QUASIORDERS , PRINCIPAL TOPOLOGIES , AND PARTIALLY ORDERED PARTITIONS

The quasiorders on a setX are equivalent to the topologies onX which are closed under arbitrary intersections. We consider the quaisorders onX to be partial orders on the blocks of a partition of X and use this approach to survey some fundamental results on the lattice of quasiorders on X.


INTRODUCTION
The study of topology on finite sets can be framed entirely in terms of equivalence relations, partitions, and partial orders.A natural construction is to partially order the members (blocks) of a partition of an arbitrary (finite or infinite) set X.It is easy to see that such a partially ordered partition of X is equivalent to a reflexive, transitive relation on X, that is, to a quasiorder on X.In 1937, Alexandroff showed that quasiorders on X are equivalent to the topologies on X in which arbitrary intersections of open sets are open.Such topologies are called principal topologies.
For Hausdorff topological spaces, the associated quasiorder relation is equality.These connections between topology and discrete mathematics have not been as widely recognized as they deserve to be, largely due to the historical connections between topology and analysis, with the latter branch being primarily concerned with Hausdorff topological spaces.Non-Hausdortt topological spaces are finding numerous applications today in areas of computer science.(See the references in Kong et al. [14] and Kopperman  [15].)Considering them as partially ordered partitions, we will study the principal topologies (quasiorders) on a set X, with particular attention to the lattice structure of the collection of all principal topologies on X.This collection is the collection of all topologies on X if X is finite.
A quasorder (or preorder) on a set X is a reflexive transitive relation on X.A partial order on X is an antisymmetric quasiorder on X.A total order on X is a partial order _< on X in which every pair of elements x, y E X are comparable, that is, either x <_ y or y <_ x.The discrete order on a set X is Ax {(x,x) x E X}. (Thus, the discrete order relation on X is equality.)A symmetric quasiorder on X is an equivalence relatzon on X.If (X, <) is an (partially or quasi-) ordered set, the decreasing hull of a subset B C_ X is defined to be dx(B) {x X x < b for some b B}.We say B is a decreasing set in X if B dx(B).Increasing hulls and increasing sets are defined dually.A lattzce is a partially ordered set in which every pair of elements has a supremum and an infimum.A poset in which every nonempty subset has a supremum and an infimum is a complete lattice.If a lattice has a least element, it will be called the zero of the lattice, and denoted 0. Similarly, if a lattice has a greatest element, it will be denoted 1.A parttzon of a set X is a nonempty collection of mutually disjoint, nonempty subsets of X whose union is X.The members of a partition P are called blocks of P.There is a natural correspondence between an equivalence relation on X and a partition of X: The equivalence classes of an equivalence relation R form the blocks of a partition P, and a partition 7 9 determines an equivalence relation R given by xRy if and only if x and y are in the same block of 79.The equivalence class of a X will be denoted by [hi.The complement of a set A will be denoted Ac.
If 79 is a partition of a set X and is a partial order on 79, we call (79, ) a partzally ordered partition, or popartztzon of X.For example, let P be the partition {[n, n + 1) n is an integer} of the real line R, and order 79 by agreeing that In, n + 1) _ _ _ [m, m + 1) if and only if n g m in the usual order on R. Then (79, __) is a partially ordered partition of R. (In fact, since is a total order on 9, (79, __) is a totally ordered partitwn of R.) Suppose <_ is a quasiorder on X.There may exist distinct elements a, b X with a <_ b and b _< a. Define a relation R(_<) on X by aR(<_)b if and only if a _< b and b _< a.It is easy to check that R(_<) is an equivalence relation on X.Let 79 be the corresponding partition.For blocks A, B 79, put A _ B if and only if a _< b for some a A and some b B. Equivalently, for blocks [a], [b] 79, put [a] - [b] if and only if there exist a' [hi, b' [b] with a' < b'.Now is a partial order on the blocks of 79.In this manner, any quasiorder _< on X generates a partially ordered partition (79, ) of X.For example, the quasiorder _E defined on R by x _E y if and only if Ix < [yl where Ix denotes the greatest integer less than or equal to x in the usual order _< on R, generates the totally ordered partition (79, _) described in the last paragraph.On the other hand, if (79, _) is a partially ordered partition of X, the relation _< defined on X by a < b if and only if [a] [b] is a quasiorder on X.Thus, from any partially ordered partition of X, we can obtain a quasiorder on X, and from any quasiorder on X we can obtain a partially ordered partition on X. Performing these two tricks successively would carry us from a popartition of X to a popartition of X, or from a quasiorder on X to a quasiorder on X.In either case, the original structure is identical to the final one.Thus, our notion of popartition is identical to the notion of quasiorder, with only a slight disguise.
In 1937, Alexandroff [2] noted another occurence of quasiorders in slight disguise.Quasiorders on a finite set are identical to the topologies on that set!A princzpal topology on a set X is a topology on X that is closed under arbitrary intersections of open sets.Observe that any finite topology, and in particular, any topology on a finite set, is a principal topology.Quasiorders on an arbitrary set are identical to the principal topologies on that set!If " is a topology on X, the specialization order <_ on X is defined by x _< y if and only if x cl({y}), that is, if and only if every open neighborhood of y contains x.It is easy to check that the specialization order _< is a quasiorder.If _< is a quasiorder on X, the set of all decreasing subsets of X is a principal topology on X called the speczahzatwn topology associated with _<.Since complements of decreasing sets in a quasiordered set (X, <_) are increasing; sets, the closure of a set B C_ X relative to the specialization topology is the smallest increasing subset of X that contains B. Thus, the increasing hull ix(B) of B in (X, _<) is the closure of B relative to the specialization topology.
A good account of principal topologies appears in Steiner [22].Principal topologies are called Alexandroff dzscrete topologies or A-topologies in Ern and Stege [9].Principal topologies are precisely the topologies in which every point has a smallest neighborhood.Karimpour [13] defines a complementary topology to be a topology in which the complement of every open set is open.
Example 5 in Steen and Seebach [21] considers partitzon topologies, i.e., topologies on a set X that have a base consisting of a partition of X. Complementary topologies and partition topologies are identical, and are examples of principal topologies.
A topology T on a set X is a To topology if and only if for every pair of distinct points x, y E X, there is a neighborhood of one of the points that excludes the other point.Equivalently, v is a To topology on X if and only if for every pair of distinct points x, y E X, one of the points is not in the closure of the other.If the topology r is the specialization topology associated with a quasiorder _<, the definition becomes r is To if and only if for every pair x, y X, either x i(y) or y i(x), i.e., either x y or y x.Thus, we have that the specialization topology associated with a quasiorder _< is To if and only if _< is antisymmetric, i.e., if and only if _< is a partial order.
An excellent compilation of topological properties and the corresponding properties of the asso- ciated specialization quasiorder is given in Ern and Stege [9].One can find there, for example, that a quasiorder on a finite set is an equivalence relation if and only if the specialization order is 2. ELEMENTARY LATTICE PROPERTIES Given a set X, there are natural partial order relations on the set Q(X) of all quasiorders on X, on the set PoPar(X) of all popartitions of X, and on the set PrTOP(X) of all principal topologies on X.With their natural order, each set is a complete lattice, and the one-to-one correspondences between these sets discussed above are lattice homomorphisms.Below we will define these natural order relations, and examine the lattice stucture primarily by considering the lattice PoPar(X).Lattices of principal topologies and quasiorder lattices have been studied by Ern [8], Steiner  [22], and Tma [27].Many of the works on principal topologies and quasiorders focus on applications to finite sets, in which case the lattice PrTOP(X) of all principal topologies on X is simply the lattice Top(X) of all topologies on X.
For topologies '1, T2 Top(X), we say 1 _<T r if and only if '1 C_ T as subsets of the power set of X. (We would say T1 is coarser than -., or '2 is finer than '.) Now _<To is a partial order on Top(X), and (Top(X), <_T,) is a complete lattice. (PrTop(X), <_T) is also a complete lattice, though generally the supremum and infimum of a set in PrTop(X) may not agree with the supremum and infimum of the same set taken in Top(X).For quasiorders , Q(X), we say _<Q if and only if 81 ::) , i.e., if and only if the identity map idx (X, ) (X, ) is increasing.(We use reverse inclusion here as the order on Q(X) so that PrTop(X) and Q(X) will be lattice isomorphic, as opposed to lattice anti-isomorphic.) To describe the partial order _< on PoPar(X), we first consider the poset Par(X) of all partitions of X.For P, Par(X), we say P _< if and only if [a]Q C_ [a] for every a E X, in which case we say is finer than 7p, refines 7p, or P is coarser than .With this order, Par(X) is a complete lattice (Theorem 1.5 of Ore [18]).Define a relation _< on PoPar(X) by (P, ,) _< (, Q) if and only if P <_ in Par(X) and the quotient function f,p (Q, ) (P, Z,) defined by f([a])= [a], is increasing._< is a partial order on PoPar(X).PROOF.Antisymmetry: If (P, ,) _< (, ) and (Q, Q) _< (P, ,), then as members of Par(X), 7 a , and since the quotient functions f,, and f,, are both increasing, the order on 7 must agree with the order on , so (:P, p) (Q, ) as members of PoPar(X).Transztzvity: If (79, ,) < (, _--<) and (, ) < (7, r), then P in Par(X), and the composition of two increing nctions, ], f,m o f, is increing.Reflevity: P P in Par(X) d the identity map is increing.PROPOSITION 2.2.If ((Pa, o))aeA is a collection of popartitions of X, then VoeA(Pa, a) in PoPar(X) exists; its underlying ptition is P VaeA P (supremum in Par(X)) and its order is ven by [a] [b] if d only if [a] [b] for every a A. PROOF.It is ey to veri that is a partial order.With P described, Pa P d f, is clely increing for y A, so (P, ) is an upper bod of ((P, a))aeA-Suppose (, ) is so an upper bound of ((Pa, ))aea.Then elements of Par(X), P .T o see (P, ) is the let upper bod of ((Po, ))aeA, it only remns to show that f, is increing.

Suppose [a]
[b].Since (, ) (P, a) for eve a A, we have [a] a [b] for every e A, and thus [a] [b].U A poset in wch eve nonempty subset h a suprem is called a V-complete semilattce.
Any v-complete semilattice P th a let element is a complete lattice, for it is eily shown that for any S P, the supremum of the set of lower bounds of S is itself a lower bound of S and is thus the imum of S. The dual result for A-complete semilattices holds well.By Prposition 2.2, PoPar(X) is a V-complete semilattice.It h a smallest element, namely the indiscrete ptition {X}.Ts proves the following results.
COROLLARY 2.3.PoPar(X) is a complete lattice.A description of the im of a set ((P, a))aeA in PoPar(X) will be useful.In Par(X), Aaea Pa can be described (see p. 579 Ore [18]) the ptition of X whose blocks e the mimal chin coected subfilies of aeA a where a faly is chain connected if for any A,B e , there est A, (i 1,...,n) with A A, 0, A, A,+ 0, d A, B 0.
Viewing the partitions (P)eA eqvalence relatio , AeA P can be described by the relation a b if and only if there ests a chain of equivMences a o a, a, .a ,+, a b for some o, A (i 0,1,..., n).We will say a, b X e loop connect in OA Pa if there exist a, A d a, X (i 0,1,...,n) th a0 a a, and b a, for some and [a,],., , [a,+],,, for 0, 1,..., n 1.
(2.1)That is, a, b 6 X e loop connoted in UA P if there is increing loop om [a],o to [a],.which pses through Ibid,, for some i.Note that if a is chMn connect to b, then the chin from a to b follow by the chin om b to a shows that a and b e loop connected simply by ting a, in (2.1) to be equMity at eh link."Is loop coted to" is an uilence relation P on X.This defines a ptiM order on P. Since chin connectedness implies loop coectedness, P is coser than Aaen P where the i is ten in Par(X).om the conruction of (P, ), P is the est paition below eh P, d P w given the nal order necessy to me it a lower bound of ((P, a))eA.Thus, in gogar(X), (P, ) AA((P, )).
Note that the pition derlying AA((Pa, a)) (i in PoPar(X)) is not generMly the se AeA Pa (ium in Par(X)).For exple, if (Pm, ) ({A,A},A A) and (P2, ) ({A, A}, A 2 A) e poptitio of X, the im of their underlying ptitions in Par(X) is {A,A}, while the infimum in PoPar(X) is ({X}, A).However, from the description of suprema and a in PoPar(X), we see that for scretely ordered poptitions, the ptitions underlying suprema and ima in PoPar(X) aee with the suprema d ima of the underlying partitions in Par(X).Thus we have the follong result.
It is shown in Pudlk and Tma [19] that "every finite lattice can be embedded in a finite partition lattice".With Proposition 2.4, this implies that every finite lattice can be embedded in PoPar(X) for some finite set X.
3. ATOMS AND COVERINGS IN PoPar(X) We say b covers a in a poset X if a < b, and a _< c < b implies c 6 {a, b}.If X is a lattice with smallest element 0, an atom is an element that covers 0. If X has a largest element 1, a coatom is an element covered by 1.The smallest and largest elements of PoPar(X) are the indiscrete partition {X} and the discretely ordered discrete partition ({ {z} "z 6 X}, A), respectively.The atoms of PoPar(X) are the two element chains in PoPar(X), that is, the partitions of form {A, Ac} with A -< A or A -< A. The coatoms are the discrete partitions with orders of the form Axu{(Xo, Xl)} where x0 and xl are distinct elements of X.Every nonzero element of PoPar(X) is above an atom: For (P,-<) 6 PoPar(X), let A 6 P and consider d,(A) {B 6 7 9. B -< A in (79, )}.Now if i,(A) is defined dually, d,(A) i,(A) 79 would imply A is maximum and minimum in (79,-<), so (P,-<) ({A}, =) would be the zero element of PoPar(X).If (79, _--<) # 0, assume, without loss of generality, that d,(A) # P so that d,(A) and \ d,(A) are disjoint nonempty sets, the former decreasing and the latter increasing, that partition "P, and therefore partition X. (79, _--<) is above the atom PA --= ({d;(A), 79 \ d,(A)}, d,(A) < 7 =, \ dp(A)).Indeed, every nonzero popartition is the supremum of the atoms below it: If (79, Z) 6 PoPar(X) and A 6 79 is not maximum, PA gives an atom below (:P,-<), and if A is not minimum, ({i;(A), P \ i;(A)}, i,(A) > 79 \ i,(A)) =_ 7 gives an atom below (79, _).We claim V({79A A 6 T', A not maximum in 79} V { not minimum in 79}) (79, Z).The atoms involved in the expression above only partition X into partitions of the blocks of 79, so the supremum of these atoms will give a partition coarser than "P.However, since pA V 7A (or simply the one that exists, if both do not exist) gives a partition of X with A as a block, we see that the partition underlying the supremum of atoms given above Ibis, : [c],, then for B [b] we have [b], B [cluB.Thus, the order on the supremum of atoms given above does agree with the order _ on 7 .
Before investigating coverings in PoPar(X), observe that P covers Q in Par(X) if and only if is obtained from P by combining two blocks of 79.For coverings in the collection Po(X) of all partial orders on X ordered by reverse inclusion (i.e., (X, _<1) _< (X, _<2) if and only if the identity function id" (X, <_1) (X, <_2) is increasing), we have the following result.
PROOF.Suppose 8 covers .T hen 8 C .Let (a, b) \ 8. Now a, b _<0 y} is a partial order with 8 C C .S ince 8 covers , we have , and thus, if there exists another pair (c, d) \ 8, (c, d) :/: (a, b), then c _<e a and b _<e d, and at least one of these inequalitites is strict.Now ' 8(3 { (x, y)" x <_ c, d _< y} is a partial order with 8 C ' C b, contrary to the fact that 8 covers .T hus, if 8 covers , then 8 C P and [ \ 81 1.The converse is trivial. [3  Since only the covering relations are depicted in the Hasse diagram for a poset, a character- ization of the covering relations in PoPar(X) would be a useful tool.Although the following result does not characterize the covering relations, it narrows down the search for coverings and is still very helpful in understanding and describing the lattice PoPar(X) PrTop(X) Q(X), particularly when X is a small finite set, as we will see in the next section.PROPOSITION 3.2.

FINITE POPARTITION LATTICES
With the aid of Proposition 3.2, we are able to explicitly describe the lattices PoPar(X) where X has 2 or 3 elements.Recall that when X is finite, the lattice Top(X) of topologies on X is iso- morphic to PoPar(X).We will denote an n-element set {1, 2,..., n} by n.Let Sn,k represent the number of partitions of n with k blocks.These numbers, the Sterling numbers of the second kind, are discussed in Stanley [20].Let Pk denote the number of partial orders on k.The numbers P for k 1,2,..., 14 are given in Ern and Stege [9].It is easy to see that IPoPar(n_)l '=1Sn,Pk.
The coverings in Top (4) PoPar( 4) are also easily obtained with the aid of Proposition 3.2.However, since PoPar(4) has 355 elements, its Hasse diagram becomes unwieldly.The set Po(4) of all partial orders on 4 elements appears at the upper end of PoPar(4_) as the popartitions of 4_ whose underlying paitions are discrete.The Hasse diagram for Po(4_) has 219 vertices and 588 edges.We list 16 isomorphic classes of Po(4) in Table 4.5, and indicate the covering relations among these classes in Figure 4.6.The weights on the edges of the graph in Figure 4.6 represent the number of elements from the larger isomorphism class that cover each element from the smaller isomorphism class.For example, the edge from III to VI has weight 3, indicating that each partial order on 4 of type VI is covered by 3 partial orders of type III.These weights, together with the number of partial orders of each type (from Table 4.5) can be used to verify that the Hasse diagram for Po(4) has 588 edges.Types of Nonisomorphic Partial Orders on 4 Table 4.5  Covering Relations in Po(4) The set ToPar(X) of totally ordered partitions of X is not a lattice if IX > 1, for if IXI > 1, the totally ordered partitions ({A, AC}, A -< Ac) emd ({A, AC},A -< A) have no supremum.Arbitrary infima of members of ToPar(X) exist in ToPar(X) and agree with the infi, ma in PoPar(X).To see this, it suffices to show that if (:Pa,-,) ToPar(X) for c A, then (7 , _-_-<) =-AaA(:Ps, ___a) is totally ordered.Given [a],, [b], P, [a] and [b] must be related in (Vso, "<so) for each a0 A since (7so,---<so) is totally ordered.Thus, there is a chain of length 1 as in (2.1) between [a] and [b], so [a] and [b] are related in (7v, ).Thus, ToPar(X) is a A-complete sublattice of PoPar(X).
The set M of totally ordered discrete partitions of X occupies a central position in PoPar(X).The elements of M are the minimal elements of Po(X) {(P,,) PoPar(X) 7 ) is the discrete partition of X}, and they are the maximal elements in ToPar(X).Indeed, in PoPar(X), d(M) N (M) M, d(M) ToPar(X) and i(M) .Po(X).This last fact says that for every partial order _< on X, there is a total order _<t on X such that id (X, <_) (X, <_t) is increasing; that is, every partial order on X can be extended to a total order (see, e.g., Davey and Priestley [6], p.26).We also note that d(M) N i(M) M is the definition of M being order convex.Equivalently, M is order convex if x _< y _< z and x, z M implies y M.
If X n, it is easy to see that there are n! elements of M, corresponding to the permutations of n.In PoPar(3) with the notation of Table   For any totally ordered partition ('P, _--<,) 6 ToPar(X), the set d(T') {(Q, _-<) 6 PoPar(X) (Q, _--<) _< (P, p)} is a A-complete sublattice of PoPar(X) with greatest element (7 , -<9), and is therefore a complete lattice.The members of d( 7) are precisely the convex partitions of the totally ordered set (:P,-<,).For example, if (P, _--<,) is a totally ordered partition of X order isomorphic to n with the natural order, then d(P) corresponds to the convex partitions of __n.There are 2 "-1 of these (there are twice as many convex partitions of k as of k-1--you can add a new block {k}, or include k in [k-1]).Thus, below each of the n! maximal elements in ToPar(n__) are 2 "-1 members of ToPar(n_).Of course, one member of ToPar(n_) may be below several maximal elements, so we may conclude IToPar(n__)l <_ n!(2n-1).Recalling that S,,k gives the number of partitions of n with k blocks and that there are k! total orders on k, it is easy to see that IToPar(n)l .'=l(S,,k)(k!).The values of IToPar(n)l for small values of n are given in Table 4.1.
We may also generate the numbers IToPar(n)l recursively.In an arbitrary totally ordered partition of n, suppose the top block contains k elements of n.There are () ways to choose these k elements.The remaining n-k elements can be formed into totally ordered partitions in IToPar(n k)l ways.Letting k range from 1 to n, we have IToPar(n_)l Here we take IToPar(O)l to be 1.
A chain topology on X is a topology on X whose open sets are totally ordered by inclusion.
Chain topologies are considered in Doyle [7] and Stephen [23].The proposition below may be used to translate the recursive formula for IToPar(n)l in the preceding paragraph into the recursive formula given by Stephen [23, Theorem 2] for counting the number of chain topologies on n.A popartition (:P,-<) of X is a totally ordered partition if and only if the specialization topology -, is a chain topology on X. PROOF.For x 6 X, let N(x) CI{N 6 T, x 6 N}.The result follows easily from the equivalence of these three statements: (1) N(x) C_ N(y), (2) y 6 cl{x}, (3) y <_ z. [3 Doyle [7] defines a tower space on n points to be any space homeomorphic to n with the chain topology {@, 1, 2, 3,..., _n_n}.More generally, we will say a chain topology, r on X is a tower topology on X if for every U 6 " there exists V 6and x 6 X \ V with V t3 {x} U. Doyle shows that a finite topological space X is a T0-space if and only if there is a continuous one-to-one function from X to a tower space.We will show that this is equivalent to the statement that a quasiorder on a finite set X is a partial order if and only if it can be extended to a total order on X.First, we will give a well known fundamental result on functions.PROPOSITION 5.2.If (X, _<x) and (Y, _<y) are quasiordered sets with associated specialization topologies TX and Tr, respectively, then f (X, _<x) (Y, _<r) is increasing if and only if f" (X, TX) (Y, rr) is continuous.PROOF.Suppose ]" (X, _<x) (Y, _<r) is increasing.If U is open in (Y, rr), then U is decreasing in (Y, _<r), and it follows that f-(U) is decreasing in (X, _<x), i.e., f-I(U) is rx- open.Conversely, suppose f (X, rx) (Y, zr) is continuous.Then, from the definition of the topologies involved, the inverse image of any <_y-decreasing set is <_x-decreasing.Suppose a <_x b yet f(a) :r f(b).Then f(a) dr(f(b)).But since f-*(dr(f(b))) is a _<x-decreasing set that contains b, we have a 6 f-*(dr(f(b))), contrary to f(a) 6_ dr(f(b)).
Recalling that the specialization order for a topology is a partial order if and only if the topology is To, we now observe that the following statements are equivalent: (a) a quasiorder on n is a partial order if and only if it can be extended to a total order on n.
(b) a quasiorder _< on n is a partial order if and only if there is a total order _<t on n such that id (n__, <_) (n, <t) is increasing.
(c) A topology r on n__ is To if and only if there is a total order <t on __n such that id (n_n_, "r) (_n, r_<,) is continuous.
Noting that the specialization topology _< from a total order on a finite set gives a tower space, and that the identity function is trivially one-to-one, we see that the following statement is implied by those above.
(d) A topology on _n is To if and only if there is a one-to-one continuous function from __n onto a tower space.
Since the function in (d) is a permutation on __n and since any two tower spaces on __n are home- onorphic, we may assume, without loss of generality, that the function in (d) is the identity.
Furthermore, since any tower space topology on __n is the specialization topology for some total order on _n_n, statement (d) implies (c).Thus, all of the statements (a)-(d) are equivalent.The lattice PoPar(X) is isomorphic to the lattice PrTop(X) of principal topologies on X.By Proposition 5.1, ToPar(X) is isomorphic to the principal chain topologies on X.For the set M of totally ordered discrete partitions of X, we have M i(M) fq d(M) .Po(X) fq ToPar(X).Recalling that the lattice Q(X) of quasiorders on X is isomorphic to PrTop(X) .. PoPar(X) and that the partial orders on X correspond to To principal topologies, we see that M Po(X)fToPar(X) {To principal topologies on X}fq{principal chain topologies on X} {tower topologies on X}.Thus, the set T of tower topologies on X occupies a central position in PrTop(X), with i(T) isomorphic to the V-complete semilattice of To principal topologies on X, and d(T) isomorphic to the A-complete semilattice of principal chain topologies on X. Again recall that if X is finite, all topologies on X are principal and we could omit the word "principal" in the discussion above. 6. OTHER LATTICE PROPERTIES In this section, we will assume some familiarity with some standard lattice theory concepts as given in Birkhoff [4].Background material on continuous lattices can be found in Gierz et al. [11].
PoPar(X) is complemented.Some complements of (79, ) E PoPar(X) are just the discretely ordered complements of 79 in Par(X).Thus, for (79, _) a__ PoPar(X), if we let A be a complete set of equivalence class representatives from 79, then the discretely ordered partition {A} tA { {x} x X \ A} is a complement of 79.If IX[ >_ 3, then there exist popartitions (79, ) that are neither the top nor bottom element of PoPar(X) and have one block with more than one element of X.
For such a popartition 79, the complete set of equivalence class representatives for 79 is not unique, that is, 79 is not uniquely complemented in Par(X).It follows that if IX[ _> 3, then the elements of PoPar(X) are not uniquely complemented.Since complements are unique in distributive lattices, we see that Par(X) and PoPar(X) are not distributive if IX[ >_ 3.While any interval in Par(X) is complemented (Ore [18, p.596]), this result does not hold in PoPar(X), for, e.g., in PoPar(3) (see Figure 4.4), 27 has no complement in the interval [29,13].
In a complete lattice L, we say x is way below y, denoted x << y if for any directed subset D C L, y _< sup D implies there exists d 6 D with x _< d.A complete lattice L is a continuous laice if for every x 6 L, x sup{y L" y << x}.In a finite lattice x << y if and only if x _< y (See Remark 1.5, p.41 in Gierz et al. [11]), and thus every finite lattice is a continuous lattice.In particular, if X is finite, PoPar(X) is a continuous lattice.
If X is infinite, then PoPar(X) is not a continuous lattice.To see this, we will show that if X is infinite, then P << in PoPar(X) if and only if 9 is the bottom element of PoPar(X), i.e., the indiscrete (one-block) partition of X. (For simplicity, we will supress the orders on all popartitions in this paragraph.)Suppose X is infinite.Embed a copy N of the natural numbers in X.For each z X and each n N, define Tz(n) (respectively, TZ(n)) to be the discrete partition of X with the order {z} <_ {m} (respectively, {z} >_ {m}) for all m N C_ X with m >_ n.Define D {T(n) n 6 N} and D {T)(n) n 6 N}.Dz and D are directed sets in PoPar(X) with 9(n) _< 9z(m) if and only if 9Z(n) _< TZ(m) if and only if n _< m.Thus, D and D are chains in PoPar(X) isomorphic to N. Now for any z 6 X, sup D, sup D is the top element in PoPar(X), namely, the discrete partition of X with the discrete order.Thus, Q _< sup Dz sup D for any Q 6 PoPar(X).For 9 << Q to hold, we must have 9 _< T)z(nz) and 9 _< :D(rn) for some mz,nz 6 N and for every z 6 X.For z 6 X, define T {j 6 N" j >_ rn,j >_ n}.Now 9 << Q implies 9 is a popartition with [t,], <_ [z], _< [t] for every z 6 X and every t 6 Tz.Thus, in 9, every point z of X is identified with a tail Tz of the sequence N. Since all such tails overlap, P is the indiscrete partition of X.It follows that PoPar(X) is not continuous, since any nondiscrete popartition is not the supremum of popartitions way below it.
Dee on P by [a] [b] if and oy if there ests a chin (2.1) th ao a d b.
is simply 79.Suppose [b], and [c] are blocks in P with [b] _< [c],.Each atom included in the supremum above partitions 7 into an increasing set I and a decreasing set D. If [b] I, then [c], I so that [hi [c] in any of the atoms listed.Similarly, [b] [c] if [c], D. In the remaining case, Ibis, 6 D, [c], I, since D < I in each atom, we have [b] _< [c] in, each atom.Conversely, if Po()[ [Par()[ ITop(a)l [PoPar(a)l