A class of partially ordered sets: II

>e, that is, ifelJwel} is not a position in X - {e}, then we will say that e is an essential element of X. Let E(X) be the set of essential elements of X. For any element xeX and position PeP(X), we have that if x >>P then xeE(X); i.e. only essential elements can make a non-trivial contribution to the generation of a position in X. If xeX then we have xeE(X) if and only if there exists y > x• such that y ~ x. We arrive then at the idea that as a general rule E ( X ) c X ~ P(X). Obviously, by definition, the non-elementary positions (the elements of P(X) - X) are determined by the elements of X. Thus they play no essential role in determining the structure of P(X). But going beyond this trivial observation, we note further that P(X), and hence X itself, is determined by the smaller set E(X). In this sense, it becomes rather meaningless to specify whether an arbitrary element of P ( X ) - E(X) is supposed to be contained within the original set X or not. Therefore two normal sets X and X' such that E(X) ~=~ E(X') will be considered to be essentially the same set. This gives an equivalence relation amongst the set of normal posets. A simpler way of expressing this would be to say that we will confine our attention to 'essential sets', i.e. those sets X for which X = E(X). Unfortunately, this might lead to some confusion, owing to the fact that E(X) itself is not necessarily a normal poset. (Of course, though, P(X) always remains normal.)

2. G E N E R A L C O N S I D E R A T I O N S

It will be necessary for us to choose some definite rules for determining 'probabilities' in our infinitely large normal posets. Now, the standard approach in probability theory is to define in some quite arbitrary way a probability measure to be taken on a particular a-algebra of measurable subsets (assuming that the sample space is infinite--otherwise the counting measure could just be taken). After that preliminary specification is gotten out of the way, one then proceeds to calculate the various probabilistic functions associated with the space. But our problem is different; we are not concerned with defining some arbitrary probability measure and seeing what the consequences might be. On the contrary, we are concerned with deriving a reasonable probability measure in such a way that the structure of the normal sets is respected. For this, it will

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be necessary to agree on some general principles to be followed when deriving such probabilities. Perhaps the best starting point for such a discussion is to think about the structure of the essential elements. Since a normal poset X is determined by its set of essential elements E(X) ~ X, it is clear that a probability theory of normal sets must be concerned with a probability theory of the essential elements. There are two different aspects to such a theory. First of all, we might consider some definite pattern of elements which could exist as essential elements in a normal poset. For example, consider the three elements {a,b,c} and imagine one pattern being, say, that a < b while clxla and clxlb. Call this Pattern 1. Then Pattern 2 might be the situation that a el~ with y :~ e. That is, we have a certain amount of uncertainty in specifying the upper set el). Rather than just taking e~ to be the set of all elements of X which are greater than e~, we are, to some extent, free to choose the details of the upper set e~, giving m a n y possible posets which are all slightly different from one another. Clearly this introduces a further element of probability into the theory.

3. UPPER SETS OF ESSENTIAL ELEMENTS

Theorem 1. Let e~E(X) be some essential element of the normal poset X. Let y e X - {e} be such that y~ ~ e~. Then y6e~. Proof'. If eg-yl}~ then we would have some ze. • Since yl>~eimplies that y~) - e ~ is infinite, it follows that, for an essential element e~E(X), there are always infinitely m a n y elements of X which are above e~ yet not above e itself. For convenience, let us make the following definition. Definition. Let e~X be an essential element. The 'position directly beneath' e is the position generated by the set e U (see Property 4 of Section 1). Note here that if E~P(X) is the position directly beneath e~E(X) then clearly eCEl}. On the other hand, in the elementary position e, we have e itself being in both the upper and lower sets. In particular, E is distinct from, and lower than, the elementary position e. Theorem 2. If E~P(X) is the position directly beneath eEE(X) then El) = el) and e~ ~ E~, where e~ ~ E~. Proof. Let x
Definition. Let X be a normal set and let A c X. Then we will say that A is a closed upper set if, for all y~X with y~ c A, we have yeA. Thus in particular, the upper sets of all positions are closed upper sets.

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Theorem 3. Let X be a normal set and let PeP(X) be a non-elementary position which is not beneath an essential element of X. For each xlxlP, choose an element uxeP~-x~). Let V c X be a smallest closed upper set containing all the sets (ux)~, for all xlx~P. (Note that certainly V c PN and thus V> P~.) N o w let X+ = X u {e}, where e is a new element added to the poset X with ordering relations given by e~ = V and el) = P~. Then X+ continues to be a normal set. Proof'. Since Pl) < V c P~, we certainly have X+ continuing to be a poset. The only axiom which must be verified is stability. Since PI} is not the lower set of any element of X, we cannot have e• = xl) for any xeX and similarly e~ 4: x~ for all xeX. (If we had V= xl), then we must have xePn, for otherwise we would have x]×[P and hence, by assumption, there would be a uxe V - x ~ , which contradicts V= x~.) Assume that aeX with al) c el) = Pl) and a~ = e~ = V. By the way we have chosen V, we cannot have a[~P. The only conclusion is that aePl), that is, ae. • For convenience, let us say that an upper set such as V in the theorem is 'sufficiently wide' with respect to the position P. Clearly e will be an essential element in the new poset X+ if and only if V=e~ is a proper subset of P~ and, furthermore, in this case we would have infinitely m a n y elements in V - P ~ . Put another way, we can say that while el) is the lower set of a position in X, en is definitely not the upper set of a position in X. Is there some 'minimal' possible upper set for an essential element e? For this, let us make the following definition. Definition. The normal poset X will be said to have the triple intersection property if, for all triples of unrelated elements a, b and ¢, we have (al)~bn)- c~ ~ 0. Note that not all normal posets have the triple intersection property. For example, the particular 2-dimensional set described in Part 1 does not have this property. On the other hand, one might say that the more general case is that a normal poset would not be expected to be confined to 2-dimensions, and therefore the triple intersection property can be expected to hold.

Theorem 4. Let V1 and II2 be two sufficiently wide upper sets in X, constructed as in Theorem 3. If X satisfies the triple intersection property then Vtc~V~ is also a closed upper set with V, n V2-x~ ¢ O, for all xl×lP. Proof'. This follows from the observation that if UleV,-x~ and uzeV2-x~ then there exists some u > u~ (i= 1,2) yet u_> x. Then ueV, c~V2. The fact that an intersection of closed upper sets is again a closed upper set is a trivial consequence of the definition. • Thus, given the triple intersection property, we can say that the upper set of an essential element must always contain the intersection of all of these particular closed upper sets. So given some position P as in Theorem 3, let W be the intersection of all the sufficiently wide closed upper sets with respect to P. Then an essential element e whose lower set is el)= Pl) must have W ~ e~ c P~. Definition. Let F = { x ~ ..... x,} be some finite subset of X. Let E={weX:wZ}, Q~={zeX:z
Theorem 5. The position directly beneath F = {x~..... x,} is a position in X, and it is the highest position beneath F. Definition. Let F = {Xl ..... x,} be a finite subset of a normal poset X and let eeE(X) be an essential element. We will say that e is directly above F if e~ = PU, where PeP(X) is the position generated

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by F. Similarly we will say that e is directly below F if e~ = Q~, where Q~P(X) is the position directly beneath F. Note here that if e is directly above F then certainly F = el}. On the other hand, if e is directly below F then it may be the case that F ¢ e~, since e~ is properly contained in the upper set of the position directly beneath e. However we do have the following trivial result:

Theorem 6. If F is a finite subset of the normal poset X and e~E(X) is either directly above or below F (with F ~ e~), then no other essential element of X is between e and F. So now the question is whether, in fact, in the case that e is directly beneath F, we have F c e~. For this, consider the following definition. Definition. Let F = {xl ..... xn} be a finite subset of a normal poset X and let A n be the union of F and all the upper sets (xi)]. If A~ is closed then we will say that it is a finitely generated upper set. An essential element e~E(X) will be called finite if there exists a finitely generated upper set A~, such that e~ < An c e n . The whole set X will be called dense if there exists no non-elementary position with a finitely generated upper set. So let e~E(X) be a finite essential element in a dense normal poset X. Let S=X be some separating subset above e and let Xs be the set of elements of X below S. Is it true that ({e} we~)~Xs= E~c~Xs (where E~P(X) is the position directly beneath e)? That is to say, is the set difference E ~ - ( { e } we~) confined to the region above S in X?. If not, then we can add in to en all of the elements of the finitely generated upper set A~, and also the (finite number of) elements of (E~- ({e})we~))c~Xs, together with their upper sets. Finally take the smallest closed upper set containing this expanded version of e n. According to Theorem 3, this changed version of X is still normal, and since X is dense, we cannot have the new version of e~ being the whole of E n - {e}; thus in the changed version of X, the element e continues to be essential. To summarize:

Theorem 7. Given a finite essential element e in a dense normal poser X a n d an arbitrary separating subset S c X, then there is a possible upper set e~ for e which corresponds with E ~ - {e} (where E is the position directly beneath e) beneath S. That is, one can say that the 'maximal possible' upper set E ] can be realized for any finite essential element, at least going up to any arbitrarily high separating subset of the dense normal poset X. Hence any finite set F = En can be encompassed within such an essential element. Is it sensible to say that 'most' normal posets are dense, and that 'most' essential elements are finite? We will assume that this is true. The first assumption would seem to follow naturally from the fact that, for all xlx[y, we have both the sets x ~ - y ~ and y ~ - x ~ being infinite. The second assumption is related to the observation that an upper set for an essential element can be generated by elements of the form uxEen-x~, for the elements xl×le. Since X is assumed to be strongly discrete, it seems reasonable to assert that only finitely many elements of the form ux will generally be required.

4. RULES FOR CALCULATING PROBABILITIES IN FINITE REGIONS

Until now we have considered the structures of isolated essential elements in a normal poset. But when looking at such a poset as a whole, the more obvious structures are given by the local relations of the essential elements in a small region of the set. Are there characteristic patterns which we can expect to see and, if so, how do these local patterns combine to make global structures in the poset? To answer this question, it is necessary to define some specific procedure for determining probabilities in normal posets.

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So, once again, let X be a normal poset. We are concerned with describing typical structures that we might expect to find within X, assuming that X is just an 'average', 'typical' normal poset, in some appropriate sense. So what is 'average' or 'typical'? One standard approach to such a question would be to consider all possible normal posets and find out what proportion of them have one particular property or another. But the difficulty with such an approach is that all normal posets are infinite, not only with respect to 'greater than' or 'less than' directions, but also in 'horizontal' directions, along separating subsets. Thus, if we are looking for typical structures in X which consist of only finitely many elements, it is reasonable to expect that a single poset X contains infinitely m a n y examples of each of these structures. Furthermore, if we go on to examine other possible normal sets other than X, it is reasonable to assert that there is an infinity of different possible normal posets, each of which contains infinitely m a n y of the different possible finite structures. Concentrating on the specific normal poset X, for simplicity let us assume that X is complete, so that X=P(X). Given any two points a,b~X, we must have a~b~ being finite. Thus it is possible to investigate systematically finite regions of X by confining ourselves to regions between pairs of related points a < b. Since X is strongly discrete, we also have b y - aU being finite. The first step is to take some fixed positive integer n and then to consider pairs of related elements a < b with precisely n elements (that is to say, positions) of X between them. (Note that a position P=P~P~ is between a and b if a~P~ and b~P~.) For large n, we expect there to be many different possible configurations of n elements between a and b. What does this mean? Let us say that {u~..... u,} is one possible set o f n elements between some pair of elements a < b in a normal set X. Just considered by itself, {u~,...,u,} is itself a finite poset. Perhaps we can find some other normal poset Y, with two elements c < d in Y such that there are precisely n elements of Y, namely {v~..... v,} between c and d. The two configurations are 'the same' if there is an order-preserving one-to-one correspondence between {u~..... u,} and {v~..... v,}. Otherwise the two configurations are 'different'. One can think of this as providing a set of equivalence classes of certain subsets of normal posets. Clearly the main consideration at this stage is the number n. For fixed n, there are only finitely m a n y different configurations, so it is reasonable to try to calculate the relative probabilities of these different configurations simply by counting. Given some particular configuration {u~,...,u,} between a and b in X, we observe that each ui is associated with a number of essential elements in b ~ - a ~ . Thus the structure of X in a~nb~) is determined by the configuration of the essential elements in by - a~. H o w m a n y essential elements can there be in bl~- al)? If we assume that a is an essential element (and for the purposes of the present argument, we will make this assumption), then Property 5 of Section 1 certainly implies that there are no more than n essential elements in the set b~)-al~. On the other hand, it is reasonable to expect that quite different configurations of essential elements--containing different numbers of essential elements--in bU-aU can each lead to configurations of X in bl~na~ with just n elements. Indeed, different generating sets of essential elements can lead to the same configuration of elements of X between a and b. Seen in this larger framework, there seem to be two possible rules for fixing the probabilities in our theory. The first possibility would be to assert that all of the configurations of n elements of X between a and b are equally likely. The second possibility would be to assert that all of the configurations of essential elements in b y - aU leading to n elements of X between a and b should be taken to be equally likely. Which of these two quite different possibilities should we choose? The overly simple answer is that sets which are different are, after all, different, so all of the different configurations of n elements between a and b should be given equal statistical weight. But this answer is just too simple. As has already been explained in Part 1, the reason this answer runs into difficulties is that it depends on the particular number n which happens to be chosen. On the other hand, since the normal poset X is always infinite, we should look for some rule

A class of partially ordered sets: I1

1225

which is concerned with a limit of large n. Since X is strongly discrete, this implies enlarging the configurations of elements of X which we are to examine by choosing a and b to be 'further apart', so that n increases. Given a fixed number of positions in some limited region of X, we can imagine various configurations of essential elements which may be contained within the region. Which configurations contain the greatest number of essential elements? According to the argument in Part 1, the configurations with as few unrelated essential elements as possible contain the greatest number of essential elements. The reason for this is that sets of unrelated essential elements can be combined in many different ways to generate many different positions, whereas sets of related elements can combine in only few ways. Recall Property 4 of Section 1 and note that if, say, u < v are two related elements then, if v is used to generate some position PeP(X), u is also in the lower set P~ of P. On the other hand, if ul×lv, there is no such restriction; thus more positions will be generated by ul~v than by u
Definition. The normal poset X will be called probable if, for any a < b in X, it is not possible to rearrange the essential elements between a and b to obtain a new normal poset, such that the rearranged essential elements have fewer pairs of unrelated elements than in the original configuration. The idea now is to concentrate on the probable normal posets and see if we can identify typical configurations which might be expected to arise. But it is immediately obvious that a certain configuration can occur in a probable normal poset.

Definition. Let C = E(X) be a totally ordered subset with no greatest or least element C='"'
l

which is maximal in the sense that every separating subset S = X contains precisely one element of C. If C has this property then it will be called a linear chain. Thus we expect a probable normal poset to contain linear chains or, at least, long finite linear chains are expected. Can there exist other possible configurations beside these linear chains? In order to approach this question, it will first be necessary to consider further the structure of the essential elements. 5. E S S E N T I A L E L E M E N T S

IN P R O B A B L E N O R M A L

POSETS

We have seen that, for an essential element e~E(X), the upper set e(~ is somewhat smaller than the set of all elements of X above e~. We have also seen that, at least in the case that X satisfies

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G. HEMION

the triple intersection property, e~ might approach a minimal possible upper set which could be very much smaller than the upper set of the position directly beneath e. Thus we expect that there can be many different possibilities for the upper set of the essential element e~E(X). But, given some element y > e J), with y ~ e, we might ask whether it is possible to change X to give us a new poset X* which is identical with the original X except that now y (and all the elements of y~) are declared to be greater than e. In the new poset X*, we have fewer pairs of unrelated elements than in the original X. After all, the original yl~e is now replaced with y>e. In general therefore, we would expect X* rather than X to be a probable poset (assuming that X* continues to be normal). Of course, this rearrangement of the elements of X can be expected to bring with it many further consequences. But, as a general rule, it is surely reasonable to assert that in a probable normal poset, the upper sets of the essential elements are as large as possible; i.e. the difference of the sets E~-e~ is as small as possible, where E is the position directly beneath e. At this stage, it is useful to recall Theorem 7, which says that if we assume X to be dense and e~E(X) to be finite (and from now on we will assume that these are reasonable conditions to be fulfilled in a 'typical' normal poset) then below any separating subset we may take e~ (together with the element e itself) to be identical with El, the upper set of the position directly beneath e. In fact, according to our arguments, it is overwhelmingly probable that ({e} we~)nXs= E~Sx, for any given separating set SeX. So to fix our ideas at this point, a further definition will be helpful.

Definition. Let e be an essential element in the normal poset X and let S e X be a separating subset. We will call e probable with respect to el) and S if ({e} we~)c~Xs=E~nXs, where E is the position directly beneath e, and Xs is the set of elements of X beneath S. Our assumption then is that in a probable normal poset X, each essential element eeE(X) that we consider will be probable with respect to e~ and S, for all separating subsets that will be considered. In other words, for all 'practical purposes'--meaning with respect to all possible finite regions X--we have that e is 'maximal' in the sense that e~ is simply the set of all elements above 4 - Or, put another way, not only the non-elementary positions but also the essential elements of X a r e uniquely determined (at least 'locally') by their lower sets. Therefore one source of uncertainty has been eliminated from the theory; we no longer have a choice in the possible upper sets for essential elements. The upper sets are, for our further arguments, simply taken as maximal, and thus fixed. So much for the upper sets of essential elements. What of the lower sets? Let e~E(X) be some essential element whose lower set is e~. The corresponding upper set for e is then the maximal upper set e~, determined by e~). As an alternative to e, we might replace e/~ with some different lower set Q]) of a position Q~P(X). The corresponding maximal upper set Q~ is thus also determined. As we have seen, QV can be specified by saying that it is the convex set determined by some finite set of elements of X which are associated with the position Q. The question is now, which lower sets are most probable? Since the lower set of an essential element determines its upper set, we have a certain balance between the sizes of the upper and the lower sets: if the lower set becomes relatively smaller, then the corresponding upper set becomes larger, and vice versa. So the question is, what situation can we identify where the total number of pairs of unrelated essential elements is as small as possible? To begin answering this question, let us examine in detail how the size of the upper set decreases as the size of the lower set increases. Recall how the upper set e~ depends on the lower set e J) of e. Let x[x[e and let us try adding the element x to e~l, thus increasing the size of e V and at the same time reducing the size ofe~. Now recall that there must be some ux~e~-x~. Assume that x is 'directly next to' e. We take this to mean that there exists no zlxle with z :~x and e~ -z(~ ~ e~ - x ~ . Let q~ be the lower set of the position in X generated by the set {x)we~). Let Q~ be the

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corresponding maximal upper set for an essential element whose lower set is Q]). Since Q• < Q~, we certainly have uxCQN.In fact, the entire infinite set e ~ - x ~ is disjoint from Q~. So let xl×le be directly next to e. Can we have some z~xU-eU? Obviously not, since then xncz~, hence en-z~ce~-x~. Thus x~-e~=O. On the other hand, we know that there must be infinitely m a n y elements of X in e~ - x ~ . This seems to contradict our goal of striking a balance between the sizes of the upper and lower sets. We have QU containing only one single extra element in comparison with e]), namely the element x. On the other hand, we must remove infinitely m a n y elements from en to obtain Q~=e~nx~. Thus at first sight, and following our previous arguments, it would appear that the altered version of X with the essential element e changed so that its lower and upper sets are now QU and Q~ must be much more probable than the original version of X. On second thoughts however, one sees that there is a problem with this argument. To see this, just take e < f t o be any related essential elements in X. Then it is possible to go from eli t o f ~ by adding in finitely many essential elements, which can successively be taken to be directly next to the given essential element. Our argument would then imply t h a t f m u s t be more probable than e, and in fact the higher one goes in the given poset X, the more probable the essential elements become. Clearly this is nonsense. Therefore we must be more careful about what we are comparing when making such assertions about relative probabilities. So the question is, given e]) or the alternative Q~, which pattern, as the lower set of an essential element, is more probable. As before, this question can be resolved by looking at regions between elements a < e < b and counting the number of positions between a and b. In examining such regions, we note that the new positions which may be formed in association with x are above e and x. Furthermore, the old positions that were not above x but were above e and which thus disappear in the new alternative poset were also above e. But the new positions associated with x may make an infinite cascade of further new positions and essential elements in the upward direction of various alternative versions of the original poset. That is to say, the single additional element x in the new version of eU creates infinitely many new positions going upwards in X and this serves to balance the effects of removing infinitely m a n y elements higher up in X above e. Thus the addition of the single element x to Q]) may have a greater effect than the subtraction of the infinite number of elements high up in e ~ - x ~ to obtain Q~. Is el~ or Q~ more probable? Clearly there is no simple answer to this question. It just depends on the details of the rest of the poset X near to e and x. QU will be more probable if x is relatively far downwards in X, away from e and if, at the same time, the set e(~-x~ is relatively large near to e. If these relationships do not hold then the original e~ will be more probable. Up till now we have only considered adding the element xl×leinto the lower set of e to obtain the alternative version QU- It is just as possible to consider removing some element z~eg from the lower set to produce an alternative version QI} with Q~=e~-{z}. This would give a genuine alternative if this new Q~ is, in fact, the lower set of a position in X - {e} and furthermore z is directly next to QU. Again, the question of which of these versions is more probable must depend on the details of the poset X. The following definition therefore remains rather vaguely formulated.

Definition. The

essential element e~X is 'as round as possible' if the effect of either adding in or removing an element from e~ is very much the same for all x directly next to e (or, in the case of removing an element, such that the removed element is directly next to the alternative version of e). The idea is then that in a probable normal poset, all essential elements are as round as possible. O f course the word 'round' has very definite geometric connotations. I f we consider the special case o f a poset embedded in some finite-dimensional Euclidean space, then our assumption could be translated into the condition that each essential element is located at a definite point in the

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Euclidean space, and its upper and lower sets are precisely the elements of the set X in the 'light cones' above and below that point of Euclidean space. In other words, the 'non-localized' positions in Euclidean space described in Part 1 are not candidates for being possible essential elements of X.

6. CHAINS OF ESSENTIAL ELEMENTS

Definition. Let X be a normal poset and let Z c E ( X ) be a set of essential elements of X. A cross-section of Z is a set of the form y c~S, where S c X is a separating subset. The set 37 will be called a chain of order n if the following three properties hold: 1. There exists a cross-section of Z with n elements, yet no cross-section has less than n elements. 2. If xlxly are two elements of Z, then x~c~Z:~y~nZ and xl]nY.:~yl}nE. 3. If a < b < c and x are four elements of Z then either x ~>a or x ~
Lemma A. Given any chain Z, of any order, there cannot exist four elements a, b, c and d in Z with a~d. Proof'. Given that this pattern did exist in a chain Z, if we had some xeaU -bl~ it would follow that x < a < c, yet x[x[b and c[x]b. This is impossible. Therefore both a~ - b~ = O and bl) - a V= O; i.e. a~ = bl) and consequently a = b, which is a contradiction. • Lemma B. If n > 1 and x,yeE then we cannot have either yl) = {x} wx~ or y~ = {x} wxl). Proof'. Let x < y be such a pair and let a[xl {x,y}. We cannot have a zexl) - aV, for then z < x < y with z[xha and y[×la. On the other hand, a ~ - y O ~ O is impossible by Lemma A, and the only remaining possibility, namely that al)n (y~ - x V ) ~-O, is impossible since y~ = {x} uxl). Therefore a~ c xl), and we would have al)= xU, which is impossible. The argument that y~}:~ {x} wx• is similar. • It is now a simple matter to determine the structure of chains of various orders. To begin with, if the order of the chain is not zero, the chain must contain infinitely many elements and, furthermore, it must have no greatest or least element; otherwise, there would be a cross-section with zero elements. A chain of order one is just a totally ordered set. That is, given a cross-section with only one element, the second property of the definition ensures that all other cross-sections also have only one element. Thus a chain of order one has the structure Y~='"
Assume now that we have a chain of order two and let F0 = {a,b} be a cross-section with two elements; therefore alxlb. Let FI be the set of elements of Y~directly above {a,b}. That is, xeF1 if and only i f x is greater than a or b, and there is no yeY~ with x > y > a or x > y > b between x and {a,b}. We must have at least two elements, say c and d, in F1 with elxld. According to Lemma A, we cannot have, say, a {a,b} and d > {a,b}, for then we would have c]) = d{~. Thus one of these elements is above both a and b, while the other element is only above one of the elements a or b. For definiteness, let us suppose that c > a and d > {a,b}. Can there be other elements beside c and d in r~? Suppose that x is such an element. We cannot have x > {a,b}, for then xl) = 4 - Also, x > b is impossible because of Lemma A. Therefore x > a and, to avoid having xl) = el), we must have either c]~- x~ ~ O or else x{~- c{{~ O. Suppose that yeel) - x~. Then y < b < d, yet xl>~y and xlxl d, which is impossible. (The case xU - c ~ =~O is, of course, identical with this.) Therefore, F~ = {c,d}. Now take F2= {eft) to be the cross-section of E directly above F~. According to the previous argument, we must have one of these elements, say f, being greater than {c,d} and the other element e being greater than either c or d. But if e > d, then we would have el)= {d} udU, which is impossible by Lemma B. Therefore e > e and also e > b to again avoid having a < c < e unrelated to b. This construction can also be extended downwards from the original cross-section. In summary then, we have that

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1229

also chains of order two have a unique structure, namely, £ = {...,x_ ~,x0,x~,x2,...}, with xi < xj if and only if i ~ 2 and let F0= {a~..... a,} be a cross-section with n elements. As before, let F~ be the cross-section of directly above F0. Take {b~..... bn} to be some set of n elements in F~. Arguing as before, we may assume that b~> aj, for all i >~j, while bklx]a~, for k < I. Furthermore, some other element xeFl must have the same elements of F0 in its lower set as b~, for some i between 1 and n. Which i could it be? i=n is impossible, for then we would have xU = (b,)l). Next, i = n - 1 is impossible, since otherwise we would have some y < a , with x N y , thus y < a , < b , with ylx[x and b, Nx. By extension, all further choices for i are also impossible. Choosing F2 to be the cross-section directly above F~, and so forth, we again find that there is no freedom to make arbitrary choices. To summarize:

Theorem 8. Every chain of order n has the structure = {...,x

,,Xo,X,,X2

.... },

with xi 2 and a cross-section is specified, the initial specification of the order of the elements in the cross-section can be taken to be arbitrary. Thus one might imagine that such higher order chains have a 'spiraling structure' with different possible orientations, in a certain sense. For example, if the whole normal poset X is embedded in an order-preserving way in n-dimensional Euclidean space (considered as a partially ordered set, as in Part 1), these order n chains would appear as true spirals. On the other hand, chains of order higher than n could not be realized within the Euclidean geometry. Is it reasonable to imagine that non-linear chains of order n > 1 are a c o m m o n feature of probable normal posets? To answer this question, let us return to the reasoning of the last section. There we saw that a probable normal poset has as few unrelated pairs, within a given region of the set, as possible. This leads to the idea that a normal poset should contain chains of totally ordered elements, that is, chains of order one, according to our definition. Such a structure certainly has fewer unrelated pairs than some more disordered configuration. But, given a whole collection of such linear chains, we might try to bring them together into an even more compact form, reducing even further the number of pairs of unrelated elements. What reason is there for expecting such compact configurations of linear chains and what are the limits of this close-packing? As far as the first question is concerned, it is easy to see that, if we have two linear chains ---< x_ 1 • x0 < Xl < and ... < y _ 1
'

"

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set xi)nE. After all, if there exists some other essential element y in another version of our normal poset X with y ~ n E = x l ) n ~ and y is genuinely less than x (meaning, say, that x~cyl)), then surely we could say that the original Z was not closely packed. After all, a closer packing would be obtained by replacing x by y, giving in particular fewer unrelated pairs when compared with other essential elements of X outside of E. In a similar way, we can justify the condition that x ~ n Z = y l } n Z implies x=y. The other condition is that if we have a < b < c and x in Z then either x >~a or x ~ x, with z~E. This further essential element z must itself also be round. We can again take this to mean that the position directly beneath z is not necessarily associated with x. Indeed, if this concept of 'roundness' implies that very m a n y elements are associated with positions directly beneath typical essential e l e m e n t s - - m a n y more than the order of the c h a i n I t h e n it seems reasonable to assert that it is very unlikely that the position directly beneath z is, in fact, associated with x. Another way of looking at this situation is to take F1 -- {Zl..... z,} to be the set of least elements of Z in x•. Then it is not true that x is the greatest possible round essential element beneath F~. Let us summarize this argument. Roundness of essential elements implies that many elements of X outside Z contribute to defining their lower sets. This limits the possible closeness of the packing of a chain. The result is that a given element xeZ is either the smallest round essential element above F0, the set of elements of E in xU, or x is the greatest round essential element beneath r~, the set of elements of E in x~. Note furthermore that we do not expect to have a mixture of these two conditions in a single chain, since the closest packing will result if all the elements of the chain are either as low as possible with respect to one another or else as high as possible. The result is that for each order of chain we have a corresponding 'anti-chain', such that in the chain the elements are always as low as possible and in the anti-chain the elements are always as high as possible. 7. R E L A T I O N S H I P S

BETWEEN

DIFFERENT

CHAINS

In this section, a number of ideas will be sketched which relate the present theory to the ideas in previously published papers, as described in Part 1. The picture we have is that a probable

A class of partially ordered sets: 11

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normal poset consists of chains (and anti-chains) which are related to one another in some way. Even without restricting ourselves to normal sets in finite-dimensional Euclidean space, certain geometric ideas can be naturally expressed. For example, we can say that a chain E follows a 'straight path' through the poset X. For if x < y are two elements of E then, since E is as closely packed as possible, we have that it contains the greatest possible number of elements between x and y. This is analogous to the situation in Euclidean space R", where, for the purposes of the assumed partial ordering, distances are measured according to the Lorentz metric. Then the straight path in R" from x to y with x < y is the (future-pointing) path of greatest length. Note that, if we assume our normal set X to be Euclidean, the requirement that essential elements be as round as possible means that they can be represented by Lorentzian 'light cones' in R". We would have the set of possible round positions for essential elements of X being a discrete set in X and, under reasonable assumptions, we could expect this set of potential positions to be homogeneously embedded in R", at least locally. Within such a framework we would have, on average, a constant Lorentz distance between adjacent elements in a chain. Thus the assumption of an additional underlying variational principle to ensure the straightness of the chains would not be necessary. Even without a variational principle, a mechanism for the interaction between different chains and anti-chains can also be described. Imagine that we have two different chains Z~ and E> An interaction will result if we have, say, two elements x < y with xeE~ and yES22 such that x is associated with the position directly beneath y. As we have seen, this means that x serves to define the lower cone o f y in an essential way. In other words, if E~ is altered somewhat, moving x to another position, then the position directly beneath y is no longer available; hence some alteration to E2 will be a necessary consequence. The effect which we are looking for is, of course, of a purely probabilistic nature. As always, it is necessary to compare like things with one another. In this situation, we have various possible versions of Z~ and E2, sharing the property that some element yeE2 has the position directly beneath it being associated with an element xeE~. A number of mechanisms can now be proposed to describe the influence of one chain on the other (through some sort of conditional probability). For example, given that two chains interact as above, one might say that the position of the element yeE2 is much restricted by its relationship with the given element xeY~ in comparison with the case that the position directly beneath y is not associated with x. On the other hand, according to an argument in Part 1, the probability of an element being associated with a given chain is greater if the element is near to the chain (measured using a natural distance function within the poset). Perhaps different mechanisms may become relevant when Et or E2 is an anti-chain. So one sees that the theory offers much scope for drawing possible analogies with various mathematical structures in modern physics. Be this as it may, it could surely also be said that the basic definition of normal posets is so simple and general that the theory, taken by itself purely as mathematics, might also be worthy of further investigation.

8. CONCLUSIONS In the present paper, we have outlined a probability theory for posets. At this stage, we seem to have reached a point where further progress can only be made by making additional, perhaps arbitrary, assumptions. The situation may be analogous to that in chaos theory. There, it is not possible to prove that some simple and unique structure always occurs in a particular model of chaos. On the contrary, the basic idea of chaos is that many possible structures can come about in subtle ways depending on the exact specification of the model. In the same way, it seems true that a probable normal set will contain many possible structures which can only be described by

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restricting ourselves to particular special conditions imposed on the model. For example, perhaps it would make no sense to attempt to prove that infinitely long chains, as we have described them, must exist in a probable normal poset. It may be that only long finite segments of chains occur, with some other pattern at the ends of the segments. Certainly such ideas must, at least for the present, be confined to the realms of speculation. In any case though, this would give us a discrete version of chaos, which might be related to the continuous models of chaos described, for example, in [2]. By considering, not the individual normal posets, but rather the ensemble of all possible normal posets, it is conceivable that one would be led to take continuous Euclidean space for describing this ensemble. Then a theory such as that in [3] might be found to be appropriate for describing the behaviour of the chains of the individual posets in this ensemble.

REFERENCES

1. Hemion, G., A class of partially ordered sets. Chaos, Solitons & Fracta&, 1996, 7(5), 795, Referred to as Part 1. 2. El Naschie, M. S., R6ssler, O. E. and Prigogine, I., Quantum Mechanics, Diffusion and Chaotic Fractals. Elsevier, Oxford, 1995. See also Chaos, Solitons andFractals, 1996, 7(5 and 6). 3. Nelson, E., Dynamical Theories of Brownian Motion. Princeton University Press, 1967.