Pacific Journal of Mathematics ON n-ORDERED SETS AND ORDER COMPLETENESS L. G. NOVOA Vol. 15, No. 4 December 1965 PACIFIC JOURNAL OF MATHEMATICS Vol. 15, No. 4, 1965 ON ^ORDERED SETS AND ORDER COMPLETENESS LINO GUTIERREZ NOVOA In this paper, the notion of an n-ordered set is introduced as a natural generalization of that of a totally ordered set (chain). Two axioms suffice to describe an w-order on a set, which induces three associated structures called respectively: the incidence, the convexity, and the topological structures generated by the order. Some properties of these structures are proved as they are needed for the final theorems. In particular, the existence of natural ά-orders in the "flats" of an ^-ordered set and the fact that (as it happens for chains) the topological structure is Hausdorff. The idea of Dedekind cut is extended to ^-ordered sets and the notions of strong-completeness, completeness, and condi- tional completeness are introduced. It is shown that the Sn sphere is s-complete when considered as an ^-ordered set. It is also proved that En, the ^-dimensional euclidean space, fails to be s-complete or complete, but that it is conditionally complete. It is also proved that every s-complete set is com- pact in its order topology but that the converse is not true. These results generalize classical ones about the structure of chains and lattices. IL n-Ordered sets* An element of the cartesian product Xn+1 n of a set X will be called an ^-simplex and denoted by σ = (s09 s19 , sn) where s< e X for every i. The class of even permutations of this sequence is called an oriented ^-simplex and denoted by | σn | = \s09s19 , sn |. The class of odd permutations is another oriented n ^-simplex denoted by | — σ | = | — (s0, s19 , sn) |. The set of all oriented w-simplexes of X will be denoted by | Xn |. In what follows w-simplex will mean oriented ^-simplex. Λ The join of two simplexes | <7 | = | s0?s17 , sh \ and | τ* | = I *o> t19 , tk [ is the h + k — 1 — simplex ( s09 s19 , sh910, t19 , tk \ and will be denoted by | σh, τk |. An n-ordered set is a pair (X, φn), where X is a set and φn is a n function from | X \ to the set {— 1, 0, 1} and which satisfies A1 and A. n n n n A10—For every \ σ \ e \ X [ <pn \ — σ \ = — φn \ σ \. Before stating A2 we introduce the following notation: n n Φi {σ , τ ) = ψn I ti9 sl9 s2J , sn \ φn \ t09119 , ί^, s0, t1+19 •••,<„! Received March 8, 1964. 1337 1338 LINO GUTIERREZ NOVOA n A2.—If Φi(σ% τ ) ^ 0 for i = 0,1, ... n; then <pn\σ*\φn\τ*\^0. 71 1 Dlm—The simplex | TΓ " | is said to be an upper bound for the set 71 1 {xa; a 6 1} c X if <pn \ xa, π - | ^ 0 for every a el. If all the relations are strictly > then | π*1"11 is a proper upper bound. Similar definitions for lower bounds using <^ and <• D2.—The %-order φn is open from above (from below) if every finite subset of X has a proper upper bound (lower bound). Tx.—If φn is an open from above (or from below) n-order of X then the following transitive property holds: s If Ψn Iθ9 sl9 , Sι_ly x9 si+19 , sn ( ^ 0 for all i and some x e X then: n ψn σ I ^ 0. n 1 n n ι Proof. Apply A2 to the pair \x,π ~ \,\σ \ where | π ~ \ is a proper bound for {s<} U {x} EXAMPLES. (a) In the vector space Vn over the reals define: <Pn-i I ^o, v19 , vn_, I = sign of det. | vOf v19 , vn_Ύ \ n The function ψn_x is an n—1-order of V . (b) In the same space define: ψn I i>o, v19 , vn ] = sign of det. | v{ - v01, i = 1, 2, , w. cp% is an ^-order of F%. (c) The function of example (a) restricted to the sphere | V \ = 1 gives an n-1 order of the w-1-sphere. (d) Any 1-order satisfying the transitive property of Tx is equivalent to a chain if we define: φx \ a9 b \ to be — 1, 0 or 1 according to a> b9 a = b and a < b respectively. (e) A field G is said to be -^-ordered if it is also an ^-ordered set and the mappings: fa:x—>ax and ga: x —>a + x are order-automor- phisms for any a Φ 0. n n If we call: | σ \ = | SQ, S19 -, SΛ | | aσ \ = | αS0, αSx αS% |, and u I a + cr I = I a + SQ9 a + Su , α + Sn |, then the definition means n n n n exactly that φn\σ \φn\ aσ \ and φn \ σ \ φn \ a + σ \ depend only on a. The following examples can be given: (ex) The real numbers field is a 1-ordered (open) field. (This is a well known result). (e2) The complex numbers field is a 2-ordered (open) field if we 2 define for any | σ \ = | a09 al9 a2 \: ON ^-ORDERED SETS AND ORDER COMPLETENESS 1339 [111 2 2 iΔ{σ ) 2 φ2(σ ) = where Δ(σ ) ~\aQ aλ a2 A{σ2) I a being the complex conjugate of a, \ a | the modulus of α. (e3) The field of quaternions, considered as a 4-dimensional vector space over R and with the 4-order of example (b) above becomes a 4-ordered (noncommutative) field. (f) The w-order of Vn given in example (b) makes an ^-ordered n vector space out of V in the sense that the mappings fa: x-^ax and gy\ x—^x + y are order-isomorphisms for any aeR, a Φ 0 and any y e Vn. This example can be generalized as follows: (g) Let V be any linear space over the ordered commutative field K, and B a V any Hamel base for V. It N = {b19b2, •••, bn} is any finite subset of B\ we can make V into an n-ordered vector space by β defining φn(V0, V19 , Vn) = + 1, — 1 or 0 whenever det (Vί — Vξ) is >, < or = 0 in K(V( is the coefficient of b3- in the expression of Vi in terms of the base B) The independence of the axioms follows from the following examples: In the set {a, 6, c} define: φ2 \ a, 6, c | = φ2 \ b, a, c | = 1 and φ2 — 0 elsewhere. This system satisfies A2 but not Alo In the set {α, b, c, d, e} define: φ2\e, c, d\ = φ21 e, c, a \ — φ2 e9 c,b\ = φ , a,b = φz\d,b,c\ = φ2\d,c,a\ = φ2 a, c9 b I =1 and define φ2 on the remaining simplexes according to Alt This system satisfies Aγ but not A2. Ill* Consequences of the Axioms* D3.—Two elements x, y of W are said to be equivalent if for n 1 n 1 n ι n x every \π ~ \e\X ~ \ we have: φn \ x, π ~ \ — φn \ y, π ~ \. They are n ι 71 1 conjugate if φn \ x, π ~ \ — — φn \ y, π - 1. The relation between equivalent elements is an equivalence relation and the set of equivalence classes can be %-ordered in the usual way. For this set the following axiom holds. A3.— There are no distinct equivalent points. From now on we assume (X, φn) satisfies A19 A2 and A3 and call (X, φn) a reduced ^-ordered system. An easy consequence of A3 is: C3.~An element xeX has at most one conjugate cc*. k D4.—A simplex | σ |, k ^ n, is said to be singular if for every n we have: πn-k-i i — o. In particular \σ \ is singular if: Ψn I = 0. 1340 LINO GUTIERREZ NOVOA The following theorems follow easily and are stated without proof: T2.—x* nonsingular, is the conjugate of x, if and only if \ x, x* I is singular. T3o — Any simplex with repeated elements is singular. T4<,—There is at most one singular 0-simplex. n n x 1 Tho—If xΦy, for some \ π ~' |: φn\ x, π ~ \Φ φn\y, π^ \. We have also: n n n n Γ6β—// Φi (σ , τ ) ^ 0 for i = 0, 1, 2, .., n then φJ σ \ φJ τ \ S 0. (Compare A2) n n n n Cβ.—If Φi(σ , τ ) = 0 for i = 0, 1, 2, . , n then φn\ σ \ φn \ τ | = 0. n n TΊ.—If Φi(σ , τ ) ^ 0 for i = 0, 1, 2, = 0 then: Φi(σn, τn) — 0 for every i. IV* Flats and relative orders* k Dβ.—Given a nonsingular fc-simplex | π |, k <n, the set F x, TΓ^ I is singular} will be called the flat determined by πk n ι n T8.—If S; e F\ π ~ |, i = 0,1 n then \σ \ is singular. n 71 1 Proof. Apply C6 to the pair | a |, | x, π " 1 where the last simplex is nonsingular (Such an x exists by Z?4) n n C8.—If I σ I and | τ \ are both nonsingular, then for some i: I ti9 s19 s29 - , sn I is not singular. k k Γ9.—If \μ ,π \, h + k = n — 1, is nonsingular, the function h I σ\ τrfc I is α reduced h-order defined on the h-simplexes φh I a I = w h fe h I σ I o/ ί/^seέ F| /i I, φh is called the order of F\ μ \ relative to I πk \.