A class of partially ordered sets

Chaos,

A Class of Partially GEOFFREY Faculty

of Mathematics,

University

Ordered

Solitom

& Fractals Vol. 7, No. 5, pp. 795-819, 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved o%w779/96 $15.00 + o.cu

Sets

HEMION of Bielefeld,

33615 Bielefeld,

Germany

Abstract-This paper is concerned with a theory of discrete partially ordered sets which satisfy a number of additional axioms. In particular, the sets will be assumed to be ‘stable’. That is, if a and b are two elements of such a set, with the property that all elements above b are also above a, and all elements below a are below b then-necessarily-we must have (as expected) a c b. It is found that such sets have certain geometrical structures which are analogous to the usual Euclidean geometry. A probability theory for a class of such sets is proposed, and it is found that sets which contain particle-like chains of elements are most likely. Copyright 0 1996 Elsevier Science Ltd.

1. INTRODUCTION

How many axioms are necessary to specify the usual kinds of mathematical analysis which are common in the description of physics? Surely many tens, if not hundreds of axioms are needed! Even a definition of the real number system requires a surprisingly large number of assumptions. Then more axioms are needed to specify Euclidean space. Still more are required to specify the complex numbers and the rules of complex analysis. Many more are needed if we delve into differential manifolds. If algebraic topology is also brought into the picture, then a great number of further axioms are again needed. Has anyone ever gone to the trouble of making a complete list? If so, perhaps it would be worthwhile to check the logical connections between all of these axioms. It may be that some small set of seemingly primitive axioms have unexpected consequences that are in conflict with-or perhaps imply-some of the axioms which appear more towards the beginning of the list. In this paper, we will deal with only a very few, extremely primitive axioms. In particular we will consider the theory of discrete partially ordered sets. Now it is certainly true that discreteness and ordering are things which are not usually considered to be at the beginning of the subject of mathematical analysis. On the other hand, in modern physics these concepts are usually adjoined somewhere onto the main body of the whole theory in a way which confuses the strength of the assumptions which are being made. Therefore, to clarify matters, we will examine these concepts here in an abstract way, free of all peripheral assumptions. 2. BASIC

2.1.

DEFINITIONS

Confluence, discreteness,stability

A partially ordered set (or ‘poset’) is a set X, together ‘c’ C X x X with the properties (1) (4 a) E S, for all a E X, (2) if (a, 6) E G and (b, a) E c then a = b and (3) if (a, b) E G and (b, c) E c then (a, c) E C. 795

with an ordering

relationship

Of course it is more common to write ‘a s h’, rather than ‘(a, h) E 5:‘. The expression n ~1h explicitly excludes the possibility that o = h. In addition to this, we will use the notation ‘17 :?=Ci h’ to denote the idea that a and h are unrelated: that is. a Dd b o u $ h and a 8 h. Most of the posets with which we shall be dealing will be infinite, and so it will often be necessary to find a shorthand way of describing specific examples. For this the following definition will be helpful. Definition. Let (X, 5) be a poset and let 3 LCC XxX. generates c: if s is the smallest partial ordering containing E.

We will say that E

Definiticm. Let CJE X be a typical element of our partially ordered set. Then we will denote by +, the set of elements of X which are less than, but not equal to, a, i.c. a,, - {u E X: ci
A class of partially ordered sets

797

(3) Let R” be n-dimensional Euclidean space, together with the partial order specified by requiring that (x1, . . ., x,) < (yl, . . ., y,J if x1 < yl and (x1 - yl)* b z;=2(xj - yi)*. Th en, for all it 3 1, R” is stable, but not discrete. (It is also common practice to denote the coordinates of a point in R” by (x0, . . ., x,-~) so that the special nature of the first coordinate is emphasized.) Definition.

This partial ordering will be called the natural ordering of R”.

(4) Let 2” C R” be the set of points of n-dimensional Euclidean space with integer coordinates, together with the induced partial ordering. Then 2” is both stable and discrete, for all n 2 1. (5) If a E X is such that uv = 0 then we will call a a least element; the definition of greatest element is analogous. Given that X is stable, then there can be, at most, one single greatest or least element. Similarly, if X is confluent then it is clear that there can only be one single element b of X which is directly above the least element. (x is directly above y if x > y and xU n yfi = 0; in this case, we can also say that y is directly below x.) Also only a single element can be directly below the greatest element if X is confluent above. Can it be that X is finite? If so, then X is certainly strongly confluent, and in fact it is a totally ordered set: X := {Xl < - * * < x,}. (6) Assume that X is discrete, confluent and stable, and that there is some element a E X with X = {u} U av U a+. Then X is isomorphic either to 2, or to some subset of 2. This can be seen using the same arguments which applied to the last example. Thus if an infinite stable set is discrete and confluent and has a least element, then it must be isomorphic with the positive integers. Similarly if it has a greatest element and is confluent above, then it is isomorphic with the negative integers. Since these sets are rather trivial, we will from now on assume that the sets which we consider are infinite, with no greatest or least elements. Theorem. Let X be discrete and stable, and let a rxl b be two elements of X. Then either au - b, or b, - ufi is infinite. Proof. If, say, the set a U - b, is finite, then there is some lowest element z E uv - b, such that zv fl (a+ - b,) = 0. (If av - b, = 0 then take z = a in the present argument.) Since z Q b, yet zu C b,, we must have b, - zfi #0. Let u > b, with u > z. But since u > b, we have uv > b,, hence zv C uv, and since u > z, there must be another element u > u, with u > z . Of course we also have u E b, - z+. The argument can then be repeated with the element u substituted for u, and by extension we conclude that the set b, - .zfi c b, - a+ is infinite. QED Certainly the sets 2” are such that if a DU b, then both uU - b, and ufi - b, are infinite. In fact, to show how it is possible to experiment with these basic definitions, consider the following. Definition. The partially ordered set X is strongly stable if for all a, b E X with either uv G b, or a+ 2 b,, we have a 6 b. Clearly strongly stable z. stable. Also 2” is strongly stable for each n 5 1. But on the other hand, the idea of strong stability brings with it an overly large amount of ‘rigidity’, which we do not want to assume. For example, looking at the proof of the last theorem, it becomes clear that strong stability always implies that both uv - b, and ufi - b, are infinite, for all a D<1 b. Thus, even if we remove some large, but finite, subset of X, the

7’1%

G. HEMION

‘geometry’ remains completely unaffected, in some sense. On the other hand, if X is only stable -not strongly stable-then we might have two elements a Ml b, with only a single element in a+ - b,. Removing this element suddenly brings us to the situation of having u* c 6,. Thus the removal of a single element might have consequences for the relationships between other elements of X: one might say that a ‘stable’ set still contains a certain degree of ‘instability’ which allows for more interesting ‘geometrical’ structures. Another reason for us to concentrate on this idea of ‘stability’ for partially ordered sets is that it allows us to specify a stronger type of ‘discreteness’ condition, which in turn will allow us to develop a probability theory for this class of partially ordered sets. But such ideas will become more clear as the theory is further developed. Suffice it to say now that the condition of stability-but not the condition of strong stability-is of central importance for our work. 2.2.

Strongly discrete sets

Let a P-G h in some stable (confluent and discrete) partially ordered set X such that either aul - h+, or afi - bql, is finite; to be definite, assume that av - bti is finite. As we have seen, this implies that 6, - uO is infinite. But looking at the argument which led to this conclusion, we see that it is also true that if a < h, and b, - uG is finite, such that there exists some c‘ E (6, ‘-- a&) with c W1 u, then we must have u,+ - b6 being infinite. Let us then think about the case that for all pairs of elements u, b E X, we have either u,, - b,, or afi - ho being finite. Could it be that there is some pair a and b with u; - bu finite, and some other pair c and d with c+ - d, being finite? If we assume that X is strongly confluent, then we can find some 24 E X below all of a, h, c and d, and also some u E X above all of these points. Then we must have both u,~ - uv and u,,+ - 11~ being infinite, which is impossible. Therefore we have: Theorem. If X is stable, discrete and strongly confluent, such that either a+ - h, or h, .- udl is finite for all 11, b E X. then we have either (i) a{,, - b, is finite, for all a, b E X or (ii) u,,. - b,, is finite, for all u> b E A’. In order to deal with the case of merely confluent (not necessarily strongly confluent) sets. we make the following definition. Definition . The partially ordered set X will be called strongly discrete if for all u, li, E X we have u+ - b+ being finite. (Note that since a* i? h, can only be non-empty if a < b. and in this case rl h fl b, c uJi. - h,, it follows that strongly discrete =3 discrete.) Theorem. Let the partially ordered set X be stable, confluent, non-trivial (i.e. there exist two elements x M J’ in X) and strongly discrete. Then we have that for any pair u, h E Xi either ali - b+ = ,D (,that is, (1 2 b), or otherwise lt3 - b, is infinite. For us, a set is ‘tri
A class of partially

Let X be stable, confluent, Theorem. separating subset is infinite.

ordered

non-trivial

sets

and strongly

799

discrete. Then every

Proof. Assume to the contrary that Y C X is a finite separating subset. Since X is confluent, there exists some element u E Y- which is below all elements of Y, i.e. u < Y. However, this implies also that u < Y+. On the other hand, since X is non-trivial, there must exist some other element u E X with u DU U. But then our assumptions imply that VII - Ufi is infinite. However, since X is discrete, there must exist some u’ E ufi - u,-, which is not below any element of Y. That is U’ is unrelated to all elements of Y, which is a contradiction. QED We see then that the partially ordered sets with which we are dealing are always infinitely ‘wide’ at all ‘levels’. 2.3.

The existence of non-trivial, confluent, stable, strongly discrete posets

As we have seen, the idea of ‘stability’-when combined with discreteness-is more restrictive than one might at first think, and it can easily lead to rather degenerate, trivial examples of posets. Thus the question arises as to whether, in fact, there exists any example of a non-trivial poset which is confluent, stable and strongly discrete. Further on, we will assume that there are very many possible such sets, and we will investigate the ‘probabilities’ that various characteristic structures might occur in a ‘typical’ confluent, stable, strongly discrete poset. Assuming that such a set does exist, then, as we have seen, each separating subset must be infinite. Let us therefore consider some abstract set S, which we imagine to be a separating subset of some poset X which we would like to construct. In particular, a I><] b, for all a, b E S. Now S is infinite, and also we are looking for some strongly discrete, confluent set X; thus it is sensible to assume that S is countable. For simplicity then, let us assume that S = Z, the integer numbers (but now without their natural ordering). As a first example, consider the following set. Let A.E R be some fixed irrational number. Then we have X,

= {(n, m + An) E R2: m, n E Z, n 2 0}

X-

= {(-2”,

rn2”+l - 1) E R*: m, n E Z, n 2 0}

and our poset is then X = X, U X-, with the ordering relations inherited from the natural ordering of R2. Clearly X is discrete, and strongly confluent. Is it strongly discrete? To show this, begin by noting that all points of the form (0, k), with k E Z are elements of X, and it is clearly sufficient to show that for any a = (0, k) and b = (0, I), we have av - b, is finite. But this, in turn, will follow if we just consider the cases a = (0, k), b = (0, k + l), and a = (0, k + l), b = (0, k), for various k E Z. If a = (0, k) and b = (0, k + 1) then if an element p = (-2”, rn2”+l - 1) of X is in av - bu, we must have 0 s rn2”+l - 1 - k + 2” c 1.

This certainly implies that 1(2m + 1)2” - (k + 1)1 s 1.

So the question is, can we have this condition being satisfied for infinitely many pairs of integers m and n 3 O? Obviously not, since the number 2m + 1 is always odd. The case a = (0, k + 1) and b = (0, k) is similar.

ROO

G.HEMION

Is X stable? Owing to the way we have constructed X,, it is clear that for any two points a, h E X we have a d b if and only if h+ C I+. Thus we need only check that for Again, we begin by checking the particular case a = (0, k), crfh we have ~+fh~. b = (0, k + l), for some arbitrary integer k. If k is an even number, then we have (- 1, k - 1.) E a@ - h,, . if k is an odd number, then (-1, k + 2) E b, - av. More generally, let * = (-p, p&y+1 - 1) and b = (-2”. (m + 1)2”+’ - 1). Then again, if M is even, we have (-y+l, m2tl+l - 1) E a+ - b,, otherwise, (-2”+‘, (pn + 1)2”+’ - 1) E b+ - LQ #0. From this it follows easily that if a and b E X , then a + h implies that av # bb. What happens when a or 6 is in X, 3. Here we need only examine the case where a = (n, m + An) and b = (n, (m + 1) + An) with y1b 1. But then, depending on the value of d, we have an m’ E Z with ((n - 1). (m’ + A(n - 1)) E X+ being either in uuL - h,, or else in h 1 - LI:, Mare generai constructions. This last example was certainly very particular, and its construction gives no plan for constructing large classes of confluent, strongly discrete, stable posets. We will therefore indicate a few general principles which could be applied in the construction of more examples. Our idea in the particular example was to start with a separating subset S, which was then taken to be the integer numbers 2 in their natural embedding in the two-dimensional Euclidean plane R’, i.e. we had S = ((0, k) E R’: k E 2). But to proceed, let us now ignore such possible embeddings of our starting set S in some Euclidean space. Instead, we will simply take S to be some abstract, countable set. We will consider a sequence S,, i = 0, 1, 2, . . , of sets of subsets of S. To begin, let $ = S. where we identify each element x E S with the singlet subspace (x} C S. Then S,, is the set of all such singlet subsets. Definition, Let S be a countable set, and let S,, i = I). 1, 2, . be a sequence of sets of subsets of S with So = S as described above. Then (S,),,, will be called a separating sequence if for all i E RI,, we have that if II. h E S; with neither a c h nor h C a, then there exists some j ;. i and 12: (’ E S, with either a C c’ and b $? c or else h C L‘ and a $ c. In addition we require that for fixed a and b, only a finite number of c satisfy this condition and furthermore we require that if s E Sk and i E St, with k <: I then we must have t 2 .Y. Sow it is not difficult to see that if we have a separating sequence as in this definition, then it represents a particular strongly discrete. stable poset X. For let X._ be the set of all elements of the ,I’,, for all i. Given 11. h E X-, then WC define a cc h if and only if b C CI. Thus X is the ‘lower set’ (below the separating subset S) of our proposed set X. We still need to specify some ‘upper set’ X,.. For this, we have a great deal of freedom, provided only that our X., is sufficiently ‘dense’. The meanings of all these rather vague ideas will become clearer as we proceed. But to be specific, perhaps the simplest plan would be to again identify 5’ with the integers % as before. and then take X.+ to be our two-dimensional upper set in the previous example. As we will see, in some given region (i.e. some finite set of elements) of a strongly discrete. stable poset. the structure is completely determined by the elements below the region; for all practical purposes, the elements above the region can be ignored. Thus our identification of lower sets with separating sequences is usually sufficient for our purposes. On the other hand, given that we have constructed some appropriate X-, then we are free to attach almost arbitrary discrete extensions X+, and we continue to obtain a poset satisfying our conditions. It is only necessary to ensure that for any a Q: b there is always some c in X. with L’ E h, .- ai,.

A class of partially ordered sets

2.4.

801

Positions

In order to proceed, let us consider the following further definition. Definition. Let A C X be such that we can separate A into two disjoint subsets A = Au U Ai,, with A+ A+ *y e A,-then we will call A a position in X. A position is therefore a maximal double cone in X. In addition, for each element x E X we will call the set {x} U xG U xfi a position; namely an ‘elementary position’. Now the idea of an elementary position is certainly clear enough; it is just another way of looking at an element of the set X. But why think about this more general concept of ‘positions’? Theorem. Let the partially ordered set X satisfy our conditions (stable, confluent, strongly discrete), and let x E X. Consider now the smaller set X’ = X - {x} with the element x removed. Then within this set X’ there is a position A C X’ with A, = xU.

Let a E X’ with a < xfi. If a 4 xu, then, since X is stable, we must have So let u E X’ with u E av - xv. But then also u < x6, and so repeating the argument with u in the place of a, we obtain a further element u E X’ with u < u and u C x, and so on. We conclude that a+ - xv must b e infinite, which contradicts the definition of strongly discrete. QED On the other hand, it is definitely not true that xfi is necessarily the ‘upper set’ of a position in X’. We will make much of this difference in our further arguments. But before proceeding on, it is important to note that not all positions are ‘associated’ with elementary positions, in the sense indicated in this theorem. For example, consider the set R3, consisting of three-dimensional Euclidean space with its usual partial ordering. Of course this set is not discrete. Nevertheless we can still consider the concept of positions in R 3. Let A C R3 be the following position. Let

Proof. aU $ xv.

A = ((0, s, 0) E R3: -1 s s s l}.

That is, A is a ‘horizontal’ line through the origin of R3, with length 2. Now take A, to be the set of all points of R3 which are less than or equal to at least one point of A. Let A, be the set of all points of R3 which are above all points of A+. Then A = A+ U A, is a position in R3 (with respect to the natural ordering of R3). To see this, we need only check that A is maximal, i.e. let x < A*; then we need to show that x E A,. If not, then x is beneath no point of il. Let H, be the ‘horizontal hyperplane’ in R3 through x = (x0, x1: x2), that is, H, = {(x0, y, z): y, z E R}. Clearly, H, n Au is a convex set. Thus there exists a circle o on H, which contains H, n A, completely, but does not contain x. But then there exists some point p E A, such that H, rl pv is precisely the set of points within cr. Thus x 4: p, a contradiction. The important idea here is that the position is generated by a convex set of points. We can consider this idea in the context of stable, strongly discrete posets. Let {x1, . . ., x,} be any finite set of elements of X. Let Z!.+ be the set of elements of X which are above all of the xi and let %+ be the set of elements which are below all of the elements of the set 9+. Call % = %+ U 9+ the position generated by the xi. (It might then be sensible to call 94 the convex hull of the set {x1, . . . , x,} .)

8x!

G. HEMION

Theorem. position.

Given a finite subset of X. the position generated by the subset is, in fact. a

Proof. We certainly have .%;, s 3+,, and by definition, ;9+ is maximal. Is I,, also maximal? That is. let y E X be such that y :> &, In particular. y is greater than all elements in the finite set. Thus y E gda. QED. Let X be any arbitrary poset. Let Y(X) be the set of all positions in X. G!!(X) is then itself a partially ordered set, with ordering given by I G A o I, C A+ and A+ c I,, for I. A E ?P(X). Clearly, by identifying elements of X with elementary positions in X we have X C Y(X). as a partially ordered set. 11efinirion. The partially ordered set X will be called complete if X = Y(X). That is to say. X is complete if all its positions are elementary. Can every poset be embedded in a complete set? The following theorem answers this question in the affirniativc Theorem.

Let X be an arbitrary partially

ordered set. Then ?P(X) is complete

Proof. Let 2 E ?!(9(X)) be an arbitrary position in the set of all positions of X. The problem is to show that 5 is an elementary position in Y?(X). For this. let ,2,, = (.r E X: 31’ F Z.i with .r E I‘,, }. Similarly

let .A - {v E X: !J’ F: 2, with J* E Y,}.

Let A = A, U A,. Then A is clearly a position in X. and in the identification with Y’(X) we must have 3 being identified with A. QED

of 9(9(X))

Theorem Let X be a confluent. stable, strongly discrete poset. Then 9(X) contluent. stable and strongly discrete.

is also

Proof. The confluence and stability of Y(X) are obvious, owing in particular to the tact that for any A E Y(X) we have A,, = U, s ,.L ~, and A+ = UxP,.C+. As for strongly discrete. let A < 1 in Y(X). Choose .V .: A and ,V ‘=- c in X. Since X is strongly discrete. there are only finitely many positions between .Y and y, thus between A and c. But we have seen that each position in P(X) is really just a position in X. QED In light of these results. one might think that it would be a natural thing to consider only complete sets-that is, one could always take the completion, G!(X) of any given set X. In fact though. this is not really the most sensible way of going about things. Rather than complicating a given situation by adding in more elements to a set X, it is better to recognize the fact that most sets contain a great deal of needless information. It is therefore better to discard this useless information, and thus reduce things to just the essential elements of a given set. These ideas will be developed in the next two sections. 2.5.

Associating elements with positions

Positions are defined in terms of elements of the given partially ordered set. Nevertheless, given some specific position A, it seems clear that some elements are more closely associated with the position than others. It will be useful for us to make this idea more precise

A class of partially ordered sets

803

Definition. Let A C X be a position in X and let u E X be an element of the set. We will say that the position A is associated with the element a if in the smaller partially ordered set X’ = X - {a} we have that A’ = X’ rl A is not a position. We will also say that the element a is associated with the position A in X and we will write a >> A to indicate this situation. Given that a >> A, then, since A - {a} is not a position in X - {a}, even though it is a double cone, it must follow that A - {a} is not maximal, i.e. it is possible to add in some element x E X, with x 4 A either to Au or else to Ati, and we still have a double cone in X. Now if we had a 4 A, then there would be no hindrance to adding in this element x to A in the original set X, contradicting the fact that A is a position. The only conclusion is that we must have a E A. Theorem. If a >> A then a E A. Given that a >> A, then A - {a} is not maximal in X - {a}. It can be made to be maximal by filling it out with some extra elements from X - {a}. What possibilities are there? Since a E A, we have either a E Au or else a E A,: that is, either a is below or above A, respectively. We will then write a < A or a > A, respectively. Let us assume that a < A (the case where a is above A is similar). Could it be that there exists an element u E X - A with u < Afi - {a}? But we have Afi - {a} = A1,, thus even in X we would have u Ab}. Theorem. Let a >> A. Then in X - {a} there is a unique position A’ which contains A - {a}. Furthermore, A’ is also a position in X, with a 4 A’ and A’ # A. What happens when the set X is stable and strongly discrete? Can it be that a E Afi, where a >> A? If so, then there must be a u 4 A, with u < (A+ - {a}). In other words, u 4: a. But then we have seen that there must be an infinite number of elements in the set a* - ufi. On the other hand, each of these elements must also be elements of A*, and this is a contradiction. Theorem. Let the partially and let a >> A. Then a < A.

ordered

set X be confluent,

Corollary. Given the conditions of the Theorem, be associated with any position in X. Theorem. a LX b.

Let I E Y(X)

stable and strongly

discrete,

then only finitely many elements

can

be associated with both a and b E X, where a # 6. Then

Proof. This is rather trivial. If, say, a < b and I is associated with u, then there exists a y > I+ - {a} with y > a. But then y > b gives a contradiction. QED Theorem. Let x E X and I E P(X). wedohavexecIfi*x>r.

Although

we do not have x* C I,

* x < I, still

RI4

G. HEMION

Proof. The first statement follows immediately from the observation that X is strongly discrete. As far as the second statement is concerned. if x > r then there exists some z, E TU with x > z. But then x0 - ;+ C x*, - rfi must be infinite. QED Finally a theorem about convex hulls of finite subsets. Theorem. Assume that the subset (.r,, . . . x,,} C X (n > 1) has the property that for each i. there is an element y, E X with y, > x,, yet y, :> x, for all j # i. Then all of the x, are associated with the position 9. generated by the subset. and no further elements of X are associated with this position.

Prooj’. The fact that each x, is associated with the subset follows from the fact that y, is in the position generated by the subset which results when x’, is removed. Can it be that some other element L $ { R’i . . . .Y,,j is also associated with 9? If so, then there is some further element CI’2 {.rl, . ., x,), with NJIp z. But then z $9,+, according to the definition of 1. This contradicts the fact that all elements associated with 2 must be in 9 b.

2.6.

Essential ancl inessential elements

Definition. Let x E X C b(X). x is called essential if it is associated with itself, otherwise x is called inessential. How can we recognize the difference between essential and inessential elements in confluent. stable, strongly discrete posets? The answer is quite simple. If x is essential, then xii U x9 is not a position in X - {x}. But. as we have seen, xU U xfi can be made into a position by adding in new elements of X - {.x) to it. Where are these elements of X - (s 1 which can be added in to xcL U s+ without destroying the property of the set being a double cone? Can we have some y E X - {s) with y < .Y*, and yet y $x+‘? But then, considered in the original set X, we have y 4 x, therefore there are infinitely many elements of X in x+ - ye. In particular there exists some z E x* with y -4~t. This is a contradiction. Therefore all elements of X - {.r} which can be added in to xu U X~ must bc above .V,, . Theorem. In a confluent. stable. strongly discrete essential o there exists an element v I .Y with 1’ ... .r:

poset X.

an element

x E X is

Definition. For each confluent. stable, strongly discrete poset X, let x(X) be the set of essential elements of X. The set X itself will be called essential if X = x(X). Our goal now is to show that for confluent, stable. strongly discrete posets 8(X) contains essentially all of the ‘information’ contained in the original set X. Theorem.

k(X)

I.et .K. v c X be arbitrary elements.

If .Y.~ - y , f L??then also (x4, - J,)

c7

f i’i.

Prooj. Let us say that ,- 6 .r4) -- J, If : I$ Z(X) then there must exist some further element ; i c. ; with z, Q ,J. (This follows because : is inessential. thus if zti C y then we must have y E zfl. ) Since X is strongly discrete we must end with some essential element In I-*: - 1.. QED Theorem. Assume that the position A E 9’(X) is associated with the element x E X (which is a confluent, stable and strongly discrete poset). Then ,r is an essential element.

A class of partially

y py;

g=,;:;;

ordered

sets

805

x2 1s . not a position in X - {x}, there must exist some element *. Since xv C Au, we have y > xv. Yet y > x. Therefore x must

be esszntial. QED Let S be some separating subset of X, and let X = S- U S+, where S- is the set of elements which are below (at least one element of) S, and S, is the set of elements which are either in S, or above elements of S. Then let Zs(X) = (S- fl Z(X)) U S,. Theorem.

Let A E 9(X).

For any separating

subset S C X we have A fl Z,(X)

E

S@,(X)>. Proof. If A fl gs(X) were not maximal, then either (i) there exists some x < A+ n g,(X) with x $ A, fl Z,(X), or else (ii) there exists some y > A, n ge,(X) with y $ A, fl Z,(X). In the first case (i), the fact that x 4 Au means that there must exist some z E A, with x 4 z. But since X is strongly discrete, we can assume that z E S,, and this contradicts the assumption that x < A, fl Z,(X). In case (ii) we must have A+ - yv # 0. But then (Au - yu) n Zs(X) # 0, which is again a contradiction. QED Theorem.

For any separating subset S C X we have 9(%s(X))

= 9(X).

Proof. By the previous theorem there is a natural inclusion 9(X) C S@,(X)) which can be represented by the mapping 4: 9(X) + 9(%s(X)) taking each A E 9(X) to the position An 2?,(X) E 9(Zs(X)). This mapping is certainly onto. Is it one to one? Let I #A in 9(X). Then we have either (I, - A+) n %(X) # 0 or (A, - I,) n %(X) # 0. Therefore $(I) # $(A). QED

Corollary. Let x E X (a confluent, stable, strongly discrete poset). Then x E ?P(%e,(X)), for every separating subset S C X. Let X be confluent, stable and strongly discrete. Can we assume that the set of essential elements 8(X) also has these properties? Strongly discrete is of course trivial, but stability might not be preserved. For example it is conceivable that above some separating subset S C X, all elements of X might be inessential. But this would surely be a rather ‘pathological’ situation. The more ‘normal’ case would be that Z(X) is also stable and confluent. Theorem. Assume that X is confluent, stable and strongly discrete and that %(X) is also stable and confluent. Then 9(%(X)) = 9(X).

Proof. According to our previous results, for any x C y in g(X), we must have x*) tl S(X) # 0. The proof is then analogous to the proof in the previous theorem. (Ye -

QED

We see then that the information about the ‘structure’-namely a specification of all of the ordering relations-of a confluent, stable, strongly discrete poset X is contained in its set of essential elements, g(X). Generally speaking, we have Z(X) c X Z 9(X). But going to the trouble of specifying which of the inessential positions of 9(X) are supposed to be included in X gives us only a mass of uninteresting ‘trivial’ information which is irrelevant to the structure of the set. Thus, given some arbitrary such set X, it seems sensible to just take its completion 9(X) and work with that-that is, we assume that our set X is complete. But then, our arguments will generally concentrate on the essential set g(X), since that is where the relevant information is concentrated.

Cr. HEMION

m-l

Definition. A confluent. stable, strongly discrete, complete poset is a normal poset. Before going on to the next section, let us once again repeat and emphasize the fact that-assuming that we are interested in discrete mathematics-all of these words in no way imply any particularly unnatural restriction on the class of posets which we are going to consider. Confluence essentially means that our posets are connected through their ordering relations. Note that we are by no means requiring strong confluence (both above and below). On the other hand, it must be admitted that confluence below for all pairs of elements is certainly more restrictive than the weaker condition that arbitrary pairs could be related either above or below. Stability means that we are ruling out the possibility that the structure of the set might be totally altered by removing, or adding in. some particular element. The ‘full’ structure of a poset is given by its completion in terms of the set of its positions. Strongly discrete ensures that this completion is itself discrete.

3. GEOMETRY,

DIMENSION

In this section some ideas will be outlined which might prove useful in an investigation of the global properties of normal posets. There seem to be various possibilities for defining functions on a normal poset which are analogous to distance functions in more usual theories of geometry. Little will be proved here since each possible theorem would involve a number of further assumptions. and this might tend to confuse things in the framework of the present paper. i j

I:lIciidt’NIl

p0.w

The idea of ‘geometry’ is certainly at the centre of much mathematics, but it is seldom associated with abstract, discrete posets. As the name suggests, geometry is concerned with measuring distances between objects, and most people feel comfortable with taking the real numbers when doing the measuring. Mathematical tradition then tells us to take the Cartesian product R”. giving n-dimensional Euclidean space which we can order using the natural Euclidean ordering. But R” is far from being discrete. so it is certainly not the geometry that we are really looking for. On the other hand, perhaps it would be nice to be ;tble to use our familiar Euclidean geometric intuition to think about normal posets. This \vould bc possible if ;I given such poset allowed an order-preserving embedding in R”. for iome appropriate II E ;\;. Now as wc have seen. our ‘normal’ pose& are infinite, and thus thcrc is no reason to expect that such an embedding exists at all. Nevertheless. it will bc 114ul to take ;I quick look at this concept. Ikfinition . Let X be an arbitrary poset and n E h’ a natural number. X will be called an n-dimensional Euclidean poset if there is an order-preserving embedding X - R”. \\ here R” is considered together with its natural ordering. Obviously any n-dimensional Euclidean poset is also m-dimensional Euclidean for all 171 . II. When is ;I set n-dimensional Euclidean’? Specific sufficient conditions are difficult to find. and they are somewhat irrelevant for our further investigations. More interesting are necessan conditions. fk@ition. Assume the poset X contains a set of nz distinct. mutually unrelated points .s = {p,. . . . p,,) with p, ;2<1 pi for all i # i such that for every subset T C S there exists an element yi- E X with s, <: )‘, for all p, E T and p, 4~ ,v,- for all pI E .Y - T. Then we will calI S a differentiated horizontal subset of X of order m.

807

A class of partially ordered sets

Theorem. Assume X contains a differentiated not n-dimensional Euclidean, for any n < m.

horizontal

subset of order m. Then X is

Proof. Otherwise, there must exist a set of IZ + 1 mutually unrelated points {pi, . . . , P,,+~} in R” which are differentiated, in the sense of the above definition. Let H, = {(t, x1, . . .) x,-~): xi E R} be a horizontal hyperplane (of dimension it - 1) in R” which is below all pi, and in fact which is so far beneath the set {pl, . . ., P~+~} that there exists more than one point q E H, which is beneath all pi. Now for each i = 1, . . . , n + 1, the set of points below pi in H,, namely (pi)+ fl H,, is an (n - l)-dimensional ball in H, whose boundary is an (n - 2)-dimensional sphere, Si C H,. Let us concentrate on the particular sphere Sn+l and note that, since the set {pl, . . ., JJ~+~} is differentiated, each Si fl Sn+l is an (n - 3)-dimensional sphere in S,,, , for i = 1, . . . , n. Consider next the set of II - 1 spheres (all of dimension IZ - 4) Si fl S, n Sn+l, 1 G i G IZ - 1, and so forth. Eventually we arrive at a set of four one-spheres (or circles) on a two-sphere. The fact that the original set is differentiated means that these circles are differentiated on the two-sphere. It is now easy to prove by elementary means that this is impossible. QED Remark. On the other hand, for each y1E N, R” contains a differentiated horizontal subset of order n. For example, taking a typical point of Rn to be x = (x0, x1, . . ., x,,-~) with x < y = (y,,, y,, , . ., y,-1) if x0 < yo and ci”=i(x; - yi)2 s (x0 - Y~)~, then the set

{(O, * * -7 O), (0, 1, 0, * * *, 01, . . -3 (0, . . -, 0, 1)) is differentiated. Definition. Assume that the poset X contains n unrelated and differentiated elements, but no set of IZ + 1 such elements. Then we will call X a set of combinatorial dimension rz. Does the converse of our previous theorem hold-namely that a poset X of combinatorial dimension n is necessarily also n-dimensional Euclidean? This is certainly untrue. A rather trivial example is the finite poset {a, b, c} with a txl b and a, b > c. Then according to our definitions, the combinatorial dimension must be one, but clearly this set is not one-dimensional Euclidean, since it is not totally ordered. For each it we can find similarly trivial counter-examples. But finite posets are not really of interest to us. After all, we have already decided that all finite posets are ‘trivial’. What is the situation with respect to normal posets? Does a given combinatorial dimension bring with it the same Euclidean dimension? This hypothesis seems to be untrue, but why bother with a search for specific counter-examples? Instead, the main purpose of our definitions till now was to exhibit examples of normal Euclidean posets of every dimension greater than 1. In the last chapter we have seen an example of a two-dimensional Euclidean normal poset. The higher dimensional examples which we will now investigate are constructed in a similar way to this two-dimensional example. So let it > 2 be some natural number. As before, we will construct our set X C R” (considered with the natural ordering induced from R”) in two parts, X = X, U X-, which are in the ‘upper’ and ‘lower’ half-spaces of R”, respectively. To begin with, X, consists of the points X+ = {(m,

kl, . . ., k,-1): m, i, ki E 2 with m, i > O}.

The ‘bottom layer’ of X, is the set of points with a vanishing first coordinate. This can be put into one to one correspondence with the integers Z, and then associated with the real line {(x, 0): x E R} in R 2. Then we can simply take X- to be the same set as in our

two-dimensional example. Proceeding in this way, we cannot assert that the set X which we have constructed is n-dimensional Euclidean, but it is n-dimensional combinatorial. As before. the problem is to show that X is stable and strongly discrete: but stability follows from the two-dimensional case of the previous section. Strong discreteness follows from the fact that we have chosen two-dimensionality in the lower sets and proved strong discreteness there. Finally. it is important to mention the geometric meaning of the relationship of association between positions and elements of X. As we have seen, even if the elements of the poset X can be embedded in a strictly order-preserving way in Euclidean space, still, many positions in X may not be representable by Euclidean double cones. But let us consider a position I‘ which is. in fact, nearly a Euclidean double cone in the given embedding in Euclidean space. Let u E X be associated with T. That means that there is some ~1> Fib - {a} with y > a. But this can only occur if a is near to the boundary of the lower cone of I-. Put another way. we expect that the lower cone ([!i is contained within. and yet it is near& tangent to, the lower cone I‘, 3.2

Near1.v Euclidean p0.w.~

When thinking about various examples of posets and the question of whether or not they are Euclidean, it soon becomes apparent that in many cases a Euclidean structure might apply if we could allow ourselves to ‘distort’ the ordering structure in some way. For example the physicist might think of the Schwarzschild metric around a star (but not going down to a ‘singularity’): such a metric can bc thought of as being a slightly distorted version of the more simple. flat Lorentz metric of Minkowski space. By taking some discrete set of points in this distorted space. we obtain a discrete poset which is not strictly Euclidean, but rather it is ‘nearly’ Euclidean. ln fact, it seems possible to describe this situation without reference to Euclidean space itself. Surely this is a good thing, since Euclidean space is, after all, a rather complicated and artificial mathematical object. Our starting point is the observation that a discrete one-dimensional Euclidean poset X is nothing but a totally ordered set. Or expressed in a rather pedantic stvle, one can say that there is a single totally ordered subset Z of X (namely the set XV itself: E =: X) with the property that no two different points of X have the same set of ordering relationships with the set Z:. The more general definition is the following. Definition. .4 subset Y C X of the poset X will be called a distinguishing anv 11f h in X. we do not have at n Y = h. f? Y-.

subset if for

Definition. The discrete poset X will be called nearly Euclidean of dimension n if there exists a distinguishing subset B C X such that any separating subset of E contains at most n elements. One might think that it would be a good thing to have the ordering structure of the set following from the ordering structure of the distinguishing subset. For example, given s and .V in X then it is a simple consequence of the ordering axioms that if x s y then it follows that all elements of the distinguishing set Z which are above y are necessarily also above I. Similarly the elements below s in Z must also be below .v. But can we expect the converse to hold? i.e. if s and J have this property with respect to the distinguishing set 2 then we must have .Y < y. In general this is not true, as one can see by thinking about the ordering structure of n-dimension Euclidean space for II 2 3. As far as four-dimensional space is concerned, we might also describe the situation by

A class of partially ordered sets

809

saying that X contains two disjoint totally ordered subsets A = {. : * < u-~ < a0 < aI < . . m} and B={.-< b-i < b0 < bi < +- e} which are distinguishing in the sense that no two different points x # y in X have the same ordering relations with A and B. This is analogous to the situation in R4 where we can choose two skew lines, for example A = {(t, 0, 0, 0): T E R} and B = {(t , t/2, 1, 0): t E R}. (Although here, we generally have two distinct points of R4 having identical ordering relations with the lines A and B.) In some cases it may be a reasonable technique to argue that if we are interested in some ‘local’ region of X, such that the elements of the distinguishing subset B are ‘far away’ (perhaps measured using one of the ‘natural’ distance functions of the next section), then the local embedding of X in Euclidean space will be nearly order-preserving.

3.3.

‘Natural’ distance functions

The work to now only serves to demonstrate the fact that Euclidean geometry is seldom very useful in the investigation of normal posets. Instead of searching for complicated embeddings in our preconceived idea of what ‘geometry’ should be, it is surely better to define distances between the elements of a given poset in a more direct and natural way.

Definition. Let X be a normal set and let a < b be two elements of X. The retarded distance d,(a, b) from a to b is then the number of essential elements of X in the set b, - av. On the other hand, the positional distance d,(a, 6) is the number of positions of X which are between a and b. Note that if a rxl 6, then we have left the distance between a and b undefined. There are various possibilities for defining d(a, 6) in this case as well. For example, it could be taken to be the number of essential elements of X in (au - bti) U (b& - a+); another reasonable possibility would be to define it to be the smallest possible value of d,(c, d), where cEa+rlnb, and dEa*nb,. A number of further, equally plausible definitions could also be proposed, and we could proceed to prove many theorems concerning such definitions in the case a DU b. (In fact, we could also list many further plausible alternative distance functions in the case a < b.) But for our present purposes, and to keep matters within reasonable bounds and directed towards very specific goals, let us restrict ourselves here to the two distance functions of this definition. No effort is made to bring the idea of a metric into play here. For example, referring to the positional distance, if we have three elements a < b < c, then since every position which is either between a and b or between b and c is also between a and c, while a position cannot be both above and below b, it follows that d,(a, b) + dp( b, c) c d,(a, c). This is just the opposite of the triangle inequality for metric spaces. But this reflects the situation with respect to the natural ordering of our usual Euclidean space R”, which is best described using the usual Lorentz (pseudo)-metric, which, of course, also fails to satisfy the axioms for a metric space. On the other hand, it is equally easy to see that we do have d,(a, b) + d,(b, c) 2 d,(a, c), for all a < b < c. What if our poset X happened to be n-dimensional Euclidean? Could this situation be identified by examining either the retarded, or the positional distance functions within the set? Certainly one would expect that it would be possible to prove some theorems in this direction, but there does not seem to be a simple relationship which can be easily proved. Going in the other direction, we can think about how our ‘natural’ distance functions might be formulated in n-dimensional Euclidean space R”. Let a = (ao, a,, . . . , u,-~) and

h -: (h,,,

h,.

. . I.

h,. , ) with

N -: h. Then we might identify d(a. h) with the volume of R”

between a and h. Let

be the Lorentz distance from a to 6. Then the volume of Euclidean space between a and h is not directly proportional to the Lorentz distance. Instead, it is the Lorentz distance taken to the power n, where n is the dimension of the Euclidean space under consideration. It may be useful to attempt to adopt this result to the theory of discrete posets. Arguing 111analogy to the situation in Euclidean space. we might specify that three points u < h < c in the discrete poset X are ‘equally spaced along a line’ if &(a, h) = d,(h, C) and there is no : e X between a and c with both d,(a. Z) 2 d,(a. h) and n,(;, c) 2 d&b. c). So given that (I < 0 I. c are equally spaced along a line in ,Y. then we can proceed to compare the numbers d,(a, h) and ~!,,(a. c). Let us say that there is a number k 2 1 such that rl,,(a, c) = -“d (a . b) for arbitrarily widely separated equally spaced elements along a line in .Y. Then o:e might specify that the ‘natural’ dimension of X is the number k. Such ideas would undoubtedly lead to many further thoughts which. however. we will not pursue here. What is the situation with respect to the ‘retarded distance’ in R”? If n < h, then there is simply an infinite volume of Euclidean space in the set b, -- lzg, so our definition would seem to be quite inappropriate here. But what is the reason for thinking about these volumes in Euclidean space in the first place ? We are really interested in discrete posets .%‘, which might be embedded in R” in some order-preserving way. As we have seen, the normal’ posets which we are interested in are always infinite-though discrete-and in fact they are also infinite in ‘horizontal dimensions’ (see Section 2.2). Thus we might expect that it would be possible to embed X in R” such that, when taken ‘on the average’ according to some appropriate definition, the number of elements of X which are expected to be in some region of R” would be proportional to the volume of that region. But this simple expectation is bound to be disappointed. Our set X is strongly discrete, so there can only be finitely many elements in h+ - aJl. and this is at variance with the infinite volume of that region in Eudidean space. On the other hand, our example of a normal set in Section 2.3 shows that we can still have an embedding in Euclidean space. but we must expect that the ‘density’ of the embedded points decreases rapidly-exponentially-when going ‘downwards’ with respect to the natural Euclidean ordering. A simple calculation shows that such an exponential decrease will generally allow us to preserve the property of having strong discreteness.

3 4.

Whv few dimensions~)

The world seems to favour four dimensions, so perhaps it would be a useful exercise to set if, in some sense, four dimensions are special in the theory of normal pose% Of course we cannot assert that an arbitrary such set X is four dimensional with regard to any of the definitions which have been dealt with in this section. Our constructed examples preclude any such result. On the other hand, it is interesting to pose the question in another way. Is it true that a ‘typical’ normal poset could be expected to have some specific dimension, or. remembering that such sets are always infinite, could we expect that a ‘typical region’ of a single normal poset would have-a characteristic dimension? So what is ‘typicai’; how can we investigate the ‘probabilities’ of possible normal posets? Such questions will be dealt with in the next section. For the moment. let us think about the following question.

A class of partially

ordered

sets

811

Question. Let x E X be a ‘typical’ element in a ‘typical’ normal poset X and let 9 C X be a position associated with x . What is then the retarded distance d,(x) 9) from x to 9? Of course this question is, on the face of it, practically meaningless. Given any positive number, one might expect to find elements associated with positions whose retarded distance from one another is that number. So this is not the thing that we are really interested in. Rather it is the question of how the ‘probability’ that a given element x E X is associated with positions in X in some distant region of X varies with the retarded distance from x to that region. We will derive a simple relationship between the retarded distance and the probability of association between elements and positions. In fact we will derive this relationship in two different ways, and the fact that these two derivations must lead to the same result will give us a statement concerning the dimension of the set. To begin with, it is important to restrict ourselves in the kinds posets which are to be considered, and in the kinds of positions within those posets which are to be brought into question. In fact, we only want to investigate essential posets (X = Z(X)) which are Euclidean, and positions which are very nearly Euclidean double cones. One way to do this is to restrict ourselves to positions which are ‘directly below’ elements of X. Definition. The position I E Y(X) is directly below the element (I E X if a > I and there is no position A between a and l? (i.e. u > A > I). Since positions are determined by their lower sets, it is clear that there is a unique position I+ which is directly beneath the element a E X. (This is not the elementary position a, owing to the fact that a is an essential element of X.) Given some typical a E X, what can be said about the position I directly beneath a? We know that only a finite number of elements of X are associated with I. Can we say that there is some definite number which is the ‘average’, or expected number of elements associated with positions like I-averaged over the whole set of essential elements of X? In general this is certainly not true, but let us assume that X happens to belong to the class of sets where there is, in fact, some definite such expected value. (Of course some method of averaging would have to be specified in order to make this definition more precise, but for our present purposes a less formal style of argumentation seems more appropriate.:) Going beyond this, we will further specify that there is some definite real function p: iV + R which is such that for a typical position such as I which is directly beneath an element u E X, the probability that a given element y of X which is beneath a and yet on the boundary of uv (that is, yfi fl uU = 0) is associated with I at the retarded distance r E N from a is p(r). So the question is, what is this function p? (I) First derivation We have our given set X, containing the essential element a and the position I which is directly beneath a. Let us now remove some element z E X from X. If z is not associated with I then this action will have little effect on the situation, but if z >> I then the removal of z will cause the position I to disappear, and instead some lower position will now become the position which is directly beneath a. The next step is to remove not one, but many elements from X, so that it becomes uniformly ‘thinned out’. What is meant here is that-on average-some fixed proportion of the elements of X are removed to obtain a ‘thinner’ but ‘similar’ poset X’ C X. Remember though, X is infinite, so we will have to remove infinitely many elements to obtain X’, which remains itself infinite. We take the idea that X’ is ‘similar’ to X to mean that there is some ‘thinning factor’ cr, which is just a real number, such that if u and u are two ‘typical’ elements in X’ , then if the retarded distance between them in X was

ti,(u, LJJ. in the thinner set X’. the retarded distance now becomes N x d,(u, u). (Note that it is easy to prove that the removal of any finite subset of an essential poset leaves us again with an essential poset.) But since X’ is similar to X. it has again the same function /J. where the retarded distance is now measured within X’. All of this sounds very much like a ‘gauge invariance’ argument in physics. In particular. we have p(r) = cup(ar). But now we can choose N = l/r to obtain that p(r) = p(l)/r. i.e. the expected number is inversely prqportional to the retarded distance. Of course one should bear in mind the limitations of this argument. The fact that X is discrete means that continuous ‘gauge transformations’ are impossible. Thus-at the most-the argument can only be expected to be of limited relevance, and in particular for small values of r. where the discreteness of X becomes most noticeable. it becomes irrelevant.

I‘hi:, ttme we would like to think in more geometric terms. Assume that the set X is embedded in an order-preserving way in the Euclidean space R”. Assume furthermore that ,I’ really is n-dimensional, in the (rather vague) sense that it is distributed more or less homogeneously throughout horizontal hyperplanes in R”. and its ‘density’ in R” varies smoothly in the ‘vertical’ direction. Again. let u E X be some typical element, and let 1’ .:’ N be directly below U. The question is. what is the probability that some given element 1’ ‘1 N with the retarded distance d,(a, J)) .= r which is on the ‘boundary’ of a;! (i.e. \ n (1\, =--!Z) is also associated with T? As we have seen. both a and J are represented by double-cones in R”, and in fact )‘; is represented by a downwards pointing cone which is nearly tangent to the downwards pointing cone representing a,. Generally speaking, we expect the vertex of yti to be some (Euclidean) distance down along the cone a% from the vertex of (I :, . This represents the retarded distance from JJ to a. .4t this stage it is sensible to take a different view of these things. Let us take a horizontal hyperplane ff, C R” with t being sufficiently small that it is below both a and )‘. The space H, can be identified with the n - 1 dimensional space R”. ‘. Then we examine the intersections u,, n H, and J* fl H,. They can be thought of as being n - 1 dimensional balls in R”.. ’ whose boundaries arc II - 2 dimensional spheres. The sphere representing ?: is smaller than the sphere representing a-- the greater the retarded distance from J’ to (I in X. the smaller the size of the sphere representing ,Y-and it is contained in the ball representing 0. In fact. the two spheres are nearly tangent to one another in R” ‘. But remember that we are considering various possibilities for positions which are directly beneath a typical element such as (1. Thus we should investigate all of the different elements such as !J which are nearly tangent to a. Since X is strongly discrete, there are only finitely many such elements and so we can choose If, to be so far ‘downwards’ in R” as to be beneath all these elements. Therefore they are all represented by II - 2 dimensional spheres of different sizes in If ,. spaced around, and nearly tangent to, the sphere representing U. What does all of this mean? Let us begin by asserting that since we know nothing particular about the way that X is embedded in R”. then the possible spheres representing elements like y are nearly tangent to the sphere representing a. and furthermore they are ‘randomly’ distributed around the sphere representing u. What is the ‘probability’ that a position like l- is associated with an element like J? One way to investigate this question is to note that if .v >> T then in particular there would be an element MI E A.’ with y Q MS and vet T,: - {y 1 C MIX,.. Thinking in terms of the n - 2

A class of partially

ordered

sets

813

dimensional spheres in H,, we have that the sphere representing y will not be completely contained within the convex hull of the set of all the spheres representing the other elements which are associated with I. So the question now becomes, how does the probability of this situation occurring vary with the diameter of the sphere representing y? We will assert that this probability is proportional to the area of the sphere representing y which is near to the sphere representing a. Translating this into purely geometrical terms, we consider the element y as being represented by the PZ- 1 dimensional ball B, in H, and the element a as being represented by the n - 1 dimensional ball B,, with B, contained within B,, yet nearly tangent to it. We will then assert that the probability that y is associated with I is proportional to the area of the boundary of B, where B, approaches it nearly. To be more specific, let Y2 E Rn-’ be the standard unit (n - 2)-sphere: n-1

s”-2 =

(x1, . . .) x,-1): {

xx;

= 1 .

i=l

1

Furthermore, let r > 0 be some small number (with r being, in any case, much smaller than unity) and let S, be the (n - 2)-dimensional sphere in R”-l with radius 1 - r, centred on the point (r, 0, . . ., 0). Finally, let 0 < 6 < r, with S being much smaller than r. So the question is, what proportion of the unit IZ - 2 sphere Y2 is within a distance 6 from S, for varying values of r? The answer depends on the dimension of the spheres in question (that is, the dimension IZ - 2), and in fact an elementary geometrical argument shows that it is proportional to

But, according to our first derivation,

we expect the relation

l/r.

This can only be true if

n = 4. 3.5.

Reformulating

this argument

in terms of natural distance functions

The argument of the previous paragraph depended on a number of very restrictive conditions on the partially ordered sets X. It was assumed that the sets were Euclidean of some particular dimension, and then given this, it was further assumed that they admitted a ‘homogeneous’ embedding in some R”-in some appropriate sense. Furthermore it was assumed that a certain definite probability distribution existed to describe the elements associated with positions directly beneath essential elements. Now this last condition was formulated in terms of the retarded distance in the poset X. This is a natural distance function in the sense that it is defined in terms of the set X itself; it does not depend on some particular embedding in a Euclidean space. Is it possible to formulate the rest of the argument in a more ‘natural’ way, in this sense? To begin with, let a E X be some particular essential element. Let A be the set of elements of X which are on the boundary of av. Recall that this means A = {y E X: y < a, av 17 yfi = 0}. Let us now examine A on its own, seeing if we can find some natural geometric structure. Given any two elements y , z E A we must have y rxl z. But we have only defined the positional and the retarded distances for pairs of elements which are related to one another. For this new situation with y DU z we might define a new kind of natural distance-called d,(y) z) -where either the number of elements in yv - zv or else zv - yv is taken, whichever is smaller. This new function is certainly not a metric. For example, in general we will not expect the triangle inequality to hold. On the other hand, if n, y and z E A with d,(x, y) equal to

the number of elements in .t+ .- y4!. d,(y, z) equal to the number of elements in y,,, - z4, ;) equal to the number of elements in xc - I+ then the triangle and finally d,(.r. inequality n,(s, .v) + d,(y. z) 3 d,(x. ;) will hold, since each element of x9 - z& is tither in x+ .- .vc or in (xII fl yG) - zU. On the other hand, given some specific y E A, we might examine the set of other elements of A which are near to y, measured using the d, distance function. It is found that elements t E A can be near to y even if they are far away. measured according to the retarded distance function. Thinking in terms of Euclidean space, one might say that the distance function d, measures the projections of the points, projected downwards along the surface of the cone from the point a. projected onto an tt - 2 dimensional sphere. .At an!’ rate. we can take the distances tf, between all pairs of elements of A and this is simply a set of numbers for each cone of points A beneath each essential element a E X. Here it is again necessary to make some assumptions on the average behaviour of typical posets like X. There is no reason to favour any particular y E A over other elements of A. Thus we might expect to have some typical distribution of distances to other elements of A. independent of the particular element. (Note though, that the retarded distance of y to a might also be expected to play a role here.) Given all of this, we proceed to construct a geometry in this ‘projected’ space by seeing how the distances between various sets of points fit together. It would seem reasonable to expect that such ideas as curvature could be formulated in this geometry of A. and we would expect to have a constant curvature since, after all, no particular assumptions are being made about the elements of A: thus there is no reason not to expect a constant curvature. This would then lead us to postulate :I spherical structure of some dimension. Then the question arises as to when an element \’ k. .Z is likely to be associated with the position directly beneath n. ‘Thinking about the purely geometric argument of the last paragraph, we might postulate that this probability is proportional to the number of elements of A which are near to T. measured using the d, distance function. This will then lead to the same argument as before. All of these ideas could bc taken still another step further. The argument in the second derivation of the last paragraph shows that the retarded distance is related to the ‘density’ of elements of X below the region of X which is being measured. What happens if this density is not homogeneous. as we assumed in the previous argument, but rather it varies through horizontal hyperplanes through X (with respect to some perhaps only approximately order-preserving embedding in R”). Given a greater density of the elements of X beneath some element a E X then, following the argument through, we must assign a greater retarded distance in the Euclidean geometry. when compared with the retarded distance calculated within the set X. This would lead to something analogous to the curved ordering structure which can occur in a partially ordered manifold which has a non-vanishing curvature. 4. PROBABILITIES d. I

Genmd

IN NORMAL POSETS

considerariott3

The considerations of the last section show that much might be gained if we formalize the ideas of statistics or probability within the class of normal posets. As we have seen, all such posets have infinitely many elements and furthermore they are infinitely ‘wide’ in the sense that all separating subsets are infinite. So what is a sensible way to formulate a theory of statistics in such infinite posets? To begin thinking about the problems involved, it might be interesting to consider the set of integer numbers, which surely must be the simplest imaginable infinite poset. One might

A class of partially

ordered

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ask, what is the probability that a ‘randomly chosen’ integer be even? Surely the sensible answer is that this is l/2, but why is this so? If we write 2 = (0, +l, +3, +2, +5, +7, +4. . . .} then it might seem that, in fact, the probability of an integer being an even number is only l/3. But it is clear what the ‘mistake’ is here; we ignored the essential structure of the integers, namely that they form a totally ordered set. So let X be some ‘random’ normal poset, and let a < b say be two related elements in X. What is the expected structure of X between a and b? Since X is discrete, there can be only finitely many elements of X between a and b; therefore-at least-we are not confronted with the arbitrary choices which are always involved in assigning a probability measure to an infinite sample space. But on the other hand, something is still missing. If we think about the integer numbers again, we might take two integers a < b and count how many even numbers there are in the interval between u and b. This is quite straightforward, owing to the fact that the specification of a pair of numbers brings with it a knowledge of the size of the finite set which is to be considered. Things are different in X. Given only the information that a < b in X, then we do not know how many elements are between them; thus it is not clear how to answer our question about the structure of the set a,, n b,. The obvious thing to do here is to simply specify the number of elements which are between the two given elements a and b. Thus the question becomes: let a < b be two abstractly given elements, and let n E N be some positive integer. Consider the set of all possible normal posets with two such elements and n elements between them. What are the relative probabilities for the various possible configurations which can arise? Translating this back into a statement about the integer numbers, surely this is a very sensible framework. The actual values of a and b are no longer relevant; rather the distance between the numbers, i.e. b - a, is the only thing of importance. Although foi any 12, there are infinitely many pairs u and b with b - a = 12, still for many reasonable questions which can be asked about the probabilities of numbers, that will lead to no But of course many questions concerning the probabilities of particular complications. numbers-for example the prime number theorem-do not fit into this framework. One sees then that in order to speak sensibly about probabilities, it is most important to think first about the basic structures of the objects at hand. Within abstract probability theory, we can simply assign any arbitrary probability measure to a given sample space, and then take things from there. But the question that interests us here does not concern possible rather absurd and non-sensical probabilities which might be assigned to the class of all normal posets. Rather it is, what is a sensible probability theory which respects the ordering structures in the class of sets which we are considering? Now we have seen that for any normal poset X, we can examine the set of essential elements Z(X), and also the set of positions 8(X). (Recall that E(X) c X C 9(X) but if X is genuinely between the two sets, then this is only due to an arbitrary choice of positions which are being added onto g(X). Allowing this arbitrary choice would certainly only serve to confuse the whole question of probabilities!) Thus given a < b and IZ E N, one could look at all possible sets with n elements of Z(X) between a and b, or -alternatively-one could look at all possible sets with IZ elements of 9(X) between LI and b. Which of these sets should we choose? In the work which follows, I will assume that all of the elements of 8(X) are to be considered. Put another way, it will be assumed that the given set X is complete. The alternative would be to say that X is essential, but this would be the false alternative, leading to a theory which does not properly respect the structure of X. Why is this? To take a most extreme example, let us imagine that we have a specific set X, together with two (essential) elements a < b with a0 n 6, rl E(X) = 0. This would lead us to

believe that a and b are close together. However, we have seen that the ‘natural’ distance functions are the retarded distance and the positional distance, and these are not at all represented by the number of essential elements between a and 6. Many further, more or less philosophical ideas could be brought into play here, but rather than losing ourselves in senseless speculation it is better to simply specify a definite rule, and for us the rule is that the set X is complete, and given a number n, we will examine all possibilities for two elements N
Onlv finitelv

many

sets to consider

Our original question has now become, given a ‘typical’ complete normal poset X (with X = .;P(X)j. and two elements a C: h in X with n elements of X between a and b. what i:an we say about the structure of the set a,, fl h,, (for simplicity, assume that a, h E ‘6(X))? Since X is strongly discrete, WC must have h, - ati being finite. In fact, we can strengthen this statement to say that, given that there are at most n elements (i.e. positions) of X between II and h and given that u is essential, then there are at most n essential elements of X in by, --- (i+. ‘This follows from the observation that any finite collection of elements generates a position, as was seen in paragraph 2.4. Thus, remembering that a position is completely determined by its lower set. where only the essential elements play a role. we see that it is not necessary to consider all of the infinitely many complete normal posets which contain n elements between two given elements a <: b. Instead. we need only consider the finite number of possible sets h, - aU which can lead to this situation. And in fact we only need consider the set of elements (at most n) in ill,, - ub j fl %(Xj which lead to the situation of having n positions between a and 6. What is a sensible rule for assigning probabilities to this finite set? Surely the only possible rule is to assume that all of the finitely many different configurations which are possible for the set bV - Q. leading to n positions between a and h, are equally likely-they should all be assigned equal probability. After ail, we have no particular information about the set X which would to lead us to choose some other rule. This lack qt’ information brings with it-one could even think of this as being the definition of the term ‘lack of information--the requirement that all possible configurations are equally likely. J.?

Taking

a larGger GM.

of thin,es

I‘he ruic which was formulated in the last paragraph is certainly quite clear, but it may bring with it a certain difficulty. Let us say that a CCh are given in X, together with a number n of positions between the two elements. Let us now say that we have a further element (‘ > h and a number n ‘. and our task is to investigate the probabilities for the various possible configurations of n’ positions between h and c. This brings with it also the question of the possible configurations between u and c. Jt is obvious that the calculation of the probabilities for the pair u c: b is related in some way to the calculation of the probabilities for the pair a c. c. Is there some particular effect which we can identify which might cause us to change our initial specification of the probabilities? Before going into this question. let us be clear about the rules which are to be followed in assigning probabilities. When looking at a pair a < b, the rule is that all possible configurations of essential elements in h, - a, which lead to some definite number n of positions between a and h are equally likely. This rule was formulated in order to describe the lack of any further definite information which we have to describe the set X. However. if we look at some larger framework LI < c. then for each of the possible configurations with n positions between a and b. there is some finite set of possible configurations of

A class of partially

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cv - av giving n’ positions between b and c, and which contains the given set of n positions between a and b. Our rule then is that for each fixed n’, the probability of a given configuration in b, - av with it positions between a and b is proportional to the number of configurations in cv - av giving n’ positions between b and c, and containing the given configuration in bv - av. In general, as it’ increases (with PZ fixed) we will expect to have changing probabilities for these configurations in b, - av. But it will be assumed (and this is a condition on the class of sets which are to be considered) that in the limit of large IZ’, definite limiting probabilities for the configurations in b, - av are reached. These then are to be taken as the true relative probabilities. 4.4.

The role of the essential elements and the formation

of chains

Only the essential elements in b + - av can be associated with positions between a and Therefore two possible configurations of b, - av are different if and only if F(X) rl (bv - av) is different for the two configurations. Thus, in order to make a list of all possible configurations leading to IZ positions between a and b, we need only look at the possible different configurations of the essential elements in b, - av, where two configurations are the same if and only if there is an order preserving isomorphism of the the sets %(X) 17 (bv - au). (Note that the relations of these essential elements to other elements of X outside b, - av play no role in this.) So which configurations of essential elements lead to n positions between a and b? To answer this question, we should recall that any finite collection of essential elements of X determines a unique position in X. Now we are assuming that the particular element a E X is also essential. Therefore, given some set {x1, . . ., x,} of essential elements in b, - av, we may choose any subset {Xi(l), . . . , Xi(p)} say, and we then have a unique position between a and b, determined by the set {a, Xj(l), . . . , Xj(p)}. Recall though, that the elements which are within the ‘convex hull’ of the other elements of this set are not associated with this position (see paragraph 2.5). When does the situation occur that one of the xi(i) is contained within the convex hull of the other elements? We can certainly identify a standard situation which always brings with it this condition. The situation is that there is some Xi(k) in the set with > Xi(i), for then Xi(i) is certainly within the convex hull of any set containing xi(k). This implies that if we have a set of essential elements F(X) rl (bv - au) with many elements of the set related to one another, then the number of positions between a and b which that set generates will be less than for some other set containing the same number of essential elements in b, - av, yet with few elements related to one another. In other words, for a fixed number IZ of positions between a and b, a set of essential elements Z(X) rl (bv - au) containing many ‘chains’ of totally ordered subsets will contain more elements than another set where few relations between the elements occur. So let us now look at things in a larger framework a =C b < c. Let some given set of essential elements in bv - av be fixed, and we take some n’ to be given as the number of positions which are between b and c. We examine how many different sets of essential elements in cv - av are compatible with this situation. There are essentially two opposite cases to consider: (i) the essential elements of b, - av form chains of totally ordered subsets, and (ii) the essential elements of bv - av have few ordering relationships between themselves. In either case, we have a certain number of essential elements in cv - b, which are not in b, - av and which also serve to generate the set of positions between a and c. If FZ’ is much larger than n, then we will have many more essential elements in cv - b, in comparison with those in the smaller set b, - av. Again we examine the mechanism for creating positions between a and c, based on finite sets of essential elements in cd - av. Many of those finite sets will contain no elements in b, - av, and b.

xi(k)

many more will contain a few elements from h, - uu. combined with other elements not in that set. In general then, we can assert that for very large n’. a vanishingly small proportion of the positions between u and c are associated with more than one essential element in 6, - a:, Thus for both cases (i) and (ii), we expect nearly the same number of essential elements in cti - 6,. But then, in case (i), where chains of totally ordered subsets occur, there are altogether more essential elements in cu - uv, therefore more possible different configurations. Therefore case (i) is more probable than case (ii). And in fact, since we are taking a limit of ever larger values of n’, we conclude that case (i) is overwhelmingly more probable than case (ii). 1.5

C‘hcrin-like wtwturcv

We have reached the conclusion that within a class of normal posets which admits a sensible probability theory, chain-like totally ordered subsets of essential elements are likely to occur. What properties do these chains have? To begin with. it is clear from the argument that the chains should be as ‘dense’ as possible. That is. if say s: c x2 cr.1. . ‘:. I,,, is a segment of such a chain, then that segment should be maximal in the sense that there is no y E x(X). not in the chain, with .t- -: \‘ a’: I,*,, for some appropriate index i. That is, at every point of the chain we have 1 ‘_ .\, Li. with no essential elements between Furthermore. given a longer segment 1, 7 .y; ,, .y’: .I ,: x ,H .T 6. say. then the situation is most likely where there is no other possible totally ordered segment between the fixed elements a and h. with more than M elements. Thus-recalling our discussion of the geometry of normal posets-we can say that a *straight’ segment. and thus a chain which is as straight as possible, is most likely. Would it still be likely that a ‘randomly given’ normal poset X can have gaps in its chains: that is. segments such as x’] ‘:. rz c .:. x,,. where the retarded distances d,(.r,, x,, , j between pairs of related elements are not all approximately the same size? Of course such a chain is possible. but if we examine the probability theory which must follow from our assumptions, then we would expect ‘on average’ a certain average retarded distance between adjacent pairs on a single segment. Nevertheless, in order to ensure more regular chains, further assumptions on the class of posets which are to be considered must be made. It is possible to have more general ‘compound’ chains, which are not simply totally ordered subsets of X? The idea here is that it may be likely that two or more chain-like structures run near together within X (measured using the ‘natural’ distance functions within the poset) for a longer distance. It may even be possible to show that more generalized ‘pseudo-chains’ appear, which are not strictly totally ordered subsets, but rather they could appear in a spiral-like pattern of self-replicating structures. Such a structure would be expected to be self-similar when shifted upwards or downwards in the ordering of X. But it may be possible to construct such ‘pseudo-chains’ which are not symmetric with respect to an order reversal of the pattern. It is clear that these ideas give scope for many further developments which would go far beyond the framework of the present paper. For example, it is possible to speculate that the probabilities to be encountered in these chain spaces may be subject to the rules which were described in an earlier paper [4). NOTES ON THE LITERATURE Considered purely \ubiect of any earlier

as mathematics. mathematical

this paper investigations.

is concerned with a subject which seems There is a literature concerning abstract

not to have been the partially ordered sets.

A class of partially ordered sets

819

considered on their own, but this is mainly concerned with finite sets. On the other hand there are investigations of partially ordered structures which are more motivated by questions of physics. An example of this is [I]. My previous papers [2], [3] and [4] were concerned with developing the ideas in the present paper, but motivated more by questions of modern physics. It is an old problem to explain why our normal, everyday world of physical reality is 3 + 1 dimensional. If we leave aside the currently fashionable speculations on the geometric significance of the extra degrees of freedom involved in non-Abelian gauge theories, then a good general survey is given in [5]. Inspired by two papers of Mirman, namely [6] and [7], I wrote the paper [8]. More recent investigations into the question can be found in [9] and [lo]. Finally, the last section of the present paper is concerned with establishing a theory of probabilities in discrete, but infinite, partially ordered sets. The statistical effects which we were looking for are very specific to the details of the definitions chosen for specifying the class of partially ordered sets considered. These have certainly not been described in any earlier paper. Nevertheless there are a number of recent books such as [ll]. There it is shown that probability theory can be developed in a discrete framework, freed of the usual complications brought in with measure theory. 1. P. M. Alberti and A. Uhlmann, Stochasticity and Partial Order. Reidel, Dordrecht (1982). 2. G. Hemion, A quantum theory of space and time. Found. Phys. 10, 819 (1980). 3. G. Hemion, A discrete geometry: speculations on a new framework for classical electrodvnamics. ht. .Z. Theor. Phys. 27, 1145 (19%). - 4. G. Hemion. Ouantum mechanics in a discrete model of classical nhvsics. ht. J. Theor. Phvs. , 29. 1335 (1990). \ I 5. J. Dorling, ?‘he dimensionality of time. Am. J. Phys. 38, 539 (19?Oj. 6. R. Mirman, The reality and dimension of space and the complexity of quantum mechanics. Znt. J. Thor. Phys. 27, 1257 (1988). 7. R. Mirman, The dimension of space-time. Lett. N. Ciment. 39, 398 (1984). 8. G. Hemion, Reality as complex space. Znl. .Z. Theor. Whys. 28, 1371 (1989). 9. M. S. El Naschie, Average symmetry, stability and ergodicity of multidimensional Cantor sets. N. Ciment. 109,149 (1994). 10. K. Svozil and A. Zeilinger, Dimension of space-time. Znt. J. Mod. Whys. A 1, 971 (1986). 11. K. Jacobs, Discrete Stochastics. Birkhluser; Base1 (1992).