A FI+(TxF) D

Vol.11, No.4, November1982 0097-5397/82/1104-0011$01.00/0

THE CATEGORY-THEORETIC SOLUTION OF RECURSIVE DOMAIN

EQUATIONS*

M. B. SMYTHt AND G. D. PLOTKIN"

Abstract. Recursivespecifications of domainsplaysacrucialrole in denotational semantics asdevel- oped by Scott and Stracheyand their followers. Thepurposeofthe present paper isto setup a categorical frameworkin which the knowntechniquesfor solving these equations find a naturalplace.The idea is to follow the well-knownanalogybetweenpartialordersand categories,generalizing from leastfixed-points of continuous functions over cpos to initial ones of continuous functors overo-categories.Toapplythese generalideas we introduceWand’s O-categorieswhere the morphism-sets have a partial order structure andwhich include almost all the categories occurring in semantics. The idea is to find solutions inaderived category ofembeddingsandwegive order-theoretic conditions which are easytoverify and which imply theneededcategorical ones. The main tool is a verygeneralform of the limit-colimit coincidence remarked byScott. Intheconcludingsection we outlinehowcompatibilityconsiderationsare tobe included in the framework.Afuture paperwillshow howScott’suniversal domainmethodcanbe included too.

Keywords. Domains,semantics, data-types,category,partial-order, fixed-point,computability

1. Introduction. Recursivespecificationsof domainsplaya crucial role in denota- tional semantics asdeveloped byScottandStracheyand theirfollowers(Gordon

[13],

Milne and Strachey

[26], Stoy [39], Tennent [40], [41]).

Forexample, the equation

(1)

D

-At+(D-D)

is justwhat isneeded forthesemantics of an untypedA-calculus for computingover adomain,

At,

of atoms. Again, thesimultaneousequations

(2)

T-AtxF,

(3) FI+(TxF)

specify^adomain,

T,

of allfinitarily branchingtrees andanother,

F,

offorestsofsuch trees. And recursively specified data types are also very useful

[10], [19], [20].

The firsttoolsforsolving such equationswereprovided byScottusinghis inverse limit constructions

[33].

Later he showed how the inverse limits could be entirely avoided by using auniversal domain and the ordinary leastfixed point construction

[34]. A

systematic expositionof the inverse limitmethodwasgivenby Reynolds

[301,

and the categorical aspects (already ^mentioned by Scott) were emphasized by Wand

[43].

All of these treatments stuck to one category, such as, for example,

CL,

the category of countably based continuous lattices and continuous functions, although thedetails did notchange muchinother categories. Then Wand

[44],

gaveanabstract treatmentbasedonO-categories where the morphismsetsareprovidedwith a suitable order-theoretic structure.The relationbetween the category-theoretictreatmentand the universal domainmethod has,untilnow,^remainedratherobscure.

The purpose of the present paperis toset up acategorical framework in which all knowntechniques for solvingdomainequations find a naturalplace. The idea as setoutin 2istofollow the well-knownanalogybetweenpartial orders and categories, and generalize from least fixed points to initial fixed points. These are constructed

*Receivedby the editorsJanuary2, 1979, and in revised formFebruary 17, 1982. This work was partially supportedbyagrant fromtheScienceandEngineeringResearch Council.

r

^DepartmentofComputerScience,UniversityofEdinburgh, Edinburgh,ScotlandEH9 3JZ.

761

762 M.B. SMYTH AND G. D. PLOTKIN

using the "basic lemma" which plays an organiz.ational role" Most of the solution methods considered appearaswaysofensuring that thehypothesesofthe lemma are fulfilled. Just as continuous functions overcomplete partial orders always have least fixedpoints,so continuous functorsover^o)-categories (asdefinedbelow) always have initial fixedpoints, whichcanbe constructed by usingthe basiclemma; thisseems to formalize some hintsofLawverementioned byScottin

[33].

The sameideaappears in

[1], [2].

All this is very general, and we introduce O-categoriesin 3 in order to apply the basic lemma to the construction of the domainsneededin denotational semantics.

Hereweareclearly greatlyindebted toWand

[44], [45]

who introduced O-categories, and indeed our work arosepartly as anattempttosimplify and clarifyhistreatment.

The idea is to apply the basic lemma not to a given

O-category, K,

but rather to a derived category, K

E,

of embeddings (equivalently, projections). We then look for easily verified conditions on K (whether categorical ororder-theoretic) which imply theneededconditions on K

E.

Our main tool is Theorem 2, which establishes a very general form of the limit-colimit coincidence remarked by Scott

[33]

and also gives an order-theoretic characterization of the relevant categorical limits. This improves Wand’s work by removing the needfor histroublesome "Condition

A"

(appearingin

[44]

ratherthan

[45]

whichincorporates some of the ideas ofthe presentpaper); more positively we alsointroduce^ausefulnotionofdualityfor O-categories.

With the aid of Theorem 2 (and the easy Theorem 1), one sees that simple conditions on an

O-category, K,

(mainly that ithas all

P-limits)

ensure that K^E is an o-category. Again with the aid of

(Lemma

4 and) Theorem 3, one sees how to take any mixedcontravariant-covariantfunctor overK (likethe function-space

one)

which satisfies an order-continuity property (usually evident), and turn it into a covariant-continuous one over

K E.

Section4 presents several examples ofuseful categories which may be handled by the methodsof 2 and 3.

The method of universal domains, in relation to the ideas presented here, is treated in the sequel to this paper.

An

indication of our approach may be found in Plotkin and Smyth

[28]

(which may also be of help in getting ageneral overviewof ourresults).

There is, however, one aspect of Scott’s presentation of the universal domain approach

[34]

which must receive some mentionhere:the question ofcomputability.

The results presentedin thispaper wouldlose muchof theirpointif we were forced to invoke a universal domain to handle computability. In the concluding 5, we indicatebriefly how thistopic caninfactbe handledatthe level of generality ofthis paper; foramore detailed treatment werefertoSmyth

[38].

Weassumethe readerpossesses a basicknowledge of categorytheory; anygaps can be filled by consulting Arbib and Manes

[4],

^Herrlich and Strecker

[16],

^or MacLane [21

].

2. Initial fixedpoints. Inthecategoricalapproachto recursive domainspecifica- tions wetrytoregardallequations suchas(1)or

(2)

and

(3)

aboveas beingoftheform

(4) X F(X),

whereXrangesovertheobjectsof acategory

K,

say, and

F:K

Kis anendofunctor of thatcategory. For example, in the case of

(1)

we could take X torange over the objects of

K,

At to be a fixed object of

K,

and

+

and to be covariant sum and

function-space functorsover

K;

then

F"

K Kis definedby"

At+ (X X).

(5) F(X)def

Let us spell the meaning of

(5)

out in detail. Recall that if Fi:K

Ki

^(i-1,r/) ^are functors then their tuplingF (El,

Fn): K- K1 "

^{x K, is defined by putting}

foreach object,

X,

of

K:

F(X)

(FI(X),

Fn ^(X))

andfor eachmorphism

f: ^X

^YofK:

F(f (F,(f

), F,

(f )).

Then the functorFdefinedby(5)is just

F + (KAt, --

o(idm idI))

where

KAt

^{:K-Kis the} constantly-At functor and

idI

^{:K-.Kis the} identity functor.

Simultaneous equations are handled usingproduct categories. For example,

(2)

and

(3)

canberegardedas having the form:

(6)

X Fo(X,Y),

(7)

Y

F

^{(X, Y),}

where

X,

Y rangeover acategoryK (suchas

CL)

and

F0

^and

F

are bifunctors over Kbeing definedby

X

o<KAt

^7rl>,

F

_def

F1 +o<K,,

^X

o<’rro, 7’/’1>>

def

where

At,

1 are objects of

K,

and the 7ri"

K xK-K

(i=0,

1)

are the projection functors. Then

(2)

and

(3)

are put into the form

(4)

by using the product category

K

xKand taking

F

tobe(Fo,

F).

Clearlythisideaworksfor n simultaneousequations

Xi

=Fi(X1, ,X,)(i 1,n)whereXirangesoverKi (i 1, n)andFi

:K

^x...xK,

K;

we_Letjust takeus now decide what a solution to

K

tobe

Kx " ^K,

^andF

(4)

^tomight^be

^(Fx,

be and^’,F,).whichparticular ^{ones we} want. In the case where K is a partial order, F is then just a monotonic function, solutions are just fixed points ofF (that is, elementsA of K such that

A

=F(A)), and we can look forleastsolutions. Further,we candefineprefixedpoints as elements

A

such that F(A)A, and it turns out that the least prefixed point, if it exists, is always the least fixed point ^as well. In the categorical case we need to know the isomorphism as well asthe object:

DEFINITION 1. Let

K

be a category and

F:

K- K be an endofunctor. Then a

fixed

^{point of F is a pair (A,}

^c)

^where

^A

^{is an object of K and}

^a’FA

^{-A is an}

isomorphism of

K;

aprefixedpoint is a pair (A,

a)

where A is an object of K and

a:FA A

is amorphismofK.

We also call prefixed points of

F,

F-algebras (same as F-dynamic of Arbib and

Manes [4]).

The F-algebrasarethe objectsof acategory:

DEFINITION 2. Let (A, a) and

(A’, a’)

be F-algebras.

A

morphism

f:

^(A,

^a)-

(A’, ’)

(F-homomorphism) isjustamorphism

f’A ^-A’

^{inKsuch that} the following commutes:

764 M.B. SMYTH AND G. D. PLOTKIN

FA A

FA’ A’

It is easily verified that thisgives a category: the identity and composition are both inherited from

K.

Following on the above remarks on partial orders, we look for initialF-algebras rather than justinitial fixed points ofF and this isjustified by the following lemma(whichalsoappearsin Arbib

[5],

andinBarr

[8],

whereit is credited toLambek).

LEMMA 1. The initialF-algebra,

if

it exists, isalso theinitial

fixed

^point.

Proof.

^{Let (A, a)} be the initial F-algebra. We only have to prove that a is an isomorphism. Now as (A,

a)

is an F-algebra so is (FA,

Fa)

and so there is an F-^{is an}homomorphismF-homomorphism and

f:

^(A,a so^{(FA, Fa );}a

of:

^{(A,onea)also(A, a)easilyis alsosees thatonea}and it must be^{(FA, Fa}

-

^(A,idA,

^a)

the identity on

A

as (A,

a)

is initial. Then as

f:

^(A,

a)-(FA, Fa)

we also have foa=(Fa)o(Ff)=F(aof)=F(idA)=idFA, which shows that a is an isomorphism with two-sided inverse

fi

Notethat we have to do morethan specifyanobject

A

such that

A -F(A)

when lookingfortheinitial fixedpoint. First wehave tospecifyanisomorphism^a"

FA -A,

and secondly we must establishthe initiality property. Both are vital in applications.

When giving the semantics of programming languages using recursively specified domainsthe isomorphism is needed justto be able tomake thedefinitions. Initiality is closely connnected to structural induction principles and both can be used for makingproofs about elementsofthe

specited

domains.When the equationsareused tospecifydata-typedefinitions within alanguage followingtheapproachinLehmann and Smyth

[19], [20],

the isomorphism carries the basic operations, and initiality is again essential forproofs. (The paper

[20]

also contains more information on simul- taneousequationsand onequationswithparameters; inmany ways,itis acompanion to the presentpaper.)

WhenKis apartialorder,theleastfixedpointcan,asiswellknown, be constructed as

IIKFn(_L),

the 1.u.b. of the increasing sequence

(F"(_k)),,o

where ^_L is the least element ofK. Thisworksiftheleast elementexists, the1.u.b. exists andFpreserves the1.u.b.that is,

F(IIK

F

(+/-)) IlK ^F(F

^(+/-)).

^Our

^{basiclemmamerely}generalizes these remarks tothecase of acategory.

First we give some notation and terminology which are not quite standard.

By

anto-chainin acategoryKweunderstandadiagramof theform^A

Do o _Da .

(thatis, afunctorfrom toto K); for m

=<

n, we write

f,,,

:D,, D, forthe morphism f,_ao.., of,,. Dually an toP-chain in a category K is a diagram of the form A

=Do ro _D r

(thatis,afunctor from

tooo

toK);form

->n

^wehave theevident f,,, :D,, D,. By the mediating morphism between alimiting (colimiting)/x over a diagram A and any othercone u over

A,

we mustunderstand the unique morphism given by universalityfrom

(to)

thevertex of u to (from) thevertexof

Notation. The initialobjectof acategory

K

is writtenas_t_iandthe uniquearrow from it to an objectA is written as _LA. If ^A

Do o _D1 ""

is an to-chain in K

and/x:

^A

- ^A

^{is acone over}

^A,

then A- is the to-chain

D1 ’ ^D2

^_f2...and

^-:

^A-

^A

is the cone

(/X,+l)n,o;

if, further,

F:K-L

is a functor, then FA is the to-chain

FDo -Fr _FD1 _rr,

_and

Ftx ^FA-FA

^{isthe cone}

Nowgiven afunctorF"

K--> K

we candefine the to-chain

A (F"

(+/-i),

F"

(+/-F_l_)) (if +/-_iexists)and trytojustify the calculationanalogoustothat for partial orders"

Flim A lim

FA

^{lim A-} lim

A.

Thebasic lemma gives conditionsfor this towork and characterizes lim_.

A

(with ^an appropriate morphism)as theinitialF-algebra.

LEMMA2(basiclemma). LetKbea categorywith initial

obfect

^{+/-andletF"K-->}

^K

bea

functor. Define

the to-chain A tobe

(F"

(1),

F" (+/-F+/-)). Suppose

thatboth Ix"^A-->

A

and

Fix" FA--> FA

are colimitingcones. Then the initialF-algebraexists andis (A,

a)

where^a"

FA

^-->

A

is the mediating morphism

from Fix

^to

ix-.

Proof.

^Let

a"FA’-->A’

be any F-algebra. We show there is a unique F- homomorphism

f:

^(A,a)-->(A’,

a’).

First suppose

f

^{is such a}homomorphism. Define acone

u’A->A’

by puttingvo=^{+/-A, +/-}

A’

and v,+l=a’oF(v,). Tosee v is acone weprove byinduction on n thatthefollowing diagramcommutes.

Fn(+/-.+/-)

F"(+/-) Fn+l(+/-)

vn+1

oF"+l(+/-+/-)

This is clear for n 0. For n

+

^{1 we have: v,+2}

=a’oF(v,+loF"(+/-l.))=a’oF(v,)

(by induction assumption) v,+l Now the uniqueness of

f

^willfollow when we show it isthe mediating morphism from_ix tov;

here we use induction on n to show v,

f

^{Ix,. This is clear for n 0. For n}

⁺

^{1 we}

have’foix,+l=fOaoF(ix,) (by the definition of

a)=a’oF(f)oF(ix,) (f

^is

ahomomorphism)

a’ F (f

ix,)

a’

v, (byinductionassumption)

Secondly, toshowthat

f

exists, letitbe the mediating morphismfrom_ix tov

(so

that v,

fo

^{ix, forall integers,n)where v isdefinedas above.Wewillshow that}

f

^oa

and

a’o Ff

^are both mediating arrows from

Fix

^to

v-,

thus demonstrating that they areequal and that

f

^{is an}F-homomorphismasrequired.

In

the first case, (foa)oFix,

=fOixn+

^{(by definition of} a)’--/n+X (by definition of

f).

In the second case,

(a’oFf)oFix,=a’oF(foix,)=a’oFv,

(by definition of

f)

^v,+ (bydefinitionof

v).

Thisconcludestheproof.

Inthe case ofpartialorders,ourmethodofconstructing least fixed-points always works if K is an w-complete partial order and

F" K--> K

is o-continuous. Here an w-completepointed partial order(cpo)isapartial orderwhichhas1.u.b.s ofallincreasing w-sequencesand whichhasaleastelement;itistermedan"w-complete partial order"

or^evenjusta"complete partial order" elsewhere. AlsoafunctionF’ K^-->Lofpartial ordersis w-continuous ifandonlyif it is monotonicand preserves 1.u.b.sofincreasing to-sequences,that is,ifwhenever

(x,),,o

is anincreasing w-sequence such that exists, then

F(I It:

x,)

IIc

F(x,). Wemakeanalogousdefinitions forcategories"

DEFINITION 3.

A

category,

K,

is ano-completepointedcategory (shortenedbelow toto-category) ifandonlyif ithasan initialelement,and everyco-chainhas acolimit.

DEFINITION 4. Let

F"

K--> L be a functor. It is w-continuous if and only if it preserves o-colimits; that is, whenever

A

is anto-chain and g"

A- A

is a colimiting

766 M.B. SMYTH AND G. D. PLOTKIN

cone, then

Fix ^{:FA FA}

^{is also a colimiting cone. (The reader is} warned that this definition is dual to the notion of continuity of functors usual in category theory (MacLane

[21]);

thisisdoneinorderto maintaintheanalogywithpartial

orders.)

Clearly, when K is an to-category and

F"

K K is to-continuous,the conditions of the basic lemma are satisfied. In 3 we will give the conditions for this to be the case. Inthesequelto thispaper

(see

alsoPlotkinandSmyth

[28]),

wewill showthat, inthe presenceof a universalobject, theconditions ofthebasiclemmamay besatisfied withoutrequiringthat Kbe anto-category and thatF be to-continuous. Usually, we can completely avoid direct verification of the conditions of the basic lemma, or of whether somethingis an to-category or is to-continuous. Of coursesometimes, as in thecase of

Set,

it isalreadyknown that wehaveanto-category andthatsuch functors as

+

and are to-continuous(seeLehmann andSmyth

[20]).

Onecase in whichthere is, so far, no alternative to direct verification is with Lehmann’s category, Dora of smallto-categories andto-continuousfunctors

[18]

(inthiscaseDora-categories might provideagood general setting).

Note thatit is only necessarytocheck that thebasic categoriesare to-categories and the basic functors are to-continuous; one easily proves that any denumerable product of to-categories is an to-category, that all constant and projection functors are to-continuous, and that composition and tupling preserve to-continuity. Thus to solve

(1)

one only needsto checkthat

+

^and are to-continuous; for

(2)

and (3)one looksat

+

^and

.

The originalwork onmodelsofthe pureA-calculus

(Scott [34],

Wadsworth

[42])

did notsolve

(4)

as lim_.

(F"(_I_), Fn(IF+/-)),

but rather as lim_,

(F’(D), F"(’n’))

for an objectD and a morphism

"n"D

F(D). Itturns out that this solution is, essentially, the initial fixedpointof a functor

F=,

derived from

F,

overthe commacategory

(D SK)

of "objectsoverD"

(see

MacLane

[21]).

The analogousidea inpartial ordersis that of aleastfixedpoint greaterthan some fixedelement, d.

The commacategory

(D,K)

has as objects pairs (A, d) where

A

is an objectof

K

andd:D

A;

the morphisms

f:(A,d)(A’, d’)

^are the morphisms

f:A ^A’

^of

^K

such that the following diagramcommutes.

D

d ^)A

A’

Now the endofunctor

F,:(D$N)-(D,I,K)

can be obtained by putting F(A,d)=

(FA, (Fd)or) for objects and

F,(/) =F(/)

^for morphisms. Then one can see that an F,-algebra is a pair ((A, d),

)

where

A

is an object of

K,

and d’D

A

and

:FA A

and the following diagramcommutes.

D

, _F(D). , _F(A)

A

homomorphism

’:

^{((A, d),)}

-

^{((A’, d’),}

^’)

^{is amorphism}

^{’: A A’}

^{suchthat the}

followingtwo diagramscommute.

F(A)

A D

^a

A

F(A’) A’ A’

Let us assume, for simplicity, that K is an to-category and F is to-continuous.

Then

(DSK)

is also anto-category. Its initialobjectis (D, idD). For colimitssuppose A=((An,

d),f,)

^is an t-chain in

DSK.

^{Then it is} straightforward to check that

/z

(A, f)-

A is a (colimiting) cone in Kifandonlyif/z:A

(A,/xo

do) isin

(D K)

(and they have the same mediating morphisms). This makes it easy to show that

F

isto-continuous.

Now,

applying the basic lemma to

(D K)

and

F,

^we have to find a colimiting cone Ix"

(F

(+/-),

F (+/-F=(I)))->

(A, d). One sees,by induction on n, that

F ^(+/-(DI))

^is

(F"(D),d,)

where

d =F-l(Tr) .o’:DF(D)

and that

F(LF(+/-))=Fn(’a’).

So from the above remarks one can take to be a colimiting cone,

/x:(F"(D), F" (zr)) A,

alsodefining

A,

and put d

=/x0od0.

Then, by the basic lemma, theinitial F-algebra is ((A, d), a) where a is the mediating morphism from

F/x to/x-

(which can be taken in K). Thus we see that

A

^=lim_.

(F"(D), F"(r)),

together with its colimiting cone, determines the initial F-algebra. Thus we have characterized the originalinverse limit construction inuniversal terms.

3. O-categories. In most of the categories used for the denotational semantics ofprogramminglanguages, thehom-sets have anatural partialorder structure. When solvingrecursive domainequationsonlythe projectionsareconsidered, and they are easily defined in terms of the partial order.

Wand [44]

introduced O-categories to study such categories at asuitably abstract level.We nowpresentaview of hiswork as providing theorems and definitions which facilitate the application of the basic lemma, asoutlined intheintroduction.

DEFINITION 5.

A

category,

K,

is an

O-category

if and onlyif (i) every hom-set is apartial orderin whicheveryascendingto-sequencehasa1.u.b. and(ii)composition ofmorphismsis an to-continuousoperation with respect^tothis partial order.

Note that if Kis an

O-category,

so is K

,

^{and ifL is another so is}

^{K L.}

^Here

the orders are inherited, and inthe case ofK^pwe have

fop___ gop

if andonly if

f

^g,

foranymorphisms

f

^{and gofK (ingeneralwewill omit}the superscript when writing morphismsinopposite categories).

As

ithappens O-categoriesare aparticularcaseofageneraltheoryofV-categories whereVis anyclosedcategory (MacLane

[21]);

hereOisthe category whose objects arethose partial orderswith1.u.b.s of all increasingto-sequencesand whose morphisms are the to-continuous functions between the partial orders.

We

will not use any of the general theory but just take over the idea of endowing the morphismsets with extra structuremin this case thatofbeinganobjectin

O.

DEFINITION 6. LetK beanO-category andlet

A rB

gA be arrowssuch that g

of ^idA

^andfog___idB. Then we saythat

(f,

g) is aprojectionpair

from ^A

^{toB, that}

g is a projection and that

f

^{is anembedding.}

LEMMA 3. Let

(f,

g) and

(f’,

g’) both be profection pairs

from ^A

^to

^B,

^{in an}

O-category

K. Then

f _ f’ if

^andonly

if

^g

_ ^g’.

Proof.

^If

^f_f’

^then g_gof’og’_gofog’=g’. Conversely, if

g_g’

then

f=

fog’of’_fogof’_f’. ^V!

768 M. B. SMYTH AND G. D. PLOTKIN

So,

inparticular,itfollowsthat onehalf ofaprojection pairdetermines theother;

if

f

^{is an embeddingwewrite}

fR

^{for the} corresponding projection which we call the right

adfoint

^of

;

^if g is a projection wewrite

gL

for the corresponding embedding which wecallthe

left

ad]oint of g. (Ouruse of theterm "adjoint" isonly a matterof convenience; whentheK-objectsareposets andthemorphisms aremonotonicmaps, itagreeswith astandardusage.)

Givenany

O-category K

wecan now formthe subcategory,K

E,

of the embeddings.

For the identity morphism

idA’A-->A

^{is an embedding with}

ida=idA,

and if

A fB-)f’c

are embeddings, so is

(f’ of)

^with

(f’of)

^R

"--fRf ^’R.

^{Equally, we can}

form K

P,

the subcategoryof theprojections. Wedo nottry totakeeither ofthese as O-categories under theinducedordering;indeedthey are, in general,notO-categories.

3.1. Duality for O-categories. Ourdiscussion ofadjointsshowsthatwehave the dualityK^E

(KP)

^p (and^{so too}K^p

(KE)P).

There is anotherkind ofduality arising from the fact that an embedding in

K

^{p is} just a projection in

K.

Thus

(KP)

^E K^P (and^so

(KO)

^P

KZ).

Thusan to-chain in

(KP)

^zcanbeconsideredasantoP-chainin K

P,

and acolimitingcone in

(KP)

^{E can}be identified with a limiting cone in

K e.

^We

therefore have thefollowing dualities (for^an

O-category K):

embedding/projection, to-chain in

KZ/0oP-chain

ⁱⁿ

^K P,

colimiting conein

KZ/limiting

cone in

K P. A

further example, O-colimit/O-limit,^isprovidedbyDefinition7.

Theseobservations will beusedin theproofofTheorem2.

Ourfirst theorem is trivialbut does illustrate theidea of transferring properties from Kto

K E.

THEOREM 1. Let K be an

O-category

which has a terminal obfect,

z,

and in which every hom(A, B) has a least element, +/-A,U. Suppose too that composition is

left-strict

^{in the sense that}

for

^any

^[:A-

^{B we have}

+/-.cof

+/-A.O ^{Then +/- is the} initial object

_Proof.

_First,

of

^K

.

_iff,

_f"

_+/-

_A

_areboth embeddings,then they have acommonright adjoint,as +/-is terminal in

K,

and so, byLemma3,theyareequal.Second, -t-.+/-.A’^+/-

A

is an embedding with right adjoint +/-A.+/-. For .J_A,+/-_L+/-,A’_[_^"-)_L must be id+/- the uniquemapfrom ^_1_to+/- and +/-_.A +/-A.+/- [A,A (by left-strictness)_idA.

To make the connection with the basic lemma, we need to be able to relate O-notions (expressed in terms ofthe ordering of

hom-sets)

to to-notions (expressed in terms of

toP-limits/to-colimits).

Thisis the main purpose of Theorem 2. Another way to view this result, exemplified further in the ensuing discussion, is that it is concernedwiththecorrespondence betweenlocal properties(thatis, propertieslocal to particular hom-sets) and global properties of the category. Yet another way to regard Theorem 2is to notethatit containsthemostcomplete and generalformulation of the limit-colimit coincidence, remarked in Scott

[33],

that we have been able to develop.

DEFINITION 7. Let

K

be an

O-category and/x"

^A

A

acone in

K ,

^{where A is}

theto-chain

(A,, f, ).

Then isanO-colimitof Aprovided that

(/x,

o/x,R

),

is increasing with respect to the ordering of hom(A,

A)

and I1,,/x,o/x,

idA.

Dually an O-limit of an toP-chain

F

inK^P is acone u’B

- ^F

^{in Ksuchthat ou,), is}increasing and II,

u

^u,

^idB.

Obviously,/x is anO-colimit of

A

if R is anO-limitof

A R,

wherewe define

A

R R

to be

(A,, f,n) and/x

^tobe

(tz,

THEOREM2. Let Kbean

O-category

and ^A bean to-chain in K

.

^Considerthe

six properties

(a) A

hasacolimitin

K,

(b) ^ARhasalimitin

K,

(c) ^Ahasan O-colimit, (d) ^ARhasan O-limit, (e)

A

hasacolimitin K

E,

(f)

A

^Rhasalimitin

K P.

We have" properties (a)-(d) ^are equivalent (to each other);

(a)

implies (e); and

(e)

is equivalentto (f). Indeed,acone

" ^{A->A (a}

^cone

^v’A

^-->

^{A R)}

^{iscolimiting (limiting)in}

K

if

^andonly

if

I

(v)

is an O-colimit (O-limit);anycolimiting (limiting)cone

of ^A(A R)

inKisalsoacolimiting (limiting)cone

of ^A(A R)

in

K

^z

(KP);

and

A A

iscolimiting

R K

P"

in K

if

^andonly

if

^islimitingin

Proof.

^{We establish}Propositions

A-E

whicharejointly equivalenttothe result.

We always suppose that

A

has the form

(A,, f,).

Note that

(’,

o,

),R

^is increasing for any cones

’AA

^and

"AA’

ⁱⁿ

^K,

^{as we have’ ,o,}R

PROPOSiTiON

A. If "

^h

^A

^{is an O-colimit, then iscolimiting in both K and}

E.

Proof of

Proposition

(A).

Choose acone

"

^h

^A

ⁱⁿ

^K,

^{and suppose}

^{" A A’}

is a morphismfrom to

’

^{(i.e.,for all n,R o,}

^’).,

^{R Then isRdeterminedby:}

0 OoUp,.o. ^Lj(0 op,.)o,

’o/R

^{since the above}

Thisproves uniqueness; for existence we can define 0 as II/, remark shows that

(’

/,^R isincreasing, andcalculate:

So is colimiting in

K.

For K^E it onlyremains to show that if

’

^{is actually a}

cone in

K

then0 as defined aboveis anembedding.

By

the remark above,

(/z,

o/z

’),,o

is increasing; toshow that 0 is an embedding, we prove thatit has the right adjoint O^{R L_l}_{l. lz}^tR

Onthe onehandwehave"

vR R vR R R

(I I/.o. )o(I I/no/z.

)=1

I/.o(/z. o/.t.)o/. =U/zno/z.

=idA, onthe other handwehave"

R vR R vR vR

U

]J,

tn

l )o([[ia, .t )---[_]]. .t in i ].I pl, idA’. ^I-!

Dually,wehave PropositionB:

PROPOSITIONB.

If ^v:A

^->

^AR

^isan O-limit, then vis limitingin both

K

and K

P.

PROPOSITION C.

If

^v"

^A->

^{ARis limitingin K, then each v, is aprojection and v}

isan O-limit

of ^A R.

Proof of

Proposition (C). Foreach

A,

we candefine a conev^(’)

:Am

^-->

^AR

ⁱⁿ

^K

^by:

.,)={f..

^(m<-n),

" ^{(A) ’}

^(m

^>

^n).

(m) R

To see that v

"

^{is a cone, we first check that} if r_->max(m,

n)

then

v

For if m<-_n, then

f .R f f .R (f f

^V(m)^{;if m}

^>

^{n then}

f nRr f

770 M.B. SMYTH AND G. D. PLOTKIN

(fmrfnm)

^R

fmr--fnRm

^ln(m) ^{Nowwesee that v(")is acone, asfollows:}

Vn+l

(f(n+l)rfmr)

(bytheabove with r max(m,n

+ 1)) (f(n+X)rOfn)

^R

fmr =fnRrfmr

v.(m) (bytheabove).

Now as v"

A -

^{AR is alimitingcone}there is, foreach m, amediating morphism

(m) (m)

O,.

"A,. A

fromv tov. Sowehavefor all rn andn"

v. O.. v..

Puttingn equal to m we find that

v.

^o0,.

ida..,

^{which ishalf the proof that v,. is a projection with}

L

Next we connect up the O,.’s by showing that

O..

O,./of. ^which holds since 0,.+1of,. ^mediatesbetweenv^")and v as canbe seenfrom:

(re+l) +1)

v.o(O.,+lof.)

v, of,., (Om+lmediatesbetweenv

"

^{and v)}

(withr max

(m + 1. ⁿ⁾⁾ f nRr fmr

^Vn(m)

This, in turn, enables us to show that

(O,.ov.,),,o

is increasing: O.,ovm=

O.,+lof.,ofov,.+l=_O+loV,.+l.

Consequently, as K is an

O-category,

we may define 0"

A A

by" 0 =1 I.o,O,.ov. To finish the proof, we show that 0 ida (as then wealso have 0,. v.,^_0 idA, completingthe proof that v,.L 0.,). This follows fromthefact that 0 mediates between v anditselfas isshownby"

’rnn m_-->n mn

By

duality"

PROPOSITION D.

If/:A--> ^A

^{is colimiting in}

^K,

^{then each} I, is an embedding, and_{Iz is an} O-colimit.

PROPOSITIONE. ^A-->

A

iscolimitingin

K if

^andonly

if

^iR

^:A - ^A

^Rislimiting

in K

P.

Proof of

Proposition E. Obvious.

Thiscompletes the proofofTheorem2.

Ourfirst main use ofTheorem2willbe to establishthe evidentcorollarythat if K is an

O-category

which has all

oP-limits,

then K has all 0o-colimits. The second main use concerns functors, but first a definition and another corollary prove con- venient.

DEFINITION 8.

An O-category

Kis said to have locally determinedto-colimits

of

embeddings provided that, whenever

A

is an to-chain in K

and/:A- A

is a cone in K

,/x

is colimitinginK ifand only

if/

^isan O-colimit.

(Note

that, by Theorem 2, only halfofthe implicationcan everbe indoubt.)

COROLLARY (toTheorem

2). Suppose

that the

O-category

K has all toP-limits (i.e., every toP-chain in K has a limit). Then K has locallydetermined o-colimits

of

embeddings.

Proof.

^Suppose

that/"

^A

A

is colimiting inK

.

^Let

^{v" A’}

^AR be a limit with respecttoKfor A

R.

ByTheorem2,

v’

isan O-limitfor A

R.

Thus,

v’

^{is an}O-colimit for_and

A,

must itselfsothatbybeTheoremanO-colimit2, v(the

’

^isproperty^colimitingofⁱⁿbeing^K

.

an O-colimit is

^Hence/z

^{is isomorphic}triviallyinvariant^with

^v’

under isomorphismofcones).

Byduality, the conclusion ofthe corollaryalsofollowsfromthe assumption that K has all t-colimits, but thatis not so useful. At present welack an example of an O-categorywhichdoesnothavelocally determined t-colimits of embeddings.

As

mentioned inthe Introduction, a major reason for introducing

K

istoenable us toconsider contravariant functors on K as covariant ones on

K "

^{the remainder}

of this section develops the idea.

We

consider throughout (thatis, to the endof the section) three O-categories

K, L, M

and a covariant functor

T: KP

L

M.

Purely covariant functors are included by suppressing

K

(i.e., taking it to be the trivial one-objectcategory), contravariantonesbysuppressing

L,

and mixed onesby taking K andLtobeproduct categories as required.

DEFINITION9. The functor

T

is locally monotonic ifand onlyif it is monotonic on the horn-sets; that is, for

f,f"A

-B in K

p,

g,

g"CD

in

L,

if

f_f’

and

g_g’,

then

T (f,

g)

_ ^{T (f’,}

^g’).

LEMMA 4.

If

^{Tis locally} monotonic, a covariant

functor

^T

^K

^L

- ^M

^can

be

defined

^{by putting,}

for

^objects

^A, B’TZ(A,B)=T(A,B)

and

for

^morphisms

f,g"

T

^E

(f,

g)=

T((fR) r’,

g).

Proof.

^{First if}

f:A -

^{B inKE andg"C}

- ^D

^inL

^E,

^then

^{T(f R,}

^{g)is an embedding}

with right adjoint

T(f, gR)

as"

T(f, g)Ro T(fR,

g)=

T(fR of, gRog)

T(iaA,iac)

idT.c)

and also:

T(fR,

g)oT(f,

gR)=T(fofR, gogR)_T(idmidD)

(by local monotonicity) idrCn.o).

Secondly, T^E(idA, idc)=

T(id,

idc)=T(idA, idc)=idrg.c).

Thirdlyif

A -rA’-’ A’

in

K

^EandB ^-gB’

g’B"

_inL^E _then

TE(f ’,

g’)

TE (f,

^g)=

T(f ’,

g’)oT(f

n,

g)=

T(fRof ’R, g’

og)

T((f’of) R, g’

og)=

T(f of, g’og).

Undersomeassumptionson

K

and

L,

we cantransferalocal continuity property of

T

tothe to-continuity of

T E.

DEFINITION10. The functor

T

islocallycontinuous(equivalently,is anO-functor) ifandonlyif it is to-continuousonthe hom-setsthatis, iff,"

A -

^{B is anincreasing}

to-sequence^Note that the constantin

K

^pandg"C

-

and projection functors^{Dis one in}

^L,

^then

^T

^(,are^{f,, II,}locally continuous, and that^g,

^II,

^,oT

^(f,,

^g,).

the locally continuous functors are closed under composition, tupling and taking opposite functors.

THEOREM 3. Suppose T is locally continuous and both

K

and L have locally determined to-colimits

of

embeddings. ThenT isto-continuous.

Proof.

^Let

^{A ((A.,}

^B.),

^(f.,

^{g.)) be} an to-chain in

K

^Ex L^E and

let/x" A - ^{(A, B)}

be colimiting, where _tz (try,

-.)..

Then

(r.)" (A., f.)- A

is colimiting in K and

(z.)" (B.,

g.)-B is colimiting in L

z.

^It follows by the assumptions on K andL that

R R

withthe right-hand sides increasing.

idA

^IItr.oct, and

id

^II

-.

We

havetoshow that

T

^E

(/z)" T

^E

(A) T

^E(A,

B)

is acolimitingcone in

M E,

which wedoby showingthat it is an O-limit (andthen applying Theorem

2).

First

(TE(la.,,)oTE(lx,,)R),,., (T(o-. R,

-,,)o T(tr.

R, (T(o’R,

r,,) T(o’,,,"r,,R

)),,

R R

(T(cr.

^oct.,r.

o.r.

772 M.B. SMYTH AND G. D. PLOTKIN R

R and

(z. oz.

).,o ^are andas T islocallymonotonic.

which isincreasing ^as

(r. or.

^)..,

Next,

R R

T

z(tz.)o

T

z(l..)

^{R II} T(o’.oo..,"r.o7..

T O’nO’n, [[ Tn’l

(bytheabove)

(bylocalcontinuity) T(idA, ida) (bytheabove)

idT(A,a). ^1-1

4. Examples. In this section we present several useful O-categories where our general theory can be applied. In generalwe only sketch proofs andeven omitthem Whentheyareeitherevidentor notdirectly relevanttothemain line ofthe argument (butforExample2, see

[22]).

The firstexample iselementary, beinglittle morethan an illustrationofthe ideas. The secondexampleis the categoryofcposwhere all the needed domains for denotational semantics can be constructed. This is an approach where as few axioms as possible are imposed. That makes the axioms very easy to understand but admits domains of little computational interest. The third example illustrates various completeness conditions that are weaker than Scott’s original requirement of complete lattices, which rules out some natural and useful domains.

Thefourthexampleconsiders axioms ofalgebraicity and continuitywhichattempt to forcethedomains tobecomputationally realistic.Oneuse ofthecompletenessaxioms (cf. Example 3) is that when combined with algebraicity (or continuity), function spaces exist although they need not otherwise do so. Example 5 turnsin a different direction, suggestinga certaincategoryof continuousalgebrasasanappropriateplace for the semantics of programming languages with nondeterministic constructs.

Example 6considers acategory ofrelationsovercpos whereitispossibletoconstruct awidevarietyofrecursively specified relations; these are usefulwhenrelatingdifferent semantics.

Example 1. Partial

functions. ^We

^considerthe categoryPin of setsand the partial functionsbetween them. The partial order relation between partial functionsis just setinclusionand canalsobe definedfor anyf,

g:A

B by

f=_g---Va

A.f(a), f(a)=

g(a)

(where

f(a)$

means that

f(a)

^is defined). Clearly limits of increasing to-sequences

(f,),

^existbeing just theset-union

Uf,

^{so that} (1

f)(a)=

^b

undefined

(:in f,

(a

b),

(qn.

f ^(a)

^isundefined).

Itiseasyto see that

f: ^A -

^{B is anembeddingifand only}if it is totaland one-to-one;

in that case

fR =f-1.

^{Thus to within} isomorphism, embeddings are just inclusions.

Notethatthe totally undefined function

:A

B is theleast elementof hom(A,B) and that composition is left strict in the sense of Theorem 1. The empty set is the terminalobject, the unique mapping being A andsotheconditionsof Theorem 1 apply, and we see that is the initial objectin Pfn^z (and of course thatis trivial anyway).

Turningtoto-colimits inPfn

,

^{it isobviousthat}they exist, asto-chains A

(A., f.)

are,towithinisomorphism, just increasing sequences

Ao _ ^{A _. ,}

^{and so}

^{A t_JA.}

is the colimiting objectwith the colimiting cone of inclusions/z,,

"A,

G

A.

This also follows from Theorem 2, since Pfn has toP-limits. These are constructed as in

Set:

let A

(A,, fn)

be an

toP-chain; putA {a

1-IA,

ltn.a,, =/n (a,+l)}

^anddefine v,

:A A,

tobethe "nthprojection"a

an.

^Then

^A

^isthelimitof

A

and v isthe colimiting cone.

Turningtofunctors,wedefine theCartesian product^onmorphisms

f:A ^A’

^and

g:B

B’

by:

(f

xg)(a, b)=

(fa,

^{gb (if}

f(a)

^and

g(b),),

undefined (otherwise).

This makes the Cartesian product locally continuous and so, by Theorem 3, ^to- continuous. Amusingly, thecategorical product exists but is different from Cartesian product and is not even locally monotonic. On the other hand, the categorical sum exists and islocallycontinuous. Onobjects it is disjoint union"

A +

B

({0}

^xA)

CI ({

1

}

^xB) and formorphisms

f:A ^-A’

^and

^{g:B B’,}

(O, f(a)) (f +

g)(c

(1,

g

(b))

undefined

(]a

A.c (0, a)

and

f(a)),

(]b B.c =(1, b) ^and

g(b)+),

(otherwise).

Finally, there is a naturalfunction-spaceconstruction definedby A B=hom(A,B),

and for

f: ^{A’ A}

^andg:B

^B’

(f

^-->g)(h go.h

of.

This is not even locally monotonic, and so Theorem 3 does not apply. This is as expected, as the recursive domain equation

(1)

cannot have solutions in Pin (for nontrivial At) by evident cardinality considerations. For another example of an elementary

O-category,

the readercan consider thecategory Rei ofbinary ^relations betweensets,withthe subsetorderingon relations.

Example 2. Completepartial orders. Weconsider the category CPO (essentially introduced in 2) of complete partial orders and to-continuous functions. It is an

O-category

when we define the order between morphismsf,

g:A

B in the natural pointwisefashion

fg

----ta

A.f(a)_g(a).

Limits ofincreasingto-sequencesofmorphismsare definedpointwise. Theconditions ofTheorem 1 aresatisfied,asthetrivialone-point partialorder is the terminalobject and any given hom(A,B)has least element a

-

^{+/-B and}compositionisleft-strict.

Turning to to-limits (to apply Theorem 3), let A

=(A, f,)

be an to-chain and construct u"A A as in the case of Pintakingthepartial orderon

A

componentwise sothat foranya,

a’

inA.

a=_a’=-

tn

a,^=_a

’n.

This makes A a cpo with least upper bounds of increasing to-sequences, taken componentwise

la(n=(a. ⁾

774 M.B. SMYTH AND G. D. PLOTKIN

andwith leastelement

(I

I,=>,,fn,,

(_t_a. )).

Further v is acone of continuous functions and itis limiting, as if

v" A’-> A

is any other, then if 0 is a mediating morphismwe have, for all n, that

O(a’),, v’,, (a’),

determining0 as a continuousfunction. Thuswe have sketched the proofthatCPOhasalloP-limits.

Turning to functors, we have categorical product and function space functors.

Theproductof two cpos

A

andB is their Cartesianproductwiththe componentwise ordering; it is easily verified to be the categorical product. Its action on morphisms

f: ^A

^-->

^A’

^andg:B-->

B’

^isalsodefined asusual:

(f

x g)(a,b

(fa,

gb

).

Clearly, productislocallycontinuous.The function-space functorhasthesame(formal) definition as in Pfn; however, this time it iseasilyseen to be locallymonotonic and, indeed, continuous. The function-space functor is the categorical one and CPO is Cartesianclosed.

Unfortunately, CPO does not have categorical sums. It is therefore better to consider the category CPO_ of cpos and strict continuous functions (where for any cpos A and B a function

f:A

^{B is strict if}

f(+/-A)=

+/-B). ^Thishas acategorical sum which isdefined oncpos

A

andB by putting

A

+B

[{0}

^x

(A\{_I_})] U [{1}

^x

(B\{+/-})] LI {+/-},

with thepartial orderdefinedby

c=_c’

=-[a, a’

^ea.a=_a’

^

^c

^{(0, a)} ^ ^{c’= (0, a’)]}

v[=lb,

b’B.bGb’^c

=(1,

b)

^c’=(1, b’)]

VC=/

In otherwords,

A +

B is thecoalesced sum,that is, it is the disjoint union of

A

and

B,

but with least elements identified. The action of sum on morphisms turns out to begiven by puttingfor

f:A ^A’

^{and g:B}

^B’

/ ^{(0, f(a))}

^(:ia

^{A .f(a)}

^#_1_

^

^c

^(0,

^a

^)),

(f +g)(c)=i(l’ (otherwise),(ZlbB’g(b)+/-^c=(l’b))’

and this shows that the sum is a locally continuous functor. Now we know that

+

and, for example

E

are covariant t-continuous bifunctors on

CPO

^and

^CPO

^E

respectively. Luckilyhoweverthese latter categoriesarethesame,asboth embeddings and projections inCPOare strict.

(To

see thislet

f:A

^{B be anembeddingin}

^CPO.

Then

f(L)Gf(fR(L))G+/-

^andso

f(+/-)=

2; also

fR(+/-)=fn(f(+/-))=

_k.).

In the same vein we can considerthe smashproduct

A

(R)B in CPO+/- ^defined as

{(a,

b)

A

xB

[a

^#+/-

=-

b#

+/-}

with the componentwise ordering inherited from the product

A

xB (which happens also to be the categorical product in CPO+/-). On morphisms

f:A

_(f(R)g)(a,

- ^A’

^and_b)=

^f:B - ^{(f(a), B’}

^{theg(b))functoracts asfollows:}

(_t_, _t_)

(if

f(a)

^+/- andg(b) +/-), (otherwise).

Thisdefinitionshows thatthe smashproductislocallycontinuous.Itcanbe character- izedcategorically. Say thata function

f" ^A

^{x B C ofcpos} is bistrict ifandonlyif for any b in B, we have

f(+/-,

b)=^+/- (left-strictness) and also for any a in

A

we have f(a, +/-) ^+/-(right-strictness). Then theevident bistrict function (R):

A

B A(R)B isthe universal bistrict continuous functionfrom

A

B inCPO_.

Thestrictfunction-space functor is defined as before (formally) butthis time in CPO+/- and we denote itby-.. It also is easily seen to be locally continuous.

From

a categorical point of view (see

[21]),

smash product makes CPO+/- ^a symmetric monoidal category. Further,as wehaveanatural bijection

hom(A(R)B, C)horn(A, B

-.

^C),

CPO.

is even a closed category. This to some extent repairs the fact that it is not Cartesianclosed and explainstheappearanceofthe smashproduct.

Finallywe note auseful functor

(.).:

CPO

CPO.

calledlifting. For anycpo

D, (D) _ ^({0}

^D)13_t_,

with thepartial order defined^foranyd, d’ in(D)by:

dd’=(::lc,

c’ D.cTc’

^d=(O,c) ^d’=(O,c’))vd=_l_.

Onmorphisms

f:D -

^E

_{(f)_(d)}

^wehavefor

J ^{(0, f(c))}

^{any d in}

^(D).

^{(if d (0, c)),}

_1_ (ifd=_1_).

From a categorical point of view, lifting is the left adjoint to the forgetful functor from CPO+/- ^to

CPO.

We now have a wide variety ofcovariant continuous functors over CPO

z= _CPO;

_Lehmann and Smyth discuss many of their uses in

[20].

What ismore,allofthemarisenaturally fromacategorical pointof view.

It

doesnotappeartobeusefultouseOitself

(a

variant ofReynolds’predomains

[32])

^asO^z has no initialobject. (Toseethis, supposetothe contrarythat

D

isinitial.

Thenthereis anembedding

f:

^D

- ^,

^andwemusttherefore have

D ;

butclearly is not initial, as there is no embedding g:

.- ^X

^for any nonempty

X.)

On the other hand, ^one often sees an alternative definition of cpo where it is assumed that alldirected setshave 1.u.b.s rather than just increasingto-sequences.

(A

subset

X

of a partial order P is directed if it is nonempty and any two elements of

X

have an upper bound in

X.)

Let us call these partial orders dcpos (directed complete partial orders). Oneeasily adaptsthe above discussion to this case. Which definition totake is not a choice ofgreat significance. Onthe onehand, the restriction toto-sequences gives alarger category and is alsocomputationally natural, ^as theyarisewhen taking leastfixedpoints;ontheotherhand thedirectedsetsarenatural mathematically. The followingfact showsthedifference is essentiallyone ofcardinality.

FACT 1.a.

A

partial order with a least element is a cpo

if

^{and only}

if

^{it has all}

l.u.b.s

of

countable directedsets.

b.

A function f: ^A

^Bbetween cpos isto-continuous

if

^andonly

if

^itpreserves

all l.u.b.s

of

^{countabledirectedsets.}

Proof.

^{First note that for} any countable directed set

X

there is an increasing sequence

(x,),,o

of elementsofX such thatanyelement ofXis less than somex,.

Then we have

IIX

IIx,, and part a easily follows. For part b, suppose

f:A-B

^is

to-continuousand letX

_ ^A

^bedirected. With

(x,),,,

asabove, wecalculate

xX

andthis finishes theproof,as the other direction is immediate. ^[3

Another way to look at these matters was discussed by Markowsky

[22],

who noted that a partial order is a dcpo if and only if it has 1.u.b.s of all

(well-ordered)

776 M.B. SMYTH AND G. D. PLOTKIN

chains, and ^{a function}betweendcpos preserves all1.u.b.s ofdirectedsets ifand only if itpreserves all1.u.b.sof(well-ordered) chains.

Example 3. Completeness. We^considersome full subcategoriesof CPOdefined by imposingvariouscompletenessconditions.

DEFINITION 11. Let D be a partial order. A subset,

X,

ofD is K-consistent if and onlyifwhenever Y

_

^{X and}

^{Y[I < ,}

^{Y hasan}upper boundinD.

Any

subset,

X,

of a nonempty partial order is 0-, 1- and 2-consistent; it is 3-consistent if and only if it is pairwise consistent in the sense that any pair of its elements has anupper bound in

D;

it is to-consistent if and only ifany finitesubset of its elements has an upper bound inD. Clearly if <_-’ then every

’-

^consistent

subset is also to-consistent; clearly too, anydirectedsubsetis to-consistent.

DEFINITION 12.

A

partial order,

D,

is -complete if and onlyif it is nonempty and every-consistentsubsethasaleastupper bound.

It follows from the above remarksthat if <-_

’,

then every -complete partial order is’-complete; also everyto-complete partial orderis acpo (and^{even a}dcpo).

Clearly for^0_-<n

<

3 the n-complete partial orders arethe complete lattices and the 3-complete partial orders are the coherent cpos, in the sense of

[24], [29]

and, essentially,

[11].

We now see that a partial order is to-complete if and only if it is consistentlycompleteinthesense of

[29], [36]

and, essentially,

[11], [24].

FACT 2. LetDbe apartial order. Itisto-complete

if

^andonly

if

^{it is a dcpo with}

l.u.b.s

of

all subsets withupper boundsinD.

Proof.

^{LetD be}to-complete. Wehave already noted thatitjs adcpo. Also any subsetwithanupper boundinD is to-consistentandsohas aleast upper bound.

Withthe conversehypotheses,letXbeanto-consistentsubset. Then everyfinite subset has an upper bound inX and so has a least upper bound. The set of such 1.u.b.sisthen directedandso must itselfhavea1.u.b. which is alsothe 1.u.b.ofX.

Turning to the properties of the full subcategory of -complete partial orders, we see that Theorem 1 may be applied, as the one-point cpo is a complete lattice.

To see that toP-limits exist, let ^A

(A,, f,)

be an to-chain and define

v:A-.

A as before.

As

this defines a limiting cone in

CPO,

it only remains to show that

A

^is -complete. The proof employsan idea of

Scott,

forthecase of completelattices

(we

have already employedittoshowthat

A

has a leastelement).

FACT 3.

A

isK-complete.

Proof.

^{Suppose X}

^A

^is K-consistent. Thenfor every m, so is

{x,lx X},

and then the leastupper boundofXis

(I

l,,__> f,n

(I {X,,IX X}))n,o.

So the category of embeddings is a full subcategory of CPO^E with the same colimitingcones of o-chains. It follows that any to-continuousfunctors over

CPO

^E

whichpreserveto-completenesscutdownto to-continuousfunctorsonthesubcategory.

Thisremark appliestoallthe functorsdiscussed inExample 2 except thesumfunctor, which only preserves K-completeness for K

=>

^{3. Sums} of lattices can be defined by adding a new top element or by equating top elements as in

[34], [30],

and can be dealtwith bylocal continuity. General completeness concepts have been considered in

[3];

itwould beinterestingto see howtheyfit intothe presentconsiderations. One approach to handling nondeterminism and concurrency is to use one of several available powerdomain functors. These are available over CPO

(see [15]),

^{and the} Smyth powerdomain

[37]

is available overtheto-complete cpos.

However,

thePlotkin powerdomain

[27]

doesnotpreserveto-completeness;avery weaknotion ofcomplete- ness wasneeded,leadingtothe so-calledSFPobjects (brieflyconsidered in 5).

Example 4. Continuity and algebraicity. Nowweconsider theto-continuous and the to-algebraic cpos. Our main definitions

(13

and

15)

are formulated entirely in

"countable" terms, but we pause to show that one could just as well start from definitions(14and 16)formulatedwithoutany countabilityrestrictions.

DEFINITION 13. LetD be acpo. The countable way-below (=relativecompact- ness) relation is definedby: x <<y if and onlyif forevery countable directedsubset ZofD,

ifym_llZthenxm_zforsomezinZ.

Acountable subset, B, ofD is an to-basis ofD ifand onlyif forevery element x ofD theset

Bo,(x)=aCf{b Blb<<,ox}

is directed with1.u.b.x.

ThecpoD is to-continuous ifand onlyif ithasanto-basis.

Thisdefinition is notquite theverysimilar one considered inthecase ofcomplete latticesby Scottin

[33]

^{andmore generallyfordcposin}

[36], [23]

and several papers in

[7].

DEFINITION 14. LetDbeadcpo. Theway-below(=relativecompactness)relation is defined by: x<<y if and only iffor every directed subset Z of

D,

if y_llZ then

y_

z for somez inZ.

A

subsetB ofD is a basis of D ifand onlyif forevery element x of

D

the set B(x)

=aef{b Blb

^<<

^x}

is directed with1.u.b.x.

ThedcpoD is continuous ifand onlyif it has abasis.

To relate these two notions, we first note afew useful and easily proved facts.

In a cpo we have that x<<,oy implies

x_y

and x<<,oy---z implies x<<,oz; analogous facts with << replacing

<<o

^{hold in a dcpo; for} any elements x, y of a dcpo, if x^<<y then x<< o,y.

FACT. 4. Apartial order is an w-continuouscpo if and onlyif it is a continuous dcpowith acountablebasis.

Proof.

^{LetD be an}w-continuous cpo with o-basisB. First, wesee it is adcpo.

For ifX is any directedset then A

xxB(x)

is countable and directed (by the above facts), and so its 1.u.b. exists and is the 1.u.b. of X. Next we show for any elements x and y of

D

that x<<y if and

on13

^{if x}<<,oy. Suppose x<<,oy and y_lIZ whereZ is directed.Then y^=_llA as before,and so for somez in

Z

and a inB,o(z) we have x^=_a<<z, and so ^x=_z. This establishes x<<y. We have already noted the converse, that x<<y implies x

<<

y. Therefore

B

^{(x)= B}

^(x)

for any x, and so B is a countable basis. The converse assertionthat any continuousdcpowith acountable basis is o-continuousisproved along thesame lines. !-1

The w-algebraic cpos are a subclass of the w-continuous ones and can also be presentedin twoways.

DEFINITION 15. LetD be acpo.

An

elementx is o-finite(=w-compact) ifand onlyifx<<,ox. The cpo^isw-algebraicifand onlyifthereis anw-basis of finiteelements.

DEFINITION 16. LetD be adcpo.

An

element x is

finite

^{(=compact) ifand only}

if x << x.The cpoisalgebraic ifand onlyifthere isa basis of finiteelements.

One then sees thatD isano-algebraic cpo ifand only ifitis analgebraic dcpo with acountable basis of finiteelements. Alsoinany

(w-)

algebraicdcpo(cpo),there isonlyone (w-)basis,namelythe set of all(o-)finite elements.

Turningtothefull subcategoryofthew-continuouscpos,wenotethatitcontains the one-point cpo, so Theorem 1 applies; however, it does not have all to-limits, and we conjecture it does not have to-colimits. The same remarks apply to the o-algebraic cpos. Fortunately, however, the embedding subcategories inherit colimitsfromCPO+/-.Weneedapreliminarylemma.

LEMMA 5a. Embeddings (in CPO) preserve the countable relative compactness relation.

b. LetEbe a cpo andBbe a countablesubset

of

^E.

If

^{x is anelement}

of

^Eand

there is a directedsubsetC

of

^{Bo,(x) with l.u.b, x, then B,,(x) isdirected, with l.u.b, x.}