Download Lattice Topologies with Interval Bases

Lattice Topologies with Interval Bases
Marcel Erné
Faculty for Mathematics and Physics
Leibniz University Hannover
D-30167 Hannover, Germany
e-mail: [email protected]
Abstract
We study topologies τ on a lattice L with open subbases of intervals
[y) and (z] (y ∈ Y, z ∈ Z). As a basic result, we show that τ coincides
with the order topology (and (L, τ ) is a totally order-disconnected topological lattice) iff Y is a join-dense set of compact elements and Z is a
meet-dense set of cocompact elements; moreover, that (L, τ ) is a compact
totally disconnected topological lattice iff L is a complete lattice with Y
as a join-dense set of hypercompact elements and Z as a meet-dense set of
hypercocompact elements, iff L is complete and τ agrees with the interval
topology. Our results lead to various large classes of order-topological lattices with interval bases, to topological characterizations of bi-algebraic
lattices and, on the other hand, to algebraic characterizations of compact
totally (order-)disconnected topological lattices.
Mathematics Subject Classification. 06F30, 54A10, 54H12
Key Words. (Bi-)algebraic lattice, (hyper-)compact, interval, Lawson
topology, order topology, Scott topology, topological lattice, totally (order-)
disconnected.
0
Introduction
In this note we investigate certain topologies on lattices that admit a topological
treatment of various algebraic decomposition properties. It is well known that
the existence of join-decompositions into compact elements is closely related
to total disconnectedness of the involved topologies (see e.g. the Fundamental
Theorem for Compact Totally Disconnected Semilattices [12, VI–3.13]). Here
we generalize some classical results of continuous lattice theory to the setting
of arbitrary order-consistent topologies. For example, if such a topology has an
interval base then it agrees with the Scott topology, and the underlying lattice
is compactly generated (and vice versa).
Concerning the “two-sided” situation, one central observation will be that
for a topology τ on a lattice L having a subbase consisting of principal ideals
and principal dual ideals, τ -convergence agrees with order convergence iff (L, τ )
is a totally (order-)disconnected topological lattice whose topology is coarser
than the order topology, and that any such lattice is bi-compactly generated.
1
This leads to a topological description of bi-algebraic lattices as totally orderdisconnected topological lattices with an open base of intervals whose complements are compact in the interval topology. As another consequence, we shall
obtain diverse characterizations of totally order-disconnected compact topological lattices. Some of these characterizations are new, others are known, but
our approach essentially facilitates proofs. In this context, it turns out that in
the presence of enough indecomposable elements (atoms, inaccessible elements,
irreducible elements, etc.) seemingly weak separation axioms like T1 already entail total (order-)disconnectedness. For example, a bi-atomistic semitopological
lattice with closed universal bounds is already totally order-disconnected. Furthermore, we improve and generalize some results of Choe and Greechie [2] and
of Riečanová [17]–[19] from orthomodular to arbitrary lattices (see also [11]).
The Axiom of Choice will often apply without particular emphasis.
1
Intrinsic topologies of lattices
In what follows, L denotes a lattice and Lop its dual or opposite, obtained by
reversing the order; often, it would suffice that L is a semilattice or merely a
poset. Principal ideals
(z] = {x ∈ L : x ≤ z},
principal dual ideals
[y) = {x ∈ L : y ≤ x},
and their intersections
[y, z] = [y) ∩ (z] = {x ∈ L : y ≤ x ≤ z}
are called intervals. By an interval base of a topology τ on L, we mean an open
base for τ consisting of intervals. Notice that these intervals need not be closed
with respect to the topology τ , though they are sometimes referred to as “closed
intervals”.
A subset U of L is an upper set (upper end, increasing set, upset) if y ∈ U
implies [y) ⊆ U ; lower sets (lower ends, decreasing sets, downsets) are defined
dually. If X is an arbitrary subset of L then
S
↑ X = {[x) : x ∈ X}
is the upper set generated by X, and
S
↓ X = {(x] : x ∈ X}
is the lower set generated by X. By J L we denote the set of all ideals (that is,
of all directed lower sets) possessing a join.
The upper (resp. lower) sets form a topology τU = α(L) (resp. τL = α(Lop )),
the so-called upper (resp. lower) Alexandroff topology. Besides these two topologies, we shall consider the following intrinsic topologies on L (cf. [4]–[12], [14]):
the upper topology τu = υ(L) is generated by the complements of all principal
ideals, and the lower topology τ` = υ(Lop ) by the complements of all principal
dual ideals; the Scott topology τS = σ(L) consists of all upper sets U such that
2
W
any Y ∈ J L with Y ∈ U intersects U . Thus, τS is finer than (and sometimes, but not always, identical with) τu but usually coarser than τU . The order
topology τo = Ω(L) consists of all sets U such that
W
V
(o) for any up-directed set Y and any down-directed set Z with Y = Z ∈ U ,
there exist elements y ∈ Y in z ∈ Z with [y, z] ⊆ U .
Hence, the τS -open sets are precisely the τo -open upper sets. We call a topology
o-coarse if it is contained in the order topology, i.e., each open set U enjoys (o).
A filter F order converges (o-converges) to a point x (cf. [4, 8]) iff x is both
the limit inferior and the limit superior of F, that is,
W
V
x = {y ∈ L : [y) ∈ F} = {z ∈ L : (z] ∈ F}.
In the present note, order convergence always refers to filters. (For the definition of net-theoretical order convergence, see e.g. [1, X–9]). The order topology
is the finest topology τ such that order convergence implies τ -convergence (see
[8, Thm. 4.6]). Order convergence is said to be topological if it agrees with convergence in some topology, which is then necessarily the order topology. This
coincidence often fails; for example, a Boolean lattice has topological order convergence if and only if it is atomic (see [4, Thm. 4]). A necessary and sufficient
criterion for order convergence to be topological was given in [8, Cor. 4.13]:
Proposition 1.1 The order convergence on a lattice L is topological iff each
neighborhood filter with respect to the order topology has a base of intervals. In
particular, this happens whenever the order topology has an interval base.
From the previous topologies one may build new ones by forming joins in the
lattice T (L) of all topologies on L. The following notation will be convenient:
given any topology τ on L, we denote by τ ∨ the topology generated by τ and the
lower topology τ` ; in other words, τ ∪ τ` is an open subbase for τ ∨ . For example,
the join τu∨ = τu ∨ τ` of the upper and the lower topology is the interval topology
τi = ι(L), the coarsest topology in which every interval is closed. Of particular
importance for continuous lattice theory is the Lawson topology τS∨ = λ(L) (see
[12, III]). Sometimes the “half-open interval topologies” τU∨ = τU ∨ τ` and τu ∨ τL
are also of interest (cf. [20, Ex. 51 and 84]). Notice that each of the topologies
τu , τ` , τi , τS and τS∨ is o-coarse, on account of the inclusions
τu ∨ τ` = τi ⊆ τS∨ ⊆ τo .
Moreover, order convergence, order topology and interval topology are related
as follows (see [8, 2.4–2.6]):
Proposition 1.2 If the interval topology τi is T2 then it agrees with the order
topology τo . Furthermore, τi is compact and T2 iff every ultrafilter o-converges.
In the present context (and also in the subsequent considerations), the meaning of the word “compact” does not include the Hausdorff separation axiom T2 .
The following generalization of Frink’s theorem on the compactness of the interval topology (cf. [1, X–Thm. 20]) is often useful:
Proposition 1.3 A lattice L with a topology between the interval topology and
the Lawson topology is compact iff L is complete.
3
By Alexander’s subbase theorem, the
T complement of a set U is τi -compact
iff for each filterbase β of intervals with β ⊆ U , there is a B ∈ β with B ⊆ U .
On the other hand,
T a set U is open in the order topology iff for any filterbase β
of intervals with β = {x} ⊆ U , there is a B ∈ β with B ⊆ U . This proves
Proposition 1.4 Any τi -compact set is τo -closed. In particular, any compact
topology finer than the interval topology is o-coarse, and so is any topology τ
such that τ ∨ is compact.
In the subsequent diagram, we display seven of the previously introduced
topologies together with the indiscrete topology. They form a join-subsemilattice
of the lattice T (L).
Diagram 1.5 Intrinsic topologies on a lattice L.
half-open interval
topology
τU∨ = τU∨τ`
Alexandroff
topology
Q
Q
Q
Q
Scott
topology
Q
Q
τS = σ(L)
upper
topology
Lawson
topology
τS∨ = λ(L)
τU = α(L)
interval
topology
τi = ι(L)
Q
Q
τu = υ(L)
Q
Q
Q
Q
τ` = υ(Lop)
lower
topology
indiscrete τ = {∅, L}
in
topology
For complete lattices L, this is even a sublattice of the lattice T (L), on
account of
Proposition 1.6 In a complete lattice, the Lawson-open upper sets are the
Sott-open upper sets, the Lawson-open lower sets are the τ` -open sets, and the
τi -open sets are the τu -open sets.
For the first and the second statement, see [12, III–1.6 and 3.16]. The third
one is obtained by dualizing the second and observing that τi is always coarser
than the Lawson topology. Without the assumption of completeness, the second
and the third assertion in 1.5 fail to be true in general:
Example 1.7 The pairs (− n1 , − n1 ), (− n1 , 0) and ( n1 , 0), where n runs through
the positive integers, form a sublattice of the real plane (partially ordered componentwise by the usual ≤), and the interval topology of this sublattice is discrete.
Hence the element x = (−1, 0) generates a τi -open principal dual ideal which is
neither τu -open nor τS -open.
4
pp s
ps p app p
s &s
@
(−1, 0) s @s
@
@
s
s (1, 0)
@s
(−1, −1)
By a topological (respectively, semitopological) lattice, we mean a lattice
endowed with a topology making the binary lattice operations (respectively, the
unary operations ∧y : y 7→ x ∧ y and ∨y : x 7→ x ∨ y) continuous. Again, the
Hausdorff property is not assumed a priori in these definitions. A lattice L
satisfying the identity
W
W
x ∧ Y = {x ∧ y : y ∈ Y }
for all x ∈ L and Y ∈ J L (hence also for all directed sets Y possessing a join) is
called meet-continuous or upper continuous. Notice that often these names are
attributed to complete lattices only (cf. Birkhoff [1] or Crawley and Dilworth [3]).
The dual notion is join-continuity or lower continuity. A topological justification
of the notation meet-continuous has been given in [4, Prop. 5] and [9, Thm. 2.8]:
Proposition 1.8 The following statements on a lattice L are equivalent:
(a)
(b)
(c)
(d)
(e)
L ist meet-continuous.
The binary meet operation is continuous with respect to o-convergence.
The unary meet operations are continuous with respect to o-convergence.
The unary meet operations are continuous with respect to τo .
The unary meet operations are continuous with respect to τS .
In accordance with other authors we call a join- and meet-continuous lattice
o-continuous, since by 1.7 and the dual statements, a lattice L is o-continuous iff
its operations are continuous with respect to order convergence, or equivalently,
if (L, τo ) is a semitopological lattice. However, an o-continuous lattices need
not be a topological lattice in its order topology (see [9, Cor. 4.11]). By an
order-topological lattice we mean an o-continuous lattice with topological order
convergence (of filters). Thus L is order-topological iff it is a topological lattice
with a (unique) topology whose convergence agrees with order convergence.
For a purely algebraic characterization of order-topological lattices as so-called
bi-continuous lattices, see [4, Thm. 2].
Examples of order-topological lattices are bi-compactly generated lattices
(in which every element is a join of compact elements and a meet of cocompact elements; (see [1, Thm. 3] and Section 24) and, in particular, chain-finite
lattices (the latter having discrete order topologies); further, completely distrubitive complete lattices (hence arbitrary products of complete chains), and
finite products of arbitrary chains. However, there are infinite products of chains
which fail to be order-topological, for example, infinite powers of the chain ω of
natural numbers (cf. [5, Section 4]). Notice also that a lattice with topological
order convergence need not be order-topological (see Example 3.1).
5
2
One-sided topologies with interval bases
Let us first recall some basic notions from topology. A topological space is
said to be zero-dimensional if it has a base of clopen sets, totally disconnected
if its connected components are singletons, and totally separated if for any two
distinct points there exists a clopen set containing one of these points but not the
other. Similarly, a partially ordered topological space is totally order-separated
or -disconnected if for x 6≤ y there is a clopen upper set containing x but not y.
The compact totally order-disconnected spaces (Priestley spaces) are the duals
of bounded distributive lattices in Priestley’s duality (see, for example [16]).
The following implications are obvious:
zero-dimensional +To ⇒ totally separated ⇒ totally disconnected +T2
and it is well known that for compact spaces, these three properties coincide
(see [14, II–4.2], though there exist examples of compact totally disconnected
spaces which fail to be Hausdorff (see [20, Ex. 99]). Moreover, for compact
topological meet-semilattices, the above three properties are equivalent to total
order-disconnectedness. Indeed, if x 6≤ z in a compact totally separated topological meet-semilattice then (z] is a closed, hence compact set disjoint from x,
and we find a clopen set U containing x and disjoint from (z]. Thus ↑ U is a
clopen upper set containing x but not z.
Of course, every totally order-disconnectedness space is T1 -ordered, that is,
for x 6≤ z there exists an open upper set containing x but not z, and dually.
Apparently, the latter separation axiom simply means that the given topology
is finer than the interval topology. Note also that a semitopological lattice is
T1 -ordered iff it is T1 in the usual topological sense (singletons are closed).
Now to the lattice-theoretical notions needed in the sequel. A subset Y
of a lattice L is said to be join-dense or a join-base if each element of L is a
join of elements from Y , or equivalently, if for any two elements x 6≤ z, there
is some y ∈ Y with y ≤ x yet y 6≤ z; meet-dense subsets (meet-bases) are
defined dually. The similarity with the definition of T1 -ordered spaces and of
totally order-disconnected spaces is evident and not casual, as we shall see in
due course.
The following types of elements are of particular importance for the decomposition theory of lattices (cf. [1, 3, 4,W6, 7, 10, 12, 13]). An element y of a
lattice L is completely join-irreducible ( -irreducible) iff it belongs to each subset with join y; similarly, y is (join-)inaccessible if this condition holds at least
for each ideal with join y, and y is compact (also finite or join-intranscessible)
iff it belongs to each ideal having a joinWabove y, or equivalently, iff for each
subset X W
of L possessing a join with y ≤ X, there is some finiteVsubset F of X
with y ≤ F . The dual notions are completely meet-irreducible ( -irreducible),
meet-inaccessible, and cocompact (cofinite, meet-intranscessible).
Clearly, every
W
W
atom (that is, every minimal non-zero element) is -irreducible, and every irreducible element join-inaccessible. However, atoms need not be compact.
Let us remark in passing that not only topological compactness may be
regarded as a special instance of the corresponding lattice-theoretical notion
(viewing topologies as complete lattices), but also vice versa: by Alexander’s
subbase theorem, an element y of a complete lattice is compact iff the com6
plement of [y) is compact in the lower topology. This equivalence extends to
non-complete lattices if compactness is replaced with strong compactness: an
element y is strongly
compact if y ∈ ∆X implies y ∈ ∆F for some finite F ⊆ X,
T
where ∆X = {(z] : X ⊆ (z]} is the cut generated by X (cf. [11]).
We shall also need two other strengthenings of lattice-theoretical compactness. Call an element y of L supercompact if the complement of [y) is a principal
ideal, and call y hypercompact if the complement of [y) is a finite union of principal ideals. Supercocompact and hypercocompact elements are defined dually.
Then we have the following implications:
supercompact ⇒ hypercompact ⇒ strongly compact ⇒ compact ⇒ inaccessible
In meet-continuous lattices, the inaccessible elements are exactly the compact elements. Furthermore, in arbitrary lattices, every supercompact element
is completely join-irreducible, and the converse holds in frames, that is, in meetcontinuous distributive complete lattices. In fact, an element y ofWa complete
lattice is supercompact iff it is completely join-prime (that is, y ≤ X implies
y ∈ ↓ X for all X ⊆ L). A less trivial observation is that in join-continuous
distributive complete lattices, the compact elements coincide with the hypercompact ones (see [10, Lemma 1.4]).
A lattice L is said to be (super-, hyper-) compactly generated if the set
of (super-, hyper-) compact elements is join-dense in L. If, in addition, L is
complete then L is a (super-, hyper-) algebraic lattice. Similarly, a lattice L is
atomistic if the set of atoms is join-dense in L. The atomistic supercompactly
generated lattices are just the atomic Boolean lattices. For a thorough study of
supercompactly generated (= principally separated) lattices and posets, see [6].
If a lattice and its dual are compactly generated then we speak of a bi-compactly
generated lattice, etc.
Diagram 2.1 Compact generation in lattices
superalgebraic
atomistic
hyperalgebraic
atomistic
algebraic
atomistic
compactly
atomistic
hypercomp.
atomistic
@
Boolean
atomistic
@
algebraic
@
superalgebraic
@
hyperalgebraic
@
@
supercomp.
generated
hypercomp.
generated
compactly
generated
Notice that a lattice L is supercompactly generated iff so is its dual Lop , and
that L is algebraic iff it is algebraic and completely distributive. For this and
other characterizations of superalgebraic lattices, see [6, Lemma 4.2].
7
The following topological descriptions of inaccessible and (hyper-) compact
elements are immediate consequences of the involved definitions (for the last
statement, apply 1.6).
Lemma 2.2 Let y be an element of a lattice L.
(1)
(2)
(3)
(4)
y is inaccessible iff (y[= (y] r{y} is closed in τS (resp. in τo ).
y is compact iff [y) is open in τS (resp. in τo ).
y is hypercompact iff [y) is open in τu .
If L is complete, y is hypercompact iff [y) is open in τi .
The element (−1, 0) of the (non-complete) lattice in Example 1.7 generates a
τi -open principal dual ideal. This element is compact, but it is not hypercompact
(and not even strongly compact).
Lemma 2.2 suggest the following definitions. Given a topology τ on a lattice
L, we say an element y is τ -inaccessible if the deleted principal ideal (y[= (y]r{y}
is τ -closed, and y is τ -intranscessible if the principal dual ideal [y) is τ -open.
(In order to prevent confusion with the topological notion of compactness, we
avoid here the word “τ -compact” for “τ -intranscessible”). In accordance with
[12, II–1.16], we call a topology τ between τu and τS order-consistent. Thus,
for any element of a lattice L and for any order-consistent topology τ on L, we
have the following implications and equivalences:
τu -intranscessible ⇔ hypercompact
⇓
τ -intranscessible
⇓
τS -intranscessible ⇔compact
and analogous implications hold for the diverse types of inaccessible elements.
Notice that the set of all τ -intranscessible elements is closed under finite joins; in
particular, this applies to compact and to hypercompact elements. The observation that compact elements are join-inaccessible, and that in meet-continuous
lattices the converse is also true, admits the following generalization:
Lemma 2.3 Let τ be any topology on a lattice L.
(1) If τ is finer than τu then every τ -intranscessible element is τ -inaccessible.
(2) If y is a τ -inaccessible element of L and the unary meet operation ∧y is
τ -continuous then y is τ -intranscessible.
Proof. (1) If (y] is τ -closed and [y) is τ -open then (y[ = (y] r[y) is τ -closed.
(2) If (y[ is τ -closed and ∧y is τ -continuous then [y) = ∧−1
y [Lr(y[ ] is τ -open.
We call a lattice L τ -generated if for any two elements x, z ∈ L with x 6≤ z,
there is a τ -open principal dual ideal containing x but not z; in other words,
if the τ -intranscessible elements form a join-base. In particular, τS -generated
means compactly generated and τu -generated means hypercompactly generated.
By definition, every τ -generated lattice with τ` ⊆ τ is totally order-disconnected.
8
With the help of 2.3 one easily detects many types of τ -generated lattices.
For example, since any atom is τ -inaccessible if {0} is closed, it follows from
2.3 (2) that every atomistic lattice with a T1 –topology τ making the unary meet
operations continuous is τ -generated.WMore generally, for any topology τ finer
than the upper topology τu , every -irreducible element y is τ -inaccessible
(because (y[ is a principle ideal). Thus we have:
W
Proposition 2.4 Every -irreducibly generated semitopological lattice with a
T1 topology τ is τ -generated.
W
It is easy
W to see that a lattice is -irreducibly generated (i.e. each element is
a join of -irreducibles) iff it W
has a minimal (equivalently a smallest) join-dense
subset, namely the set of all -irreducible elements (cf. [6, 3.7 and 3.8]). Every
atomistic lattice, but also every dually algebraic lattice
W and, in particular, every
lattice satisfying the descending chain condition is -irreducibly generated (see
[1, VII–Thm. 16] or [3, 6.1]). In the case of order-consistent topologies,there are
various interesting characterizations of τ -generated lattices.
Theorem 2.5 The following statements on a lattice L and an order-consistent
topology τ on L are equivalent:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
L is τ -generated.
L is compactly generated and τ is the Scott topology.
τ has an interval base.
τ has a smallest base.
τ has a minimal base.
The lattice of open sets is supercompactly generated.
The lattice of closed sets is (super-)compactly generated.
(L, τ ) is a topological lattice whose τ -inaccessible elements form a join-base.
Proof. (a) ⇒ (b): Suppose x ∈ U ∈ τS . Since x is the directed join of all
τ -intranscessible elements y ≤ x, one such element y must belong to U . Hence
[y) is a τ -neighborhood of x contained in U , and U is τ -open. Thus, by orderconsistency, τ coincides with τS . In particular, the compact elements are τ intranscessible and form a join-dense subset of L.
(b) ⇒ (c): The principal dual ideals generated by compact elements form a base
for τ = τS .
(c) ⇔ (d) ⇔ (e) ⇔ (f) ⇔ (g): Since τ is order-consistent, each τ -open set is an
upper set, and consequently, any interval base for τ must consist of principal
dual ideals; but these are just the cores with respect to τ , where the core of
a point x in a topological space is the intersection of all neighborhoods of x.
Hence condition (c) states that (L, τ ) is a B-space in the sense of [7], that is, a
space with a base of open cores. It has been shown in [5, Thm. 2.11] and in [9,
Prop. 2.2 B] that each of the conditions (d)–(g) characterizes B-spaces.
(c) ⇒ (h): For x ∈ L, the principal dual ideal [x) is the core of x, hence the
intersection of τ -open cores, i.e. principal dual ideals. In other words, x is
a directed join of τ -intranscessible elements, and these are τ -inaccessible, by
2.3 (1). The binary meet is trivially τ -continuous, as the inverse image of a
9
principal dual ideal [x) und ∧ is [x)×[x). Concerning continuity of the binary
join, assume x1 ∨ x2 ∈ U ∈ τ . Then U is τS -open, so we find τ -intranscessible
elements y1 , y2 such that x1 ∈ [y1 ) ∈ τ, x2 ∈ [y2 ) ∈ τ and [y1 )∨[y2 ) = [y1 ∨y2 ) ⊆ U .
(h) ⇒ (a): By 2.3 (2), each τ -inaccessible element is τ -intranscessible.
As an application of this theorem, one obtains, for example, several common
topological characterizations of compactly generated lattices, by taking for τ the
Scott topology τS ; similar results are obtained for hypercompactly generated
lattices, by taking for τ the upper topology τu . Specifically:
Corollary 2.6 A lattice is compactly generated iff its Scott topology has an
interval base, and a lattice is hypercompactly generated iff its upper topology has
an interval base.
Combining 2.5 with 1.4 and [12-III, 2.16], we arrive at the following characterizations of τ -generated complete lattices in terms of the topology τ ∨ = τ ∨ τ` :
Theorem 2.7 For a lattice L and an order-consistent topology τ on L, the
following statements are equivalent:
(a)
(b)
(c)
(d)
L is complete and τ -generated.
L is compactly generated, and τ ∨ is a compact T2 topology.
L is algebraic, and τ ∨ is the Lawson topology.
(L, τ ∨ ) is a compact zero-dimensional topological meet-semilattice.
Obviously, every τ -generated lattice is a totally order-disconnected topological meet-semilattice with respect to the topology τ ∨ , but the converse is not
always true. For example, the lattice in Example 1.7 is discrete, hence certainly
totally order-disconnected in the interval topology τu∨ , but it is not τu -generated,
since there is no τu -open principal dual ideal containing the point (−1, 0), except the whole lattice. This underscores the importance of the completeness
hypothesis in the previous considerations.
Concerning the last condition in 2.5 and 2.7, respectively, it should be mentioned that an algebraic lattice need not be a topological lattice with respect to
the Lawson topology. Indeed, it follows from [12, VII–2.4], that a distributive
algebraic lattice with that property must already be superalgebraic, a rather
restrictive condition. An example of a distributive and but not superalgebraic
lattice is sketched below. Its meet-operation is not continuous at the bottom.
s
s
s
ppp
s
sp p
s&
sp
p
@
@
s
@
@
@
10
3
Two-sided topologies with interval bases
So far, we mainly have studied certain “one-sided” topologies like the Scott
topology and the upper topology; these have the advantage that the order relation is “encoded” in the topological strucutre and can be reconstructed from the
latter (via the specialization order). On the other hand, such topologies have
the drawback to violate even weak separation axioms like T1 in all non-trivial
cases. Now, we turn our attention to “two-sided” topologies such as the interval
topology and the order topology; these have better separation properties, but
they “forget” the order structure.
In this context we frequently have occasion to consider certain filterbases
consisting of intervals. More specifically, given two subsets Y and Z of a lattice
L, let βY,Z denote the filterbase of all finite intersections formed by members of
the system {[y) : y ∈ Y } ∪ {(z] : z ∈ Z}. We define a topology τY,Z on L by
declaring this system as an open subbase for τY,Z . In other words, the interval
filterbase βY,Z is a base for the topology τY,Z . However, not every topology
with an interval base has a strong interval base, that is, a base of the form βY,Z .
Example 3.1 Adjoining universal bounds to the disjoint union of the chain ω
and a single extra element x, a complete lattice is obtained whose order topology
coincides with the interval topology and has an interval base (in fact, all points
except the top element are isolated). However, the only principal dual ideal
which is a neighborhood of x is the whole lattice, so that the order topology does
not possess any subbase consisting of principal ideals and principal dual ideals.
Of course, such a failure cannot happen in semitopological lattices, because
continuity of the unary join and meet operations ensures that for any topologically open interval [y, z], the principal ideal (z] and the principal dual ideal [y)
are again open sets whose intersection is [y, z]. It is easy to see that a lattice
with a topology of the form τY,Z is (semi-)topological iff the following condition
and its dual hold:
For all x, y ∈ L and z ∈ Z V
with x ∧ yV≤ z, there
V existVfinite sets E, F ⊆ Z such
that x ≤ E, y ≤ F , and E ∧ F ≤ z.
Hence we may note:
Lemma 3.2 Every semitopological lattice with an interval base is already a
topological lattice and has a strong interval base.
Topologies with strong interval bases play an important role in the theory of
compactly generated lattices and of orthomodular lattices (cf. [2], [11],[17]-[19]),
being a helpful tool for the translation of algebraic properties into a topological
language, and vice versa. Let us start with some necessary and sufficient conditions for topologies with strong interval bases to be totally order-disconnected.
Proposition 3.3 The following statement on two subsets Y and Z of a lattice
L and the topology τ = τY,Z are equivalent:
11
(a)
(b)
(c)
(d)
Y is join-dense and Z is meet-dense in L.
τ is finer than τo .
τ is finer than τi (that is, (L, τ ) is T1 -ordered).
(L, τ ) is totally order-disconnected.
Each of these conditions implies that (L, τ ) is a zero-dimensional Hausdorff
space.
Proof. (a) ⇒ (b): For x ∈ U ∈ τo , join-density of Y and meet-density of Z yield
the equation
_
^
x = (Y ∩ (x] = (Z ∩ [x)).
Put
W
Yx = { V E : E
Zx = { F : F
is a finite subset of
is a finite subset of
Y ∩ (x]},
Z ∩ [x)}.
W
V
(If W∅ or V
∅ is not defined, omit it). Then Yx is directed, Zx is filtered,
and Yx = Zx = x ∈ U , so we find elements y ∈ Yx and z ∈ Zx with
[y, z] ⊆ U . But [y, z] is an intersection of finitely many subbasic τ -open sets,
and consequently, U is τ -open.
(b) ⇒ (c): The interval topology is always coarser than the order topology.
(c) ⇒ (d): The inequality x 6≤ z means
x ∈ L r (z] ∈ τi ⊆ τ , so there are finite
W W
sets E ⊆ Y and F ⊆ Z with x ∈ [ E, F ] ⊆ L r (z]. Hence there is at least
one y ∈ E with y ≤ x and y 6≤ z; in other words, U = [y) is a τ -clopen upper
set containing x yet not z. Since the sets [y) with y ∈ Y and (z] with z ∈ Z
form a clopen subbase, τ is zero-dimensional.
(d) ⇒ (a): Suppose x 6≤ z, and choose a τ -clopen upper set U with x ∈ U
and
W zW6∈ U . As before, we find finite sets E ⊆ Y and F ⊆ Z such that x ∈
[W E, F ] ⊆ U . Since z does not belong to the upper set U , the inequality
E ≤ z is impossible. Thus we find some y ∈ E ⊆ Y such that y ≤ x but
y 6≤ z, proving join-density of Y ; dually, we get meet-density of Z.
In the subsequent theorem, we list several necessary and sufficient criteria
for the coincidence of the topology τY,Z with the order topology τo , generalizing
diverse results of Riečanov [17]-[19] from sets of atoms and coatoms in orthomodular lattices to arbitrary subsets of lattices without any restrictions (see
also [11]).
Theorem 3.4 The following statements on two subsets Y and Z of a lattice L
and the topology τ = τY,Z are equivalent:
(a) Y is a join-dense set of compact elements and Z is a meet-dense set of
cocompact elements.
(b) τ is the order topology.
(c) τ -convergence agrees with order convergence.
(d) (L, τ ) is a (semi-)topological lattice, and τ is an o-coarse T1 –topology.
Each of these conditions implies that L is a bi-compactly generated, hence totally
order-disconnected order-topological lattice.
Proof. By 3.3, join-density of Y together with meet-density of Z is equivalent
to the inclusion τo ⊆ τ and also to total order-disconnectedness of τ . On the
12
other hand, we know from 2.1 (2) that an element y is compact iff [y) is τo -open,
and dually an element z is cocompact iff (z] is τo -open. In other words, each
element of Y is compact and each element of Z is cocompact iff τ ⊆ τo . This
proves the equivalence of (a) and (b). Under hypothesis (a), L and its dual
are compactly generated, so that by [4, Thm. 3], L is order-topological. This
together with 3.3 establishes the implication (a) ⇒ (d) and the final statement
in 3.4. Again by 3.3, (d) implies (b), because any T1 –topology making the unary
lattice operations continuous is finer than τi . Finally, since τ has an interval
base, it follows from 1.1 that o-convergence is topological on L if τo = τ , whence
(b) is equivalent to (c).
Notice that cndition (a) in 3.4 may be replaced with
(a’) L is o-continuous, Y is a join-dense set of join-inaccessible elements, and
Z is a meet-dense set of meet-inaccessible elements.
The prvious results immediately provide a common improvement of Theorem
3 in [4] and of Theorem 3.1 in [19].
Corollary 3.5 The following statements on a lattice L are equivalent:
(a)
(b)
(c)
(d)
L is bi-compactly generated.
L is o-continuous, and the order topology has an interval base.
The order topology has a strong interval base.
Order convergence agrees with τY,Z -convergence, where Y is a set of compact elements and Z is a set of cocompact elements.
(e) L is a bi-inaccessibly generated totally order-disconnected order-topological
lattice.
The example of the rationals with the usual topology shows that a totally
order-disconnected order-topological lattice need not possess any join- or meetinaccessible element. The crucial point is here the lack of enough joins and meets
ensuring compactness of the space (compare Theorem 3.13). On the other hand,
Example 3.1 shows that even a complete lattice whose order topology has an
interval base need not be meet-continuous, hence neither o-continuous nor compactly generated.
In the presence of enough irreducible elements, we can prove an interesting uniqueness statement for lattice topologies making the unary operations
continuous:
W
Proposition 3.6 Let (L, τV) be a T1 -semitopological lattice, Y a set of -irreducible
elements, and Z a set of -irreducible elements of L. Then the topology τ is
finer than τY,Z . Hence, if Y is join-dense and Z is meet-dense in L then (L, τ )
is totally order-disconnected, and τ is finer than the order topology. If, in addition, τ is o-coarse then
τ = τo = τY,Z .
Proof. The same argument as for 2.3 shows that for each y ∈ Y , the principal
dual ideal [y) is τ -open, and dually, each principal ideal (z] with z ∈ Z is τ -open,
13
whence τY,Z is contained in τ . The other assertions are immediate consequences
of 3.3.
W
We call a lattice bi-irreducible
generated if each element is a join of V
irreducibles and a meet of -irreducibles. Thus, for example, every bi-atomistic
lattice is bi-irreducibly generated. The second part of 3.6 tells us that on a
bi-irreducibly generated lattice, any T1 –topology τ making the unary lattice
operations continuous is totally order-disconnected and finer than the order
topology.
Our next aim is to establish a topological description of bi-algebraic lattices,
which may be regarded as a two-sided analogue of 2.7. For this and other purposes, it will be convenient to call a collection of subsets of a topological space
(X, τ )τ -cocompact if each member U of this collection is the complement of a
τ -compact set. Though being rather suggestive, this nomenclature conflicts a
bit with the usual convention to call the topology τ compact if so is the space
(X, τ ): in fact, the latter means that the topology τ is a τ -cocompact collection
of sets.
A beautiful theorem due to Lawson [15] states that the topology of any
compact T2 -topological lattice is uniquely determined, having a base of convex
τo -open sets. In case of topologies with strong interval bases, it turns out that τi cocompactness is a suitable relaxation of the compactness hypothesis in order to
prove similar uniqueness statements for certain non-compact situations. Crucial
for this purpose is the following
Lemma 3.7 Let Y and Z be arbitrary subsets of a lattice L. Then the base
βY,Z ∪ {∅} is τi -cocompact iff L is complete and the topology τY,Z is o-coarse.
Proof. By 1.3 L is complete iff {∅} is τi -cocompact, and by 3.7, τY,Z is o-coarse
whenever βY,Z is τi -cocompact. Conversely, assume that L is complete and τY,Z
is o-coarse. It remains to verify that
T for each U = [y, z] ∈ βY,Z and for any
interval filterbase T
β = βW,X with W
β⊆U
V, there is some B ∈ β with B ⊆ U .
By completeness, β is the interval [ W, X]. The inclusion βY,Z W
⊆ τY,Z ⊆ τo
ensures
that
y
is
compact
and
z
is
cocompact
(2.1).
Hence
y
≤
W entails
W
V
V
y ≤ E for some finite E ⊆ W
,
and
dually,
X
≤
z
entails
F
≤
z
for some
W V
finite F ⊆ X. But then B = [ E, F ] is a member of β with B ⊆ U .
Now to the announced topological characterization of bi-algebraic lattices
(cf. 2.5 and 2.7):
Theorem 3.8 The following statements on a lattice L and a topology τ on L
are equivalent:
(a) L and its dual are τ -generated complete lattices, and τ is o-coarse.
(b) L is bi-algebraic, and τ is the order topology.
(c) (L, τ ) is a bi-irreducibly generated complete T1 -topological lattice, and τ is
o-coarse.
(d) (L, τ ) is a totally order-disconnected space with a τi -cocompact strong interval base.
14
Proof. For the implication (a)⇒ (b), proceed as in proof of 2.5.
(b) ⇒ (c): By the Birkhoff-Frink Theorem [1, VIII–Thm. 16] and its dual, L
is bi-irreducibly generated, and by 3.5, (L, τo ) is a totally order-disconnected
topological lattice.
(c) ⇒ (d): See 3.6 and 3.8.
(d) ⇒ (a): By 3.7, the topology τ is o-coarse, and by hypothesis (d), it is of
the form τY,Z . Hence, by 3.3, total order-disconnectedness ensures that τ is the
order topology, and by 3.4, Y is a join-dense set of τ -intranscessible elements,
and Z has the dual property. Since τY,Z is totally order-disconnected, the
empty set is a member of the base βY,Z unless L is a singleton. Thus 3.8 yields
completeness of L.
As in similar constellations, completeness of the underlying lattice is essential
in 3.9.
Example 3.9 Let Z denote the chain of integers. Then Z × Z with the product
order is a lattice
in
W
V which every element is compact and cocompact, while no
element is - or -irreducible. Here the order topology is discrete and has a
strong interval base, but no interval has a τi -compact complement.
Nevertheless, the implication (a) ⇒ (b) in 3.9 can be extended to noncomplete lattices, replacing (co)compact elements with strongly (co)compact
elements and using the MacNeille completion (which is bi-algebraic iff the original lattice is strongly bicompactly generated; cf. [11]). On the other hand,
similar arguments as before show that a bi-irreducibly generated T1 -topological
lattice
generated (because
W with an o-coarse topology is always bi-compactly
V
its -irreducible elements are compact and its -irreducible elements are cocompact) and carries the order topology. But Example 1.7 shows that such a
topological lattice is not necessarily strongly bi-compactly generated, even if its
topology is discrete.
For an algebraic description of compact totally disconnected topological lattices possessing interval bases, the following observation will be helpful (cf. [12,
O-4.4]:
Lemma 3.10 If (L, τ ) is a compact T1 -semitopological lattice then τi ⊆ τ ⊆ τo ,
and L is an o-continuous complete lattice.
Proof. For the first claim, apply 3.7. For the second, observe that 1.3, L is
complete; now, given x ∈ L, any
W directed set Y ⊆ L is contained in the inverse
image I of the principal ideal ( (x ∧ Y )] under the unary operation ∧x . By the
inclusion
W τi ⊆ τ ⊆ τo and continuity
W
W of ∧x , the set I is τo -closed, and it follows
that Y ∈ I, hence x ∧ Y = (x ∧ Y ). This proves meet-continuity, and
join-continuity is obtained dually.
Proposition 3.11 The following statement on two subsets Y, Z of a lattice L
and the topology τ = τY,Z are equivalent:
(a) Y is a join-dense set of hypercompact elements, Z is a meet-dense set of
hypercompact elements, and L is complete.
15
(b) Y is a join-dense set of compact elements, Z is a meet-dense set of cocompact elements, and (L, τi ) is a compact T2 space.
(c) (L, τ ) is a compact space with τ = τi = τo .
(d) (L, τ ) is a compact T1 -ordered space.
(e) (L, τ ) is a compact totally (order-)disconnected topological lattice.
Proof. (a) ⇒ (b): We show that (L, τi ) is totally order-disconnected, in particular T2 : indeed, for x 6≤ z, we may choose a hypercompact element y ∈ Y such
that y ≤ x and y 6≤ z. Hence, by 2.1(4), [y) is a τi -clopen upper set containing
x but not z. By 1.3, (L, τi ) is compact.
(b) ⇒ (c) ⇒ (e): Apply 1.2 and 3.4.
(e) ⇒ (d): See 3.11.
(d) ⇒ (a): By 1.3, L is complete, and by 3.3, Y is join-dense and Z is meet-dense
in L. Hence, for each y ∈ Y
[
L r [y) = {(z] : z ∈ Z, y 6≤ z}
is a τ -open covering of the τ -closed (hence
compact) set L r [y). Thus we find a
S
finite set F ⊆ Z such that L r [y) = {(z] : z ∈ F }. This shows that Y consists
of hypercompact elements, and a dual argument applies to Z.
Parts of the subsequent characterization theorem for compact totally orderdisconnected topological lattices belong to the folklore of algebraic and topological lattice theorey (cf. [12], [14] and [16]), but we have not been able to pursue
all traces to thier origins.
Theorem 3.12 The following statements on a lattice L are equivalent:
(a) L is bi-hyperalgebraic.
(b) L is bi-algebraic, and the interval topology is T2 .
(c) (L, τ ) is a compact topological lattice for some T2 topology τ with an interval base.
(d) (L, τ ) is a compact T1 -ordered space for some topology τ with a strong
interval base.
(e) (L, τ ) is a compact totally order-disconnected topological lattice for some
topology τ .
If such a topology τ exists then it agrees with τi , τo , and τY,Z , where Y is the
set of all compact elements and Z is the set of all cocompact elements.
Proof. For the implication (a) ⇒ (b) ⇒ (c) ⇒ (d) ⇒ (a) ⇒ (e), use 3.12 and
3.2.
(e) ⇒ (b): By the Fundamental Theorem for Compact Totally Disconnected
Semilattices (see [12, VI–3.13]), L is bi-algebraic, and by [12, VII–2.8], the
interval topology is T2 . Furthermore, by 3.11, we have τi ⊆ τ ⊆ τ0 . Hence, by
1.2 and 3.5, it follows that τ = τi = τ0 = τY,Z .
As remarked earlier, in condition (e) “totally order-disconnected ” may be
substituted by “totally disconnected and T2 ”, “totally separated ”, or “zero16
dimensional and To ”. Moreover “topological ” may be replaced with “semitopological ”. Notice also that a lattice L is bihyperalgebraic iff it is linked bialgebraic (see [16]), i.e. L and Lop are algebraic lattices whose Lawson topologies
coincide (cf. the notion of bi-continuous lattices [12, VII–2.5]). The algebraic
counterparts of compact totally order-disconnected topological lattices under
the Priestley duality are the so-called catalytic distributive lattices (see [16]).
Two additional remarks are opportune. First, look at Example 1.7 once
more in order to see that a non-complete lattice whose interval topology has
a strong interval base (and is, moreover, discrete) need not be hypercompactly
generated. Second, Example 3.1 shows that even a complete lattice whose interval topology has an interval base need not be meet-continuous, all the less
(hyper-)compactly generated.
In the presence of enough atoms and coatoms, the study of semitopological
lattices becomes particularly nice. Let τA denote the topology τY,Z where Y is
the set of all atoms and Z is the set of all coatoms.
Corollary 3.13 If L is a bi-atomistic compact semitopological lattice with closed
universal bounds for some topology τ , then it is a totally order-disconnected
order-topological complete lattice with τ = τi = τo = τA . Moreover, the atoms
are hypercompact, the coatoms are hypercocompact, and consequently, L is bihyperalgebraic.
Proof. By 3.3 L is bi-atomistic iff τo ⊆ τA iff τA is totally order-disconnected.
By 2.3 (2), each atom a generates a τ -open the principal dual ideal [a), because
(a[= {0} is τ -closed. This and the dual observation yield the inclusion τA ⊆ τ ,
and by 3.7, we have τ ⊆ τo . Thus τ = τo = τA is a compact topology finer than
τi , and 3.13 applies.
This result has interesting consequences for the theory of geometric lattices
(in the sense of Grätzer [13, IV–3]; these are algebraic, atomistic and semimodular lattices (while in [1] and [3], the name “geometric” is reserved for
finite-dimensional lattices).
Corollary 3.14 Let L be a modular, complemented and atomic lattice, endowed
with a compact T1 topology τ making the unary lattice operations continuous.
Then:
(1) L is a geometric lattice in which every atom and every coatom has only a
finite number of complements.
(2) (L, τ ) is a compact totally order-disconnected topological lattice.
(3) τ agrees with the order topology, the interval topology, and with τA .
Proof. L is relatively complemented (being modular and complemented) and
atomic, hence atomistic. Using modularity and relative complements once more,
one verifies easily that L is also coatomistic, and then one may refer to 3.14. Geometric lattices arising from “usual” geometries rarely admit a compact
T1 –topology making the (unary) lattice operations continuous: indeed, in geometric lattices like the subspace lattice of a finite-dimensional real vector space,
17
every atom and every coatom has an infinite number of complements, and by
3.15, this excludes the existence of a compact topology with the desired properties.
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