other in x y To prove that x is neutral we need to show that x y x y

This isomorphism is given by u v hu v i for any u v x y

D

The fact that A and A are neutral and complement each other in Q will follow from

1 2

D

Theorem

Let conversely A A b e stable sub domains satisfying the conditions of the lemma

1 2

We prove D A A Consider two mappings D A A and A A D

1 2 1 2 1 2

y i and hu v i u v notice that this u v always exists x p where x hp

A A

2 1

Then the following claims can b e proved

If x A y A then x y hx y i

1 2

For any z D z p z p z

A A

1 2

From these two claims it can b e easily concluded that and are mutually inverse

bijections Since they are b oth monotone they establish the desired isomorphism 2

Remark We did not exploit the fact that domains are algebraic cp os in the pro of

of lemma In fact the lemma is true for any b ounded p oset P in which greatest lower

b ounds exist for all pairs of elements and least upp er b ounds exist for all pairs b ounded

ab ove Here b oundedness means that P is p ointed that is it has the smallest element

max

and each x P is b ounded ab ove by some maximal x P

General decomp ositions of domains

It is a natural question whether all neutral complemented elements of the lattice of stable

sub domains give rise to direct pro duct decomp ositions as it is the case for lattices The

answer as we are going to show is no In fact neutral complemented ideals describ e

a much more general kind of decomp osition which for example also includes coalesced

sum decomp osition

Denition A pair of stable sub domains A A of a domain D is called a general de

1 2

composition of D which is denoted by D compA A if every element x D has

1 2

unique representation as x x x such that x A and x A

1 2 1 1 2 2

The direct decomp ositions D A A and the coalesced sum decomp ositions D

1 2

A A where A A are stable sub domains of D are two main examples of the general

1 2 1 2

decomp ositions that is in b oth cases D compA A However there exist general

1 2

decomp ositions dierent from the direct and coalesced sum decomp ositions Consider the

following domain D

y y

1 2

D L

D L

D L

D L x x x x

1 2 3 4

L

L

L

L

Then D has several general decomp ositions For example D compfx x g fx x g

1 3 2 4

or D compfx x g fx x g

1 4 2 3

Obviously if D compA A then A A fg Really if x A A and

1 2 1 2 1 2

x we have two representations x x x and x x

Lemma If D compA A then the unique representation of any x D as x x

1 2 1 2

x 2 x p where x A x A is x p

A 1 1 2 2 A

2 1

Theorem The general decompositions of a domain D are in onetoone correspon

dence to pairs of neutral complementary elements of the lattice Q ie D compA A

D 1 2

i A A are neutral elements of Q complementing each other

1 2 D

Pro of Let D compA A Then A and A complement each other in Q Thus

1 2 1 2 D

we only need to prove that A is neutral in Q To do so we will show that there is an

1 D

isomorphism Q Q Q such that A h i where and stand

D A A 1 Q Q

1 2

A A

1 2

for the top and b ottom elements of a lattice Then the neutrality will follow from

Given A Q let A hA A A A i As in the pro of of Lemma intro duce

D 1 2

another map Q Q Q by hI I i I I According to Lemma for

A A D 1 2 1 2

1 2

any A Q A A A A A that is A A

D 1 2

Let I A I A b e stable sub domains Let I fx x x I x I g

1 1 2 2 1 2 1 1 2 2

Supp ose x x x I If z x x then p x p z i By Lemma

1 2 1 2 A A

i i

x p z Thus p x I and x p x p x I This shows that I is an

i A A i A A

1 2

i i

ideal It is also easy to show that I is a stable sub domain Now let x x x I A

1 2 1

Then p x x x ie x I This shows I I A I i In the

A 1 2 2 1 1 2 i i

2

other words hI I i hI I i

1 2 1 2

Therefore and are mutually inverse monotone bijections which establishes the

desired isomorphism Since A hA fgi A is neutral

1 1 1

Conversely let A A b e two complementary neutral elements of Q Then by

1 2 D

for any A Q A A A A A If A x we have x p x p x

D 1 2 A A

1 2

Hence x p x p x If x x x where x A i then p x

A A 1 2 i i A

1 2 1

x A x x A x A x A x A x Thus x p x

1 1 2 1 1 1 2 1 1 1 1 1 A

1

Analogously x p x This shows that D compA A The pro of is complete 2

2 A 1 2

2

The neutral complemented elements of any lattice L form a Bo olean sublattice often

denoted C enL Notice that if Q do es not have innite chains then C enQ is a nite

D D

Bo olean lattice and only representation of D as the comp osition of indecomp osable factors

is in the sense of the op eration of general decomp osition D comp A

AAtoms(C en(Q ))

D

where AtomsC enQ is the set of atoms of the nite lattice C enQ For example if

D D

D do es not have innite chains then the same is true of Q and the following corollary

D

holds

Corollary If D does not have innite chains D can be factored into indecomposable

subdomains in one and only one way 2

It was mentioned b efore that direct pro duct and coalesced sum decomp ositions app ear

now as two limit cases of general decomp ositions In fact the following holds

Prop osition A general decomposition D compA A is a direct product

1 2

decomposition D A A i x A y A x y exists

1 2 1 2

A general decomposition D compA A is a coalesced sum decomposition D

1 2

A A i x A y A x y x y does not exist

1 2 1 2

Pro of If D D D and A A are dened as in the pro of of lemma then

1 2 1 2

D A A and D compA A Obviously x y exists for any x A y A

1 2 1 2 1 2

Now supp ose that D compA A and the condition of holds In order to prove

1 2

D A A we must check the conditions of Lemma ie

1 2

max max max

a For any x A y A x y D

1 2

b A A are the schemes

1 2

c x is neutral in x y

max

To prove a notice that if v x y then x p v since x A Analogously

A

1

1

max

y p v and by Lemma v x y To prove b consider any v D If w p v

A A

2 1

max max

and then since w p v exists v w p v ie p v w A and w A

A A A

1 1 2 2 1

A is a scheme Analogously A is a scheme To prove c notice that for any z x y

1 2

p z z x and p z z y thus z z x z y Then x y x y

A A

1 2

given by z hz x z y i is an injective monotone map Since hz z i z z is its

1 2 1 2

monotone inverse is a lattice isomorphism and x is neutral b ecause x hx i

is straightforward 2

It follows from part that if D is a lattice its general decomp ositions are exactly

direct decomp ositions Therefore from Corollary we derive the wellknown fact that

a lattice without innite chains admits exactly one representation as the direct pro duct

of nontrivial indecomp osable factors up to isomorphism and p ermutation of factors

see

Characterization via congruences

We have shown so far that in contrast to the lattice case the direct decomp ositions of

domains are not describ ed by their neutral complemented ideals We had to intro duce

another kind of decomp osition called general decomp osition to b e in corresp ondence

with these ideals

In this subsection we examine another approach to describing decomp ositions We

will try to nd and characterize pairs of congruences such that quotient sets are exactly

factors of a decomp osition In universal algebra it is known that pairs of p ermutable

complementary congruences exactly describ e direct decomp ositions This result holds for

domains to o although domains are considered as algebras with partial op erations of nite

and innite arity Congruences also help understand the general decomp ositions b etter

It will b e proved that factors of general decomp ositions are in corresp ondence with

pairs of congruences which are complementary and p ermutable when restricted to the

principal ideals of maximal elements

Dene the following two mappings b etween the pairs of congruences and and

1 2

pairs of stable sub domains of a domain A A D

1 2

A A A A

1 2 1 2 1 2 2 1

A A x y i p x p y i

1 2 1 2 i A A

i i

To b e more precise if and are congruences then A A dened in are stable

1 2 1 2

sub domains and for a pair of stable sub domains the relations dened in are equivalence

relations However they are congruences in the sp ecial cases considered b elow

The following two theorems show that ab ove dened and set up corre

sp ondences b etween the factors of direct decomp ositions or general decomp ositions of a

domain and the sp ecial pairs of congruences on this domain

Theorem Let D be a domain Then and form a onetoone correspondence

between the factors of the general decompositions D compA A and pairs of congru

1 2

max x x

ences such that for every x D the congruences and are permutable

1 2

1 2

x x

and is a complement of in C onx

2 1

Theorem The mappings and A A form a onetoone corre

1 2 1 2

spondence between the factors of the direct product decompositions D A A and pairs

1 2

of congruences such that and are permutable and is a complement of

1 2 1 2 2

in C onD

1

Before going to the pro of of these two theorems we formulate and prove one corollary

which claries why in the intro duction to this subsection we sp oke of quotient sets rather

than equivalence classes of b ottom element

Corollary In the previous theorems the factors A A the domain D is decomposed

1 2

into are isomorphic to D and D respectively

1 2

For example we have a p erfect analogy of the congruence characterization of the

decomp ositions of algebras In fact all the decomp ositions of a domain D are of form

D D D for pairs of complementary p ermutable congruences

1 2 1 2

Pro of Let D A b e given by x p x This denition is correct by

1 1 1 A

1

It also follows from that is injective and since z A z z it is

1 1

surjective as well Let x y in D Then x y x thus p x p y

1 1 1 1 A A

1 1

p x which proves the monotonicity of A D given by z z is the

A 1 1 1

1

inverse of and it is also monotone Thus A D The pro of of A D is the

1 1 2 2

same 2

Pro of Two claims b elow are needed in the pro ofs of b oth theorems After proving these

claims we outline the pro ofs of the theorems themselves

Claim and dened in are congruences if D compA A or D

1 2 1 2

A A

1 2

It is enough to show is a congruence Obviously is an equivalence relation

1 1

V V

Since p x x for any set of indices I preserves arbitrary meets p

A i i 1 A

iI iI

1 1

y Really since A is a neutral element x p x y p If x y exists then p

1 A A A

1 1 1

y then z x p of Q by Theorem if z A such that x y z p

A D 1 A

1 1

A x y A x A y p x p y p x p y z a

1 1 1 A A A A

1 1 1 1

contradiction We conclude that the ab ove equality holds and preserves nite joins

1

W W

p x To prove the reverse inequality x Then p x Now let x

A i i A

iI iI

1 1

W

supp ose y p x and y is compact Then y x and y x where I I is

A i 0

iI

1

0

W W W

x Since any element is the p x p x nite Thus y p

i A i A i A

iI iI iI

1 1 1

0 0

preserves join of compact elements b elow it this proves the reverse inequality Thus p

A

1

arbitrary joins and so do es ie is a congruence The claim is proved

1 1

Claim Let be permutable congruences on a domain D Then their join in

1 2

C onD is

1 2 1 2

Since it suces to show that is a congruence

1 2 C on(D ) 1 2 1 2

The only nontrivial part of the pro of is to show that preserves arbitrary existing joins

W W

y exist and x y for all i I Then for each x and y Assume that x

i i i i

iI iI

i there is a z such that x z and z y Then x x z From z y we have

i i 1 i i 2 i i 1 i i i 2 i

z x z x y Analogously we have y z x y z and y y z Let

i 2 i i i i i 1 i i i i 2 i i

W W W

y z they exist since x and y exist x y z u x z w v

i i i i i i i

iI iI iI

Then x v w u y b ecause C onD Since they are p ermutable we obtain

1 2 1 2 1 2

xy The claim is proved

Now let us come back to the pro of of Theorem Let D compA A Then

1 2 1 2

max x x

are congruences and it is easy to show that for any x D complements in

1 2

x x

C onx To show that they are p ermutable note that if a c b then d p a p b

A A

1 2 2 1

x x

b d exists and a

1 2

Conversely let b e congruences dened in Theorem and let A A b e ob

1 2 1 2

tained as in Then A A are stable sub domains We need to prove D compA A

1 2 1 2

x x

The idea of the pro of is the following Since is the total relation by claim for

1 2

x x x x

any y x y z for some z Then z A and y z y thus y p y Similarly

1 A

1

1 2 1 1

x

y p y If x y A and x y then x y From this we can conclude that

A i i

2

2

x x

x y p x p y Having proved this we can use the fact that and com

i A A

i i 1 2

plement each other to demonstrate that y p y p y is the unique representation

A A

1 2

of y as the join of two elements from A and A Thus D compA A which nishes

1 2 1 2

the pro of of Theorem

Let us give the sketch of the pro of of Theorem If D A A then it is not hard

1 2

to show that and dened in are complementary and p ermutable

1 2

x x

If and are complementary and p ermutable in C onD then so are and

1 2

1 2

max

in x for any x D Thus for A A dened as in we have D compA A

1 2 1 2

and x y p x p y by Theorem Let x A y A Then x y Since

i A A 1 2 2 1

i i

the congruences are p ermutable for some z x z y Thus p z x p z y

1 2 A A

1 2

and z x y Therefore x y exists and by Prop osition D A A Theorem

1 2

is proved 2

A simpler result can b e stated for qualitative domains Recall that a domain D is

called qualitative i x is a Bo olean lattice for every x D Since all congruences are

p ermutable in a Bo olean lattice we have

Corollary General decompositions of a qualitative domain are given by pairs of con

x x

gruences such that and are complementary elements in C onx for every

1 2

1 2

max

x D 2

Schemes and semifactors in qualitative domains

a database p oint of view

It has b een mentioned several times b efore that the concepts of semifactor and scheme

rst app eared as two dierent attempts to dene an analogy of scheme for the domain

mo del of databases In this section we will apply our decomp osition results to nd out

when these denitions coincide But let us give some basic facts ab out the domain mo del

of databases rst

For simplicity consider the at domain N of natural numb ers and assume that we

have a relational database which stores the information ab out the hotel ro oms such as the

ro om numb er numb er of baths in the ro om telephone extension and the date it b ecomes

free We further assume that all attributes values are taken from N We need if a

piece of information is unknown for example there might b e no phone in the ro om or

we might not know when the o ccupant is going to leave To represent the last attributes

values by natural numb ers we may store the dierence b etween the date the o ccupant

leaves the ro om and the current date decrementing it every day Then the examples of

records are

r f Ro omNo Bath Phone Free g

1

r f Ro omNo Bath Phone Free g

2

r f Ro omNo Bath Phone Free g

3

If L is the set of attributes ie L fRo omNo Bath Phone Freeg then the records

stored in the database can b e represented as functions from L to N We denote the

n

domain of these records by L N Of course L N N where n jLj In the

max

ab ove example r L N and r r

1 3 2

Since records are elements of a domain the relations are subsets of this domain Since

there is no need to store two comparable records as r and r b ecause one of them is just

2 3

a worse description of the same ob ject the relations in our example are nite antichains

in L N This observation led the authors of to the idea to generalize relational

databases as nite antichains in Scottdomains This idea was further develop ed in

where more complicated examples can b e found A generalized relational algebra was also

constructed in

One of the central concept of relational database theory is that of scheme A scheme

is simply a subset of attributes but since they are used to dene pro jections one can

alternatively asso ciate a scheme with all records that are obtained as the pro jections into

this scheme that is all records whose pro jections on attributes not in the scheme are

b ottom elements Schemes in L N therefore are stable sub domains

The way the denition of semifactors is justied in is the following If we accept

the p erfect analogy b etween our arbitrary domain and the domain of at records such

as L N we may assume that every element of a domain which is a database ob ject

can b e represented as two sub ob jects each carrying an indep endent piece of information

Hence pro jecting into a scheme is losing a certain piece of information and the lost pieces

and the pro jections are indep endent ie the lost pieces can b e added to the pro jections

to get the records back If we assume that we have two records x and y and pro jection

of x into a scheme A is less than y then adding information lost when x was pro jected

to y is p ossible and the result is more informative than b oth x and y But this is exactly

the denition of semifactor p x y S implies the existence of x y A

An alternative approach of do es not make any assumption ab out the structure

of the underlying domain Just note that if we have a complete description pro jecting

into a scheme should not result in the loss of some piece of information that the scheme

preserves that is complete descriptions are pro jected into complete descriptions ie

max max

p D A And this is the denition of scheme

A

Thus a question arises what are the domains in which the two concepts coincide or

equivalently every scheme is a semifactor According to the justications of the two

denitions we may exp ect these domains to b e similar to the domain of at records We

will prove the result for the qualitative domains We need to dene the blo cks analogous

to at domains from which these domain will b e built

A domain D is called simple if it has no prop er scheme ie if it has no scheme but

fg and itself Since schemes app ear as equivalence classes of congruences this denition

is motivated by the denition of a simple lattice ie a lattice having no nontrivial

congruences

Theorem Let D be a qualitative domain Then every scheme in D is a semifactor

if and only if D is isomorphic to the direct product of simple domains

Pro of The if part is easy To prove only if supp ose that any scheme is a semifactor

A fx D p x g Then A is a scheme to o Since Given a scheme A dene

A

S is distributive and A is a complement of A in S S is a Bo olean lattice Since it is

D D D

algebraic by the corollary to the First Decomp osition Theorem it is atomistic and J S

D

is the set of atoms of S It then follows from the First Decomp osition Theorem that

D

D is isomorphic to the direct pro duct of all elements of J S since the limit over an

D

of connecting pro jections is the direct pro duct It is an easy observation that

all schemes from J S are simple domains which nishes the pro of 2

D

Corollary Let D be a qualitative domain in which every scheme is a semifactor

Then S is an atomic Boolean lattice and the schemes of D are in correspondence

D

with subsets of the set of atoms of S 2

D

This corollary gives a mathematical description of the assumption that every database

ob ject represented as an element of a domain can b e decomp osed into two indep endent

sub ob jects namely to its pro jection onto a scheme and its complement in S

D

Acknowledgement

We would like to thank Peter Buneman for inventing the b eautiful to ol of semifactors for

us domaintheorists Together with the rst author he found a preliminary version of the

First Decomp osition Theorem while he visited Darmstadt in July We also thank

Klaus Keimel for directing our attention to the classical theory of ideals in distributive

lattices He realized that our denition of the congruence generalizes the classical

S

denition to domains G Gratzer and ET Schmidt informed us ab out recent results on

complete congruences and interrupted our vain attempts to prove algebraicity for complete

1

congruence lattices of domains

1

While a revised version of this pap er was b eing prepared G Gratzer and ET Schmidt wrote a pap er

titled On a congruence lattice of a Scottdomain in which they proved that every complete lattice is

the lattice of congruences of a Scottdomain

References

S Abramsky Domain Theory in Logical Form Annals of Pure and Applied Logic

R Balb es and P Dwinger Distributive Lattices University of Missouri Press

Columbia

G Berry Mo deles Complement Adequats et Stables des Lamb dacalculs types These

de Do ctorat dEtat Universite Paris VI I

G Berry Stable Mo dels of Typ ed calculi In Proceedings of the th International

Col loquium on Automata Languages and Programming volume of Lecture Notes

in Computer Science pages Springer Verlag

G Birkho Lattice Theory volume of AMS Col loq Publ American Mathematical

So ciety Providence third edition

P Buneman A Jung and A Ohori Using Powerdomains to Generalize Relational

Databases Theoretical Computer Science

PL Curien Categorical Combinators Sequential Algorithms and Functional Pro

gramming Pitman

G Gierz K H Hofmann K Keimel J D Lawson M Mislove and D S Scott A

Compendium of Continuous Lattices Springer Verlag Berlin

JY Girard The System F of Variable Typ es Fifteen Years Later Theoretical

Computer Science

G Gratzer General Lattice Theory Birkhauser Verlag Basel

G Gratzer The complete congruence lattice of a complete lattice Proc of Int Conf

on Lattices Semigroups and Universal Algebra Plenum Press New York

G Gratzer and E T Schmidt A representation of malgebraic lattices Algebra

Universalis to app ear

A Jung Cartesian Closed Categories of Domains volume of CWI Tracts Cen

trum vo or Wiskunde en Informatica Amsterdam

G Kahn and G Plotkin Domaines Concrets Technical Rep ort INRIALab oria

A Kanda Fully Eective Solutions of Recursive Domain Equations In J Becvar

editor Mathematical Foundations of Computer Science volume SpringerVerlag

Lecture Notes in Computer Science

K G Larsen and G Winskel Using Information Systems to Solve Recursive Do

main Equations Eectively In G Kahn D B MacQueen and G Plotkin editors

Semantics of Data Types volume of Lecture Notes in Computer Science pages

SpringerVerlag

L Libkin A relational algebra for complex ob jects based on partial information In

J Demetrovics and B Thalheim editors Mathematical Fundamentals of Database

Systems volume of Lecture Notes in Computer Science pages

SpringerVerlag

G D Plotkin PostGraduate Lecture Notes in Advanced Domain Theory incorp o

rating the Pisa Notes Dept of Computer Science Univ of Edinburgh

H Puhlmann Verallgemeinerung relationaler Schemata in Datenbanken mit Infor

mationsordnung Diplomarb eit Technische Ho chschule Darmstadt

M B Smyth Eectively Given Domains Theoretical Computer Science

P Taylor Homomorphisms Bilimits and Saturated Domains Some Very Basic Do

main Theory Draft Imp erial College London pp

K Weihrauch and T Deil Berechenbarkeit auf cp os Technical Rep ort

RheinischWestfalische Technische Ho chschule Aachen

G Winskel Events in Computation PhD thesis University of Edinburgh