Semigroup Forum Vol. 42 (1991) 77-82 �9 1991 Springer-Verlag New York Inc.

R E S E A R C H A R T I C L E

Equational Theory of Continuous Lattices

A. M S b u s

Communicated by K. H. Hofmann

It is known that continuous lattices and their morphisms can be obtained as algebras and the corresponding homomorphisms respectively for an (infini- tary) algebraic theory. In this paper we present one such theory explicitly.

I n t r o d u c t i o n

By a result of A. Day [1], the category C o n t L a t t of continuous lattices and maps preserving all infima and directed suprema is equivalent to Set ~ , the category of algebras for the filter monad. This implies, that continuous lattices can be defined by an (infinitary) algebraic theory, as has been remarked by R.-E. Hoffmann [3; p.816]. We shall specify an algebraic theory for continuous lattices, i.e. a (necessarily proper) class of basic operations and a class of equations. From [1] we know, that for any set X the set of all X - a r y operations (i.e., basic and composite operations) corresponds to the set F X of filters on X . If L is a continuous lattice and G E F X , then the corresponding X - a r y operation C,: L x ---* L is given by G( f ) = l iminf f (G) (for . f : X ~ n) . The class of basic operations is far f rom unique. The most simple choice is to take all operations as basic operations (and try to find a nice basis for the class of all equations, that hold between them). The aim of this paper is to show, that it suffices to take the inferior limits of filters induced by the Fr6chet filters of a small class of directed posets (called PFP-posets) . This class of filters seems to be more perspicuous than the class of all filters, and we are also able to write down a basis for the class of equations explicitly. Wyler [5] also characterized continuous lattices in terms of inferior limits, but his characterizations are not equational, since they

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involve the induced compact Hausdorff topology. On the other hand, continuous latt ices can be character ized among complete latt ices by equations (for complete lat t ices) [2; 1.2.3(3)]. The terms of these equations, however, are buil t up from infima and a rb i t r a ry suprema. Since a rb i t r a ry suprema are not preserved by C o n t L a t t - m o r p h i s m s , they cannot be continuous lat t ice operat ions , and the equations are not equations for continuous latt ices in the sense of algebraic theories. As usual , d i rected suprema (resp. posets) are always assumed to be non-empty.

The re la t ion of this note to the paper of Krasner was invest igated fol- lowing the suggestion of a referee.

1. I n f e r i o r l i m i t s for f a m i l i e s i n d e x e d b y a p o s e t

Let (P, < ) be a directed poset. In any complete la t t ice L we can define a P - a r y opera t ion (P~ <) - l im in f , or for short P - l imin f : L P -~ L by

P - l iminf((Yv)vEP) = V A Y~ (1) pEP q>p

We shall also use the nota t ion

l im in f yp for P - l iminf((yp)pep ) pEP

If L is a continuous lat t ice, then P - l imin f is a continuous lat t ice opera t ion , since the sup is a directed one. (In fact, it is the l iminf opera t ion for the (generalized) Fr4chet filter on P . ) Trivially:

(CL1) for e v e r y r E P l imin f y ~ > AYq p6P q>r

(eL2) l im in f Yv = l im in f ~ yq pEP pEP q>_p

w > l im in f (Yv A w) -- pEP

(CL3)

and in a continuous latt ice:

(CL4) A l imin f yi,v, < l imin f A Y~.p, iEI PiePi -- (pJ)JqIEYI PJ i 6 i

because infs d i s t r ibu te over di rected sups and over infs.

We also note

(CL5) If f : P ~ Q is an i somorphism of posets, then

l imin f yq = l imin f y~,(p) qEQ pEP

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So there is essential ly one opera t ion for each i somorphism class of di rected posets. If G is a filter on a la t t ice L , we consider (G, D), the di rected poset of the elements of G with their converse order. Then

l i m i n f G = (G, _D) - l i m i n f (( A re)Meg). m E M

Therefore P-l iminfs and infima suffice to generate all operat ions of continuous latt ices. In the next chapter we shall show tha t liminfs for a small class of di- rected posets ( together with infima) suffice to give an equat ional ax iomat iza t ion of continuous lat t ices.

2. A n e q u a t i o n a l t h e o r y o f c o n t i n u o u s l a t t i c e s

L e m m a 1. Let Q be an arbitrary class of directed posers. Then complete lattices with P- l imin f s for all P in Q as defined by (1) regarded as additional operations are given by the following algebraic theory."

operations: for each cardinal c~ an c~-ary operation 'inf' denoted Aie~

and for each poset P in Q a P-ary operation denoted

P - l im in f

equations: (CL0) the inf's are idempotent and satisfy all commutativity and associativity laws

(CL1), ( eL2) , (CL3) a~ above for e'oery P in Q

P r o o f . In any complete la t t ice define P - l imin f by (1). Then clearly all equations are satisfied. Conversely, let L be a model of the given algebraic theory. Then L is a complete la t t ice with respect to A. It remains to show, that P - l im in f is in fact the l iminf-operat lon of L as defined by (1). So let z = liminfpEp yp in L. By (CL1) z >_ Aq>_rYq for every r E P . Let w be another element, such tha t w >_ Aq>_r yq for every r E P .

By (CL3) w > l imin f (yp A w) -- pEP

= l imin f A ( y q A w ) ( b y CL2) pEP q>p

= l i m i n f ( w A A yq) pEP q>_p

= l imin f A y~ = z. pEP q>p

So z = V~eeAq>_, yq. �9

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R e m a r k . The proof shows, that in any lattice the operations P - l iminf are uniquely determined by the equations (CL1)-(CL3). Therefore (CLh) is a consequence of (CL0)-(CL3). The converse inequality of (CL4) is also easily derived from (CL0)-(CL3).

For a set X let Pr denote the poset of finite non-empty subsets of X . P is called a PFP-poset (= product of finitary-powerset posets) if P is of the form P = lqiez Pf(X~)

T h e o r e m 2. C o n t L a t t ~s equivalent to the category of models of the fol- lowing algebraic theory:

operations: for each cardinal a an a - a r y operation 'inf', denoted Aie~,

and for each PFP-poset P a P-any operation, denoted

P - l iminf

equations: (CL0) the inf's are idempotent and satisfy all commutativity and associativity laws

(CL1), (CL2), (CL3), (CL4) as above.

P r o o f . In any continuous lattice define P - l iminf by (1). Then clearly all equations are satisfied. Conversely, let L be a model of the given algebraic theory. Then L is a complete lattice with respect to A and by Lemma 1 the P - l i m i n f s are the liminfs as in (1). So it remains to show:

(i) L is a continuous lattice (it) a homomorphism for the given theory preserves directed sups.

For S C_ L directed and M C PI(S) choose ~s(M) C S, such that xs (M) > m for every m E M . (this is possible, since S is directed). Then

l iminf (xs (M)) = V A xs (M) = V s (2) MePI(S) N6P/(S) MDN, M6P~t(S)

t

A homomorphism g : L --* L' preserves order. So g(xs(M)) >_ g(n~) for every m 6 M, M 6 Ps (S) , and (ii) follows easily.

Now let a E L be arbitrary. Let G = {S C_ L [ S directed, V S = a}.

So a = A l iminf (xs(Ms)) (by2 ) SeG Ms6PI(S)

< liminf A xs(Ms) (by CL4) - (Ms)s~a~Hse~P,(s) s~c

Since hseG xs (Ms) << a, we have a < sup I a. This proves (i).

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3. R e d u c t i o n t o f i n i t a r y - p o w e r s e t p o s e t s

Continuous lattices may be axiomatized with an even smaller class of operations than that of Theorem 2. Infima and P-liminfs for finitary-powerset posets suffice, but then (CL4) has to be replaced by a more complicated equa- tion.

Let (P,_<) be a directed poser, and let X : t f ( P ) --~ P be a choice function, such that for all finite, non-empty M C P and for all m E M )r >

m. T h e n Aq_>p yq < AMePI(P),p6M Yx(M)

So l imin fyp = V A y q p6P p6 P q~p

NEPI(P ) MEPf(P), MDN

= l iminf (Yx(M)) MEPI(P)

For P = I]jel Pj we obtain for continuous lattices

A l i m i n f ( y , p ) < l iminf ( A Y4.• 461 PlePi ' " --MePI(I]P') i61

If the Pj are finitary-powerset posets X : Ps(1-[ Pj) ~ [I Pj can be chosen canonically (componentwise the set-union: X(M) = (U{rm [ (mA E fvl})~,) and

(CL4' ) A l iminf Y4,v~ < l iminf A Yl,u{'~d(mJ)6 M} i6I pi6Pi -- MePI(YIP~) 46I

Now it is easily checked, tha t (CL0)-(CL3) and (CL4' ) for finitary-powerset posets provide another axiomatization of continuous lattices. In particular, all operations of continuous lattices can be obtained as compositions of infima and finitary-powerset liminfs. To be more precise, let P, O be directed posets and f : P ~ Q be a map, such that

VqeQ 3,ep V,_>~ f(r) _> q. (3)

Then trivially l iminf yq = l i m i n f A yt (4)

q6Q p6P t>f(p)

If Q is an arbi t rary directed poser, we may choose P = PI(Q) and f : P ~ Q such that I ( M ) > m for every m E M . Then f satisfies (3) and the Q - l i m i n f is reduced to a finitary-powerset liminf. If f is a homomorphism of directed posets, (3) is equivalent to the cofinality of f ( P ) in Q. If Q is of the form

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(G, D) for a filter G and f is chosen canonically (i.e., the intersection), then f is in fact a homomorphism.

The special role played by finite-powerset posets in the theory of di- rected posets was observed earlier by Krasner [4], who showed that for every directed poser Q there exists a finitary-powerset poser ( ' tour ferm6') P and a monomorphism f : P ---* Q with cofinal image.

[1]

[2]

[3]

[4]

[5]

R e f e r e n c e s

Day, A., Filter monads, continuous lattice.~ and closure ~ystems, Canad. Journ. Math. 27 (1975), 50-59.

Gierz, G. et al., A Compendium of Continuous Lattices, Springer 1980.

Hoffmann, R.-E., The Injective Hull and the CL-Compactification of a Continuous Poset, Canad. Journ. Math. 37 (1985), 810-853.

Krasner, M., Un type d'ensembles semi-ordonn~s et ses rapports avec une hypothkse de M. A. Well, Bull. Soc. Math. France 67 (1939), 162-176.

Wyler, O., Algebraic Theory of Continuous Lattices , in: Continuous Lattices, Proc. Bremen 1979, Springer Lecture Notes in Math. 871, 390-413.

Rechenzentrum der

Heinrich- Heine- U niversit~t Dfisseldorf~

D-4000 Diisseldorf, F. R. Germany _

Received October 16, 1989

and in final form November 7, 1989

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