International Journal of Mathematics and Mathematical Sciences Volume 2008, Article ID 321761,6pages
doi:10.1155/2008/321761
Research Article
Continuity in Partially Ordered Sets
Venu G. Menon
Department of Mathematics, University of Connecticut, Stamford, CT 06901, USA
Correspondence should be addressed to Venu G. Menon,venu.menon@uconn.edu
Received 12 September 2007; Revised 8 October 2007; Accepted 4 December 2007
Recommended by Pentti Haukkanen
The notion of a continuous domain is generalized to include posets which are not dcpos and in which the set of elements way below an element is not necessarily directed. We show that several of the pleasing algebraic and topological properties of domains carry over to this setting.
Copyrightq2008 Venu G. Menon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Continuous lattices and their generalizations, continuous domains, have been studied for more than three decades. Continuous lattices are complete lattices where each element is the supre-mum of elements way below it. For a poset to be a continuous domain, it needs to have sups
of only directed sets in addition to the following two conditions:ieach element is the sup of
elements way below it, andiifor each element, the set of elements way below it is a directed
set. A continuous poset1is any poset in which the conditionsiandiiare satisfied. In a
complete lattice, in fact in any sup-semilattice, the conditioniiabove is automatically
satis-fied. In2, the authors have studied lattices which are not complete but satisfy the conditions
iandiiabove. The purpose of this paper is to study posets which need not be dcpos or
lattices but which satisfy the condition that each element is the sup of elements way below it. The exact definition will be given in Section2.
Here we recall some basic definitions and terminology from domain theory; more details
can be found in1. Forx, y∈P,a partially ordered set, we say thatxis way belowywritten
xy, if whenevery≤supD, for a directed setD,there existsd∈Dsuch thatx≤d.A
contin-uous poset is a partially ordered setPin which the following two conditions are satisfied.iFor eachx∈P,xsup{y:yx}; andiifor eachx∈P,{y:yx}is a directed set. A
contin-uous poset in which every directed set has a least upper boundsuch posets are called dcpos
satisfied, one needs to verify only the first condition. In any continuous posetP,the way below relation has the interpolation property, that is, forx, y∈Pwithxy,there exists an element
z∈P,such thatxzy.For more information about continuous lattices, and continuous
domains, the standard reference is1. A subsetSof a posetP is called up-complete if for any
directed subsetD ofS,for which supDexists inP, supD is contained inS.We use the
fol-lowing notations:↓ A {x : x ≤ afor somea ∈ A};⇓ A {x : x afor somea ∈ A};
and⇓x⇓ {x}.A subsetJof a posetPis called a lower set if and only ifJ ↓J.Recall that a subsetUof a posetPis Scott open if and only ifUis an upper set satisfying the property that
if the supremum of any directed setDis inU,thenDitself intersectsU.The lower topology
on a posetPhas subbasic closed sets of the form↑x, x∈P.The join of the lower topology and
the Scott topology onPis the Lawson topology and is denoted byλP.
2. C-posets
The following simple but crucial result is the motivation for our definition of a C-poset.
Theorem 2.1. LetLbe a continuous poset. Then for eachx, y∈L,withxy,there existsu∈Land an up-complete lower setJ⊆Psuch thatuy,x /∈J,and↑u∪JL.
Proof. Suppose thatLis a continuous poset. Letx, y∈Lwithxy.Then∃uxwithuy.
By the interpolation property,∃v∈Lsuch thatuvx.Clearlyvy.DefineJL\ ⇑v.
IfDis a directed subset ofJ, thenv is not way below any element ofD.Ifv supD,then
there existsw such thatv w supD.This implies that∃d ∈D such thatv w ≤ d,a
contradiction. This shows that the supremum ofDbelongs toJ,and thusJis up-complete. It
is immediate thatuy,x /∈J,and↑u∪JL.
Definition 2.2. A partially ordered setPis called a C-poset if for anyx, y∈P,withxy,∃uy
and an up-complete lower setJsuch thatx /∈J,and↑u∪JP.
Example 2.3. iIfSis an infinite set and ifPis the poset of all finite and cofinite subsets ofS,
thenP is a C-poset. Suppose thatA,B are inPsuch thatA B.Letx∈Asuch thatx /∈B.If
we defineJ {A:A∈P, x /∈A},andu{x},then thisJandusatisfy the conditions in the definition of a C-poset.
iiLetXbe aT0space for which the lattice of open sets is a continuous lattice. HereX
is endowed with the specialization order. LetLbe a continuous domain with a least element
endowed with the Scott topology. LetP X→Lbe the set of all Scott continuous functions.
It can be shown thatPis a C-poset but not a continuous poset, see1, page 200.
iiiLetPi, i∈Ibe a collection of C-posets with 0 and 1. Then the product ofPisis a
C-poset. Ifxiyi,then there existsj ∈Isuch thatxj yj.SincePjis a C-poset, there exists
a∈Pjsuch thatayjand an up-complete lower setKofPjsuch thatxj/∈KandPj↑a∪K.
Defineu ui, whereui 0 ifi /janduj a.Also defineJ ↓ S, whereS{xi:xi1 if
i /jandxj∈K}.It is easy to see that the aboveuandJsatisfy the conditions in the definition
of a C-poset.
The next theorem establishes that C-posets do satisfy the approximation property, that is, each element is the supremum of elements way below it.
Proof. We need to prove this only forx /0. Ifx /0, then there existsy ∈ P such thatx y.
Therefore, by the definition of a C-poset, there existsu y,and an up-complete lower setJ
such thatx /∈J,and↑u∪JP. We will show thatux.Suppose thatDis a directed set inP
such that supD≥x.IfD∩ ↑u∅,thenD⊆Jand sinceJis up-complete, supD∈J.SinceJis a lower set, this forcesx∈J,a contradiction. This shows thatux.Thus the set{z:zx}is
nonempty, andxis an upper bound of it. Letwbe any upper bound of that set. We will show
thatx≤w.Indeed, ifxw,then∃uwand an up-complete lower setJsuch thatx /∈ J,
and↑u∪JP.But thenux,which contradicts the assumption thatwis an upper bound
of the set{z:zx}.Thereforex≤w.This completes the proof of the theorem.
Corollary 2.5. If a C-poset is a sup-semilattice, then it is a continuous poset; if a C-poset is a complete
lattice, then it is a continuous lattice.
Theorem 2.6. LetPbe a C-poset, and let{xi,k:i∈I, k∈Ki}be a nonempty family of elements of
Psuch that{xi,k :k∈Ki}is a directed set for alli∈I.Then the following identity holds whenever
the specified sups and infs exist:
i∈I
Ki
xi,k f∈M
i∈I
xi,fi, 2.1
whereMis the set of all choice functionsf:I→i∈IKiwithfi∈Kifor alli
Proof. Since the right hand side is always less than or equal to the left-hand side, we need to
prove only the reverse inequality. By Theorem2.4, it is enough to show that wheneverx
i∈I
Kixi,k,we havexf∈M
i∈Ixi,fi.
Suppose that x i∈I
Kixi,k.Then,x
Kixi,k,for all i ∈ I. Therefore, by the
definition of the way below relation, we can choose agi ∈Kiwithx≤xi,gifor alli ∈I.
This implies thatx≤i∈Ixi,gi,and hencexis less than or equal to the right hand side of the
identity. This completes the proof of the theorem.
Letg : P → Q,and letd : Q → P be monotone functions between posets. The pair
g, d is called a Galois connection if forx ∈ P, andy ∈ Q,gx ≥ y if and only if x ≥
dy.Heregis called an upper adjoint, anddis called a lower adjoint. A monotone function
g : P → Q between posets is called an upper adjoint if there exists anecessarily unique
monotone functiond:Q→Psuch thatg, dis a Galois connection. Basic properties of Galois
connections can be found in1.
Definition 2.7. A function from a C-poset to another C-poset is called a homomorphism if it is an
upper adjoint which preserves sups of directed sets. A subposetSof a C-posetP is called a
subalgebra if the inclusion mapi:S→Pis a homomorphism.
Theorem 2.8. aLetP be a C-poset and letQbe any poset. Iff :P →Qis a surjective homomor-phism, thenQis a C-poset.
bIfPis a C-poset and ifSis a subalgebra ofP,thenSis a C-poset.
Proof. aSuppose thatgis the lower adjoint off.Lety2y1inQ.Pickx1∈Pwithfx1 y1.
Sinceg is a lower adjoint off,y2 fx1impliesgy2 x1.Then there existu∈P and an
fx≤fximpliesgfx≤x,which impliesgfx∈J.Thusfx∈g−1J I.Ifx /∈J,then
u≤ xwhich impliesfu ≤ fx.Thus↑ fu∪I Q.This completes the proof thatQis a
C-poset.
bLeti :S→ Pbe the inclusion map and letdbe its lower adjoint. Letx, y ∈Ssuch
thatx y.Then there existsu ∈ P, and an up-complete lower setJ ⊆ P,such thatu y,
x /∈J,andP ↑ u∪J.Sinceuyiy,duy.DefineI :i−1J.Sinceipreserves sups
of directed sets,I is an up-complete lower set, andx /∈I.Fort∈S,t /∈ J,impliesit t /∈ J,
which implies thatu≤t.Thusdu≤dt t.Therefore,↑Sdu∪I S.This completes the
proof thatSis a subalgebra.
Notice that the proof ofaabove did not use the assumption thatfpreserves the sups
of directed sets.
Definition 2.9. LetP be a C-poset. A subposetBofPis called a basis ofPif given anyx, y∈P
withxy,there existb ∈Band an up-complete lower set J ⊆Psuch thatb y,x /∈J, and
↑b∪J P.
Theorem 2.10. IfBis a basis of a C-posetP,then for eachx∈P,xsup{b:b∈B, bx}.
Proof. We need to prove this only forx /0. Ifx /0, then there existsy∈Psuch thatxy.Then,
by the definition of the basis, there existsb∈Bsuch thatby,and an up-complete lower set
Jsuch thatx /∈J,and↑ b∪J P.It follows immediately thatbx.Thus,{b:b∈B, bx}
is nonempty andxis an upper bound of it. Suppose thatvis any upper bound of the same
set. Ifx v, then there existb ∈ B and an up-complete lower set Jsuch that x /∈J,and
↑ u∪J P. As shown aboveb xwhich contradicts the assumption thatv is an upper
bound of{b:b∈B, bx}.Thusx≤u.This completes the proof.
Proposition 2.11. 1LetP, Q be C-posets with B as a basis ofP. If g : P → Q is a surjective homomorphism, thengBis a basis ofQ.
2IfSis a subalgebra of a C-posetP,and ifBis a basis ofP,thendBis a basis ofS, whered
is the lower adjoint of the inclusion map.
Proof. 1Letdbe the lower adjoint ofg.Suppose thaty1, y2∈Qsuch thaty2y1.Letx1∈P,
such thatgx1 y1.Sincey2 gx1, dy2 x1.SinceBis a basis of the C-posetP,there
existb ∈Band an up-complete lower setJ ⊆P such thatb x1, dy2∈/J,and↑b∪J P.
LetId−1Jandcfb.Since the lower adjoint preserves sups,Iis an up-complete lower
set, andy2∈/I.Ifx∈J,fx≤fximpliesdfx≤x.This implies thatdfx∈J,and hence
fx∈d−1J I.Ifx /∈ J,thenb≤x,which impliesfb c≤fx.Thus↑ c∪I Q.This
completes the proof.
The proof of2is similar and hence omitted.
Though algebraic lattices were studied for several decades before the introduction of continuous lattices, algebraic lattices, and their generalizations, algebraic domains, have played an important role in domain theory. Algebraic domains are continuous domains in
which every element is the sup of compact elementsan elementxis compact ifxxbelow
Theorem 2.12. IfLis an algebraic poset, then forx, y ∈ L,withx y,there existu ∈ Land an up-complete lower setJ⊆L, such thatuy, x /∈J,↑u∪JL,and↑u∩J∅.
Definition 2.13. A C-poset is called an A-poset if the condition of Theorem2.12is satisfied. Proposition 2.14. A C-posetPis an A-poset if and only if the set of all compact elements ofPis a basis ofP.
3. Topology on C-posets
The wealth of topological structures in continuous domains and the interplay between the
topological and algebraic properties of continuous domains are well documented1. In this
section, we look at C-posets endowed with the Lawson topology.
Theorem 3.1. A C-poset endowed with the Lawson topology is a pospace and hence Hausdorff. An A-poset endowed with the Lawson topology is totally order-disconnected.
Proof. LetPbe a C-poset. Suppose thatx, y∈Psuch thatxy.Then, there existu∈P,and an
up-complete lower setJsuch that uy, x /∈J,and↑u∪JP.LetUP\J,andV P\ ↑u.
ThenU is a Lawson open-upper set, andV is a Lawson open lower set. Clearlyx ∈ U and
y∈V.MoreoverU∩V P\J∩P\ ↑u P\J∪ ↑u ∅.This shows thatPis a pospace.
IfP is an A-poset, then forx, y ∈ P,x yimplies the existence ofu ∈ P and an
up-complete lower setJas in the previous paragraph with the additional condition that↑u∩J∅.
This means that↑uis both closed and open. This completes the proof of the theorem.
An inf-semilatticeP is called meet continuous if for all directed subsetsDofP for which
supDexists, and for allx∈P,xsupDsupxD.
Proposition 3.2. If a C-posetPis also an inf-semilattice, thenPis meet-continuous.
Proof. LetDbe a directed subset ofPsuch that supDexists, and letx ∈P.We need to show
only thatxsupD ≤ supxDsince the reverse inequality is always true. Suppose thatxsup
D supxD.Then there existu supxDand an up-complete lower setJ such thatxsup
D /∈Jand↑u∪JP.SinceJis a lower set, this means thatx /∈J,and supD /∈J.Thenx∈ ↑u,
and sinceJ is up-complete,∃d ∈ D such that d ∈ ↑ u. Thus for thisd, xd ∈ ↑ u which
contradictsusupxD.This completes the proof.
Proposition 3.3. LetPbe a C-poset which is also an inf-semilattice. Then the inf-operation is a contin-uous functionL×L, λL×L→L, λL.
Proof. First, considerV ∧−1P\ ↑ x {y, z:x y∧z}.We will show thatV is open. If
y, z∈V,thenx y∧z.Therefore, there exist au y∧zand an up-complete lower setJ
such thatx /∈J,and↑u∪JP.LetS P×P\ ↑u, u.Then clearlySis open inP×Pand
y, z∈S.We will show thatS⊆V.Ift, r∈S,then eitherutorur.Therefore,ut∧r,
which implies thatt∧r∈J.Ifx≤t∧r,then, sinceJis a lower set,x∈J,a contradiction. Thus
xt∧rwhich means thatS⊆V.
LetObe any Scott open subset ofP,and considerU ∧−1O.We will show thatU
{y, z:y∧z∈O}is a Scott open subset ofP×P.Suppose thatDis a directed subset ofP×P
D2{d2:∃d1such thatd1, d2∈D}.ThenD1andD2are directed sets such that supD1a,
and supD2 b. SinceP is meet-continuous by Proposition3.2,a∧b supD1∧ supD2
supD1D2∈O.SinceD1D2is a directed set and sinceOis Scott-open, there existd1∈D1and d2∈D2such thatd1∧d2∈O,which implies thatd1, d2∈U.This completes the proof of the
proposition.
Acknowledgment
The author wishes to thank the referee for several suggestions to improve this paper.
References
1G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott, Continuous Lattices and Domains, vol. 93 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 2003.
