Postulate Sets for Certain Mathematical Systems and their Complete Existential Theory Citation Levit, Robert Jules (1939) Postulate Sets for Certain Mathematical Systems and their Complete Existential Theory. Master's thesis, California Institute of Technology. doi:10.7907/e99q-m874. https://resolver.caltech.edu/CaltechTHESIS:12122024-224942949 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: (Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Bell, Eric Temple Thesis Committee: Unknown, Unknown Defense Date: 1 January 1939 Record Number: CaltechTHESIS:12122024-224942949 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:12122024-224942949 DOI: 10.7907/e99q-m874 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16920 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 16 Dec 2024 23:46 Last Modified: 16 Dec 2024 23:48 Thesis Files PDF - Final Version See Usage Policy. 9MB Repository Staff Only: item control page

POSTULATE SETS Ii'OR CERTAIN MA'I'HEMATICAL SYSTEMS AND THEIR COMPLETE .EXISTENTIAL T'.d.EORY Thesis by Robert J. Levi t In Partial Fulfillment of the Requirements for the Degree of Master of Science California Institute of Technology Pasadena, California 19.39 TABLE OF CONTENT§ I NTRODUCTION AND SUMMARY i THE NOTION Oli' COMPLETE INDEPENDENCE ii PART!. GROUPS DEFINED BY A TRIADIC RELATION l ABELIAN GROUPS 9 PART II. RINGS AND IDEALS ll DIVISION SYSTEMS 26 IDEALS PART III. i INTRODUCTION AND SUMMARY The purpose of this paper is to construct postulate sets for certain mathematical systems, establish their complete independence, and, in cases where other postulate sets are already known, to show their equivalence to ours,. In Part I groups and Abelian groups are defined in terms of a triadic relation. The existence of an equivalence relation is not assumed, as is the usual method, but proved by actual construction. The complete inde- pendence of the postulates and their equivalence to conventional gToup postulates is established. In Part II a completely independent set of postulates for rings is exhibited differing from the usual sets (for ex.ample, see Van Der Waerden, Moderne Algebra, page 35) only in that the closure of addition is not explicitly assumed, since it can be deduced from the other postulates. With this definition of the containing ring a completely independent set of postulates for an ideal can then be given. Part III contains the most novel results of the paper. A definition of divisibility was sought, vrl th properties entirely anala.gous to ari thmetic divisibility but not depending on the existence of a multiplication operation. The postulates for a Division, System as defined in this section give the properties of certain classes called principal idea.ls, which are functions of elements of the system. By means of these classes an equiv- alence relation and a relation of divisibility are defined, the latter having all the important characteristics of arithmetic divisibility. Most important, the existence of an L.O.M. of any finite set and th~ G.C.D. of any set whatever in the system is established. I wish to thank Prof. E.T. Bell for the suggestions which led to the writing of this paper. il THE NOTION QF .COMPL.§TE IlWEPEN""D.ENCE The notion of complete independence was first introduced into mathematics by E. H. Moore in 1915. ("Introduction to a Form of General Analysis, 11 by E. H. Moore, New H~ven Mathematical CollogY:tw:A, 1915, page 82.) A thorough discussion of its significance is given in the article by E. V. Huntington, l'l:?e®§.f(t;l.Qmi of the AmeriQ&D Math-t;!,c" Societ;y;. Volume 20, page 277. We might briefly mention that complete independence carries with it both co:o.sistency and ordinary independence. It further demands that no relations of implication whatever exist among the postulates. The proof of complete independence is established by an exteusion of the method used for ordinary independence. Examples a.re exhibited in which all the postulates are true, examples in which tl1ey are contradicted one at a time, two at a time, and so on until finally an example is produced for which all a.re false. It takes 'lfl- examples to establish the (?omplete independence of n pQst'Ulates. The notation used is best explained by an illustration. The character (+ ..- - + -) is used to designate an example in which t.11.e first and fourth postulate of a given set hold, while the second, third and fifth do z...,t. 1 PART ;& GROUPS DE.FIN@ BY A ThIADIC RELATION* 0.1 Definition. We consider a class K of undefined elements a, b 1 ••• and an undefined binary triadic relation R such that for any ordered triple (a,b,c) of elements of K we can either say R(abc} (a, b, and c stand, in that order, in the relation R) or i(abc} (a, b, c do not, in that order, stand in the relation R). The class Kand relation R form together a system G when they are subject to the postulates below. 0.2 Definition. Two elements a, a 1 of Kare said to be equivalent (a@ a 1) if anp. only if all the following conditions are simultaneously satisf ied: (1) R(a:xy) implies R(a 1:xy) and conversely for every x,y c: K. (2) R(xay) implies R(xa*y) and conversely for ever-1 x,y c K. (3) R(xya) implies R(:xyat) and conversely for every x,y t K. Postulate I. 3 x in K for each b,c € K such that 0.4 Postilate II. 3 y in K for each a,d € K such that 0.3 R(xbc). R(ayd). *The first definition of a group by means of a triadic relation is that of .E. V. Huntington (tr;wsactioi1s gf Am~;rican Mathematical SoCi£?:t:!;, Vol. 6, 1905, P•· 192), though the idea was suggested earlier by M. Bocher. Huntington•s first set of postulates is similar to ours and formed the basis for it. Huntington's postulates are not independent, however, as theorem 2.1 shows. Moreover., they neglect th$ consideration of the equivalence relation. (Part I) Postulate III. 0.5 2 R(a~$ ) ; R( b d' JN), R(~ K" t- ) , and r R( a f '}) ) simultaneously holding imply @ ,, provided a, f3' b ' ¥ , f ' t , v are in K. o.6 Notation. Since the only elements considered are in K, the phrase,'for ·every a of K;w111 usually be omitted. 1.1 1.2 Theorem. a @,a for every a in K. Proof: Obvious from 0 •.2. Tp.eorem. a @b implies b (§) a and conversely for every Proof: Obvious from 0.2. Theorem.. If a@ b and. b@ c, a@ c. Proof: Since a@) b, R(axy) implies R(bxy} for all a,b ill K. 1.3 x and y by 0.2. (1) Since b@ c, (R(bxy) implies R(cxy) for all x and y, R(axy) implies R(cxy) for all x and y. Since c@ b by 1.2, R(cxy) i mplies R(bxy) and since b © a R(bxy) implies R(a.xy) for all x and y, then (2) R(ax:y) implies R(axy) for all x and y. (1) and (2} of this proof together satisfy condition (1) of Definition Q:.:~. Conditions (2) and (.3) can be shown to be fulfilled by exactly analagous proof so that 1.3 follows. 1.4 Remark. The relation @ is an equivalence relation. {Part I) 2.1 Theorem*. For each p~i r a,b of elements of K there exis t s an element of K such that R(abc). Proof: (1) For each a, 3 ia such that R(iaaa) by 0.3. ( 2) F'or each a, J a-1 such that R(a-1aia) by Postula te I. (3) For each b.,] x such that R(a-1xb) by Po st ulate II. Consider now an arbitrary fixed pair of elements a.,b of K. We will show tha t the element x of step (3) is the c demanded by the theorem such that R(abc). (4) 3 p such that R(apx) by Po s tulate II. ~ (5) 3 q such that R(iaqp) by Postulate II. (6) 3 r such that R(arq) by Postulate II. (7) p@ q. =b .:: a., if This follows from Postulate III by putting a -= r, c =f =q, 'J) ::. p in steps (l), (6), (6) and (5). (8) R(iapp) by steps (5) and (7) and def. 0.2. (9) p@b by Post. III wit.11 a= a-1 , ~=a, d E = x, and iJ (10) =ia, =ia, ~ ~f::=. p, = b i n steps (2), (8), (4), and (3) • R(abx) by steps (4) and (9) and def. 0.2. *The method used iu this proof was suggested by R. Garver's work on group postulates. (Bulletin of the ,A.11erican Mathematical Society, Vol. 40, 1934, P• 698.) 4 (Par_:t, I) 2.2 Theorem. (1) R(abc) (2) R(abc') If R(abc) and R(abc'), c@ c•. by hypothesis · (3} :l x in K such that. R(xab} by Post. I, (4) :J y in K such that R(rucy-) by Th. 2.1 .. (5) 3 z in K such that R(,:ax) by Th. 2.1. (6) z@ c by Post. III, putting a ;; "6 ;; a, ~ :: x, ~ ::: y, f = z, €: == b, and V: c in steps (4), (5), (3), and (1). (7) Similarly, by symmetry, repeating steps (3) to (6) and using (2) in place of (1), z@c• (8) c@ c 1 by Th. 1.2 and 1.3. Q.E.D. 2.3 Remark. To ever-y ordered pair of elements a,b, there corresponds an element c, unique up to equivalence such that R(abc) (from 2.1 and 2.2). (Part I) .3.0 Definitign. By an abstract group we mean the well- known mathematical system consisting of a class G(a.,b,, •• ) subject to the following postulates: 3,1 Postulate. There exists an equivalence relation, binary, reflexive, symmetric, and transitive throughout G. This relation holding between a and b is symbolized, a e) b • To each pair of elements a;b ~ G there .3.2 Postulate, correspo11ds a unique element ab E G.• 3,3 Post:ulate .. For every a;b;c c G (ab)c == a(bc) For each a and b €: G there eY.ists a cor- .3,4 Postulate. responding x r:- G such that ax= b For each a,b f G there exists a. cor- 3.5 Postulate. responding y ~ G such tha. t ya= b ,3.6 Proof: Theorem., The class K(O.l - 0,5) is an abstract group. F'or our equivalence relation we chooGe that defined in 0.2, i,e., a= b if and only if a @ b. Then by 1.1 to 1.4, 2.1 is satisfied. By 2.1 for each a,b €: K there will exist a corresponding z E- K m1ch 6 (Part I) that R( abc). Designate z by ab. Let c be such that R(abc). Then by 2.2 ab@ c or ab= c The element ab may be regarded as a representative of the class of The element e = ab is thus equiva.lent elements z for which R(abz). uniquely specified (up to equivalence) by a and b, so 3.2 holds. Ta.te a, ~, "6 any elements of K and let (1) , Then (2) R(af) 6 ), R( & r t = ~o r ), R(l3 r , and· )) c ), and =G.ER(a E i ). Therefore by 0.5 (3) f- @J V or f = )) • Substituting from (l) in accordance with Th. 1.3 (4) (aJ3)6 = By postulates 0.3 and 0.4 we can find an x such that R(axb) and a y such that R(yab) corresponding to each a,b E- K. (5) Then ax= band ya= b are solvable and all the postulates a.re satisfied. Q.E.D. 3.7 Theorem. E'very abstract group G satisfies the postulates 0.3, 0.4, 0.5 for K. Proof: We shall say that R(abc) if and only if ab= c. R so defined satisfies 0.3, 0.4 since by 3.4 and 3.5 we can always find an x and y such that ax= band ya= band consequently such that R(a.xb) and (Part I) R(yab). and xy Let a~ a•. Then ax= y implies ax•= y, xa = y implies xa 1 = y, = a implies xy = a• and conversely for all x,y t G. Therefore a = a 1 implies a® a•. ')) 7 :5 a c , we have f.3.8 Remark. Filllilly, letting = ')I by 3.,3. i =at,, f ;;;. t ¥ , E ":: f3 r, Then f' .=. ')) and Post. 0.5 holds • The postulate set 0.3, 0.4, and 0.5 is equiv- alent to the set 3.1, 3.2, 3.3, 3.4, and 3.5, since each implies the other by the preceding theorems. Consequently the former set will serve as well as the latter for the definition of an abstract group. g (Pa.rt I) 4 .. 1 Thegrem. The postulates O.J, 0.4, 0.5 form a com- pletely independent set. The proof is by the usual method of examples, all of which are in this case self-explanatory. Character Class, K(a,b • .,.) (+ + +) All rational integers a+b = o (+ + ..;.) All rational integers l/2(a+b) ::: C (+ - +) Positive rational integers a=o (- + +) Positive rational integers b = C (+ ... -) Positive rational integers a-b = c (- + -) Positive rational integers b-a = C (- - +) Positive rational integers a+b = c (- Positive rational integers .2(a+b) :;: C --} Relation, R(abc) if (Part I) 9 ABELI.Al~ GROUPS Consider a class K subject to postulates II and III (0.4, 0.5) but with I replaced. by the following: ;.l Postulate IV. 5.2 Theorem. Progl: If R(abc)., then R(bac) for every a., b,e € K. Postulates II and IV imply I. Let a,b~ K. By II, there exists Jay such that R(ayb). By IV, R(ya.b) since a,b were any two eleinents,,there will always be an x ~ K such that R(xab) if we take x to bey. 5.3 Theorem. A class K subject to postulates II., III, and IV forms an Abelia.n group. Prop,£: By the preceding theorem and theorem ,3.6 it forms a group with ab = c if R(abc). ba Let R(&bc}. Then R(bac) by IV and consequently = c =ab by J.l and K is Abelian. 5.4 Theorem. Every Abelian group G satisfies postulates II, III, and IV. Propf: By theorem ,3.7 G satisfies III and IV. R(abc). But ba = c since G is Abelian. Let ab= c in G. Then Therefore R(bac) and II is satisfied since a and b were arbitrary. 5.5 Remark. for an Abelian group. II, III, and IV may be taken as the postulates (Part I) 10 5.6 Theorem. The postulates II, III, and IV are completely independent. Character Class K(a,b ••.• ) Relation, R(abc) if (+ + +) ill r ational integers a+b = c (+ + -) Positive rational integers b = c (+ - +) All rational integers l/2(a+b) = 0 (- + +) Positive rational integers a+b = c (+ - -) All ration.al integers 2a+b = c (- + -) All r a tional integers a= C (- - +) All rational integers 2(a+b) = C - -) All rational integers a+2b == c (- 11 PART II RINGS AND IDEALS 1.0 De;t:';tnition. A ring is the well-known mathematical system defined a.s follows: it consists of (1) A class K of elements a, b, ••• (2) An equivalence relation written f=,, which is binary, reflexive, symmetric, and transitive throughout K. (3) An operation, addition ( + ) , defining a one-valued func- tion of t wo variables. That is to every ordered pair a,b c K there corresponds an element (a+ b) unique up to equivalence. (4) A second operation, multiplication, indicated by ab. Thus, to every ordered pair a,b € K there corresponds an element ab unique up to equivalence. (5) The following postulates hold in K: 1.1 Postulate I. For each a,b K there exists a cor- responding x f K such that a+x=b 1.2 Postulate II. If a, b, a + b, b + a e- K, then 1.3 fpstulate rri. If a, b, c, a+ b, b + c, a+ (b + c), (a+ b) +ct K, then a+ (b + c) =(a+ b) + c (Part II) 1.4 Postulate IV. If a, b f K, then ab f K L,5 Postulate V. If" a., b, c, ab, be, (ab)c, a(bc) t- K, then a(bc) = (ab)e l.6 Postulate yr. If a, b, c, b + c, a{b + c), ab, ac, ab + ac €- K, then a(b + c) =ab+ ac 1.7 Pos:t,ulate VII. If a, b, c, a+ b, (a+ b)c, ac, be, ac +bet K, then (a+ b}c = ac + be 2.0 Remark. In the postulates for a ring given in the above section, it was not explicitly assumed that the set closed with respect to addition. A postulate to this effect is ordinarily included in the definition of a ring. This is unnecessary as the following theorem shows, and its inclusion destroys the independence of the postulates. 2.1 Theorem. The elements of a ring form an Abelian group with respect to addition. Proof: If we take R(abc) if and only if a.+ b = c, the postulates for an Abelian group, 0.4, o.;, and 5.1 of Psxt I of this paper are clearly satisfied. 2.2 Corollgy. A ring is closed with respect to addition. 12 13 (Part II) .).1 Theorem. The postulates I through VII form a completely independent set. (The equivalence relation in the following table is taken as ordinary equality for rational or integral examples while (a,b) = (c,d) if a= c and b = d for number couples. Character In the case of number couples we shall Operations Class Addition: a@ b = K( a,b, •.. ) Mu.l tiplica tion: a.ob= (+ + + + + + +) I11tegers a+b ab (+ + + + + + -) Integers a+b b (+ + + + + - +) Integers a-tb a (+ + + + - + +) Integer couples (a1+b11 a 2+b2 ) (a1b2+a2bz,a1b1+82b1) (+ + + - + + +) Rational numbers r such that 2r is an integer a+b ab (+ + - + + + +) Integers a+b ab (+ - + + + + +) Integers b ab (- + + + + + +) Positive integers a+b ab (+ + + + + - -) Integers a+b l+a+b (+ + + + - + -) Integers a+b 2b (+ + + - + + -) Integers a+b If bis evena b, If bis odd: b/2 (+ + - + + + -) Integers a+b + - + + + + -) Positive rational couples (albl+a2bl,8J.b2+a2b2) (bl'b2 ) (- + + + + + -) Positive integers a+b b 2 2 b (Part II) Character Class Addition: a EB b = 14 Multiplication: aob = (+ + + + • • +) Integers a+b 2a (+ + + - + - +) Integers a+b a, if a is even a/2, if a is odd (+ + - + + - +) Integers (+ ~ + + + - +) ' Positive rational couples a, + b 2 (8J_bl+a2bl'~b2+a2b2) (- + + + + - +) Positive integers a (+ + + ... - + +) Integer couple~ ( • albz+agb2, albl+a2b1 . ) 2 2 (+ + - + - + +) Rational couples (2a1+2b1 ,2a2+2b2) (a1b2+a2b2,a1b1+a2b1) (+ - + + - + +) Integers b a. - b (- + + + - + +) Positive integer couples ( al+11_, a2+b2) (+ + - - + + +) Rationals r such that 2r is an integer a+b (+ - + - + + +) Rationals r such that 2r is an integer b ab (- + + - + + +) Rationals r such that 2r is an a+b ab ~ 2 a ab integer (+ - - + + + +) Integers 2s. + b ab (- + - + + + +) Positive integers a+b ab (- - + + + + +) Integers a ab (+ + + + - - -) Integers a+b a-b (+ + + - + - -) Integers a+b l +a+b i£ a+b is even 2 2 l+a.+b if a+b is odd (+ + - + + - -) Integers a+b .3 l ♦ a+b 15 {Pa.rt II) Character Class Addition: a @ b == Multiplication: aob = (+ - + + + - - ) Positive rational couples {a b +a b .a b +a b ) (a,_+b ,a +b ) 1 2 2 (- + + + + - -) Positive integers a+b l+a+b (+ + + - - + -) Integers a+b b/2 (+ - + + - + -) Positive ration.al couples (- + + + - + -) Integers a+b 2b (+ + - - + + -) Integers a+b b, if bis even b/2, if bis odd (+ - + - + + -) Positive rational couples (a1b1+a2b1 ,a1b2t82b 2) 1 1 2 1 1 2 2 2 (bl2+blb2,b22+blb2) 2 (b:if 2) fOJ'bf a perfect (~, b) • §quare for.bi not a perfect square (- + + - + + -) Positive integers a+b b, for b even b/2 for b odd (+ - - + + + -) Integers 2a + b ab (- + - + + + -) Positive integers a+b 2 (- - + + + + -) Positive integer couples (a1b1+a2b1,a1b2+a2b2) (+ + + ... - - +) Integers a+b a/2 (+ + - + - - +) Integers a+b 2a (+ - + + - - +) Positive integer couples (81_b1+a2b1,8J.b2+a2b2) ( al 2+ala2'¾2+ala2) (- + + + - - +) Integers a+b 2a (+ + - - + - +) Integers a+b 2 a., for a even a/2, for a odd (+ - + - + - +) Positive rational couples 2 (a1,a2) for 61 perfect sq. (/ai',~) for 31 not a perfect. square 16 (Pa:i:'t II) Character Class (- + + - + - +) Positive integers Addition: a @ b : : : a+b Multiplications aob = a/2 for a. odd a, 2a + b a (- + - + + - +) Positive integers a+b 2 a (- - + + + - +) . Positive i nteger couples (a1b1+a2b1,8J.b2+azb2) (+ - - + + ... +) Integers (+ + - - - + +) Integers (+ - + - - + +) Integers (.... + + - •- + +) PosiM.ve integer couples for a even a-b 2 a-b b 2 aib2+a2b2,a1b1+a2b1 2 (+ - - + - + +) Integers 2a - b a-b (- + - + - + +) Po sitive integers g.+b 2a + b (- - + + - + +) Integers a (+ - - - + + +) Integers 2a + b (- + -· - + + +) Rationals r such that 2r is an integer a+b (. • + - + + +) Rationals r such that 2r is a:n integer a ab (- - - + + + +) Positive integers 2a + b ab (+ + + - - - -) Integers a+ b a - b 2 (+ + - + - - - ) I ntegers a+b a-b 2 a-b 2 3 (+ - + + - - -) Positive rational couples (- + + + - - -) Positi~e integers ab (a1b1+a2b11 a1 b2"ra2b2) (a1-bi,a2-b2 ) a+ b 2a + b 2 17 (Part II) Character Addition: a Q) b = Class (+ + - - + - -) Integers (+ - + - + - -) Positive rational couples a+ b 3 Multiplication: aob = l+a+b if a+b is odd l+a+b if a+b is . even 2 (a1+b1 ,a;tb2 ) if a1+b1 ~erfect square (✓a1+b1 ,a2+b )if a +b 2 1 1 is not a perfect sq. (- + + .... + - - ) Positive integers a+b (+ - - + + - - ) Integers 2a + b l+a+b (~ + - + + - - ) Positive integers a+b 3 l+a+b (- - + + + - -) Positive rational couples (a1b1+a2b1,8ib2+a2bz) (+ + - - - + -) Integers a+b 2 (+ - + - - + -) Positive r ation.al couples (a1b1+a2b1,a1b2+~2b2) (- + + - - + -) Integers a+b b/2 (+ - - + - + -) Integers 2a + b 2b (- + - + - + -) Positive integers a+b (- - + + - + -) Positive integer (albl+a2b1'8ib2+a2b2) couples (+ - - - + + -) Integers (- + - - + + -) Positive integers (.... - + - + + -) Positive integer couples 2 l+a+b if a+b is even 2 1+a+b if a+b is odd 2b 2a + b b/2 for b odd b for b even a.+b 2 b/2 for b odd b for b even (b1, b2) if b:i. is a .E!:,rfeet square (✓bi, b2)if ~ is not a perfect square 18 (Pc_rt II ) Character Addition: a@ b = Class (- - - + + + -} Positive integers 2a + b (+ + - - - - +) Integers 1+b (+ - + - - - +) Positive rational (a_ib1+a2°bi,8:J.b2+a2h2) (- + + - - - +) a+b couples Positive integers Mul'tiplication: aob == b 2 (+ - - + - - +) Integers 2a + b 2a (- + - + - - +) Positive integers a+ b 2a (- - + + - - +) Positive integer couples ( a1bi+a2b1, a1b2+a2b2) 2 (+ - - - + - +) Integers 2a + b a/2 for a odd a for a even (- + - - + - +) Positive integers a+b a/2 for a odd a for a even (- - + - + - +) Positive integer couples (- - - + + - +) Positive integers 2 (a1,a2) if a1 is a perfect square (~,az)if a1 is not a perfect sq. a+b a (+ - - - - + +) Integers 2a - b a-b (- + - - - + +) a+b a-b (- - + - - + +) Positive integers 2 Integers 2 (a1b1+a2b1,'½_bz+a2b2) couples Positive integers 2 a-b (- - - + - + +) Positive integer (- - - - + + +) 2 2a + b (+ + - - - - -) Integers a+b 3 (+ - + - - - -) Positive integer (a1b1+a2b1,a1b2+a2b2) a - b 2 19 (Part I I) Character Class Addition: a@b:::: Multi plication: a.ob= (+ - - + - - -) Integers 2a + b a-b (+ - - - + - -) Integers 2a + b a+b+1 if a+b is even 2 a+b+l if a+b is odd Integers 2a + p b/2 (+ - - - - - +) . Integers 2a + b e./2 (- + + - - - -) Positive integers a+b a-b 2 (- + - + - - -) Positive integers a+b 2a + b (+ - - - - + -) (- + - - + - -) Positive integers 3 a+b .3 {- + - - - + -) Positive integers a+b (- + - - - - +) Positive integers (- - + + - - -) Positive integer couples (a1b1+a2b1,a1b2+a2b2) {a1-b1,a2-b2) (- - + - + - -) Positive integer (8J_b1+a2b1,8J.b2+a2b2) {a1+b1,a;tb2) if a1+b1 couples 2 a+b 2 a+l>:tl if a+b is even 2 a+b+l if a+b is odd b/2 a/2 is a perfect square (la.1+bi,a2+b2)i£ a1+b1 is not a pe~fect sq. b1 ,_ - + - - + -) Positive integer couples (a1bi+a2b1,a1b2+a2b2) (2, b2) (- - + - - - +) Positive integer couples (8i_~+a2bl'8ib2+a2b2) ( 2 ' a2) (- - - + + - -) Positive integers 2a + b a+b+l + -) Positive integers 2a + b 2b Positive integers 2a + b 2a (- - - + - (- - - +. - - +.) ~ 20 (Part II) Character Class (- - - - + + -) Positive integers +) Positive integers (- - - - - + +) Positive integers (- - - - + - Addition: a ~ b == b/2 if b is odd if bis even b 2a + b a/2 if a is odd if a is even a (+ - - - - - -) Integers (- + - - - - -) 2a. + b Mul ti.pliea tion: ao b = Positive integers 2a - b a - b 2a + b a-b a+b 3 a-b 2 2 2 (- - + - - - -) Positive integer couples (8ibl+a2bl,8J.b2+a2b2) (al-bl (- - - + - - -) Positive integers 2a + b a-b (- - ... - + - -) Positive integers 2a + b a+b+ Positive integers 2a + b b/2 (- - - - - - +) Positive integers 2a + b a/2 - - - - - -) Positive integers 2a + b a-b (- (- - - - - + -) 2 ' a~b2 ) + if a.+b is even 2 a+b+l if a+b is odd 2 Incidental to the devising of the above examples, a nwnber of interesting elementary theorems were obtained, which greatly reduced the work. 3 .2 Theorem. Let x @ y :: ax + 'by and xoy ; nL"I: + ey where x and y are a:ny- real nwnbers and a, b, m, and n are fixed real numbers. If a+ b = l, then for aJ.l x, y, z. (1) xo(y ~ z) = xoy~ X•Z and (x e, y) z:. X•Y (?J xoa independently of m and n. holds for all x, y, z. But, if a+ b Fl, neither of the above equations (1) (Par:t II) P1-:oof: (.2) xo(y $ z) : xo(ay + bz) : mx + n(ay + bz) = mx + any + bnz. (.3) xoy$ xoz =a(mx + ny) + b(mx + nz) =(a+ b) mx +any+ bnz. Thus the first equality of (1) holds identically in x, y, z if and only ii' a+ b = 1. The second follows by symmetry. 3.3 Theorem. Let x$ y be defined as in ,3.2. Then x ~ {y@ z) : (x@ y) $ z (4) holds for all x, y, z if and only if a= 0 or a= land b = 0 orb= 1. (5) x ~ (y $ z) ;: ax + b(ay + bz) = ax + aby + b2 z. (6) (x $ y) E& z = a(ax + by} + bz = a2x + aby + bz. If (4) i.s to hold, then a2x + aby + bz :.: ax+ aby + b 2 z and (7) a2 = a, b2 = b; .3.4 Corollary. a= 0 or a= l; b = 0 orb= l . x@ y :i x, X ~ y E y, X ~ y := O, X $ y E X + y are the only linear combinations which make the operation <:B associative • .3.5 Theorem. Let K(a,bp . . ) be any abstract set in which an operation$ and an operation l O ) are defined bet~-veen every pair a,b E K. If a $ b = a. (1) or a a, b = b and (a(Ji) b) c for every a,b E- K, then {2) a.•(b $ c) ::: a•b El, a•C for every a,b 0 =a c + a b K no matter how the multiplication,(¢), i s defined. Progf: 'rake the first definition in (1) b $ c = b. ao(bEB c); aob = acb ~ aoc. a~ b = a.. (a ES b)o c = aoc = a 0 c 13, a.ob. (Part II) If the second definition of (1) is taken instead, the proof follows by symmetry from the above. In the set K of 3.5 let addition be defined in Theorem. ,3.6 any other way besides _that of equation (1). defined as follows: either =a (3a) aob Let multiplication be or (3b) aab = b. If definition (3a) is chosen, then for every a,b;c (4a) (a $ b) 0 c = aoc a, boc K ; while if (.3b) is chosen, then ao(b~ e) (4b) =a b ~ a•e, 0 but (4a) will not hold for all a,b,e. Also, when (3a) is selected, (4b) is false. Proo£• ao (b $ Choosing (Ja), (ad, b) 0 c = a EB b = a 0 c~ boc, but c) ::: a -,. ao b ~ a• c = a~ a since addition is not as in (1) of .3. The rest follows by symmetry. J.8 Theorem. In K let multiplication be defined so that a x = b (or y a = b) is solvable for all a,b E- K and such that 0 0 a•(boc) = (a•b)•c (5) Moreover, let a•b = f(b) depends only on b. a•{b •c) (6) = f(bec) (or a•b = ~"3.)) Then f(b) = b = f [ f(c)l ; where f(b) indica tes that a• b ( d) (a) = a) (a•b) o c for every b (or a) ~ K. =f{ c). From (5) f [f(c)] = f(c) Let f(c) = d; then f(d) = d for every element d such that d = f(c) = uac where u is any element of K. uoo: dis solvable inc for any 5. (Part II) elell!ent of' K ~o that for all d f(d)? d The same method will prove (I) (a) =a. 24 (Part II) IDEALS Let I(a,b, ... ) be a subset of a ring R. 4.0 Def;nition. I is called an ideal if and only if the following postulates are satisfied. 4.1 Postulate I• For each a, b t I there exists an x E- I such that a. + x == b (1) 4.2 Postulate ;tr. For every a e I and every r ~ R, r a f I 4,3 Fot3tula.te J;II. For every a E- I and every r t R, a r f I 5.1 The elements of an ideal form an additive Abelia.n Theorem. group. PrgQf: Since the elements of the ideal are in a ring the addition ~s associative and commutative. Solvability, both right and left, follows from the commutativity and Postulate I. The remainder of the proof follows the lines of Theorem 2.•1 of this section. 5,2 Remark. An ideal is usually defined by Postulates II and III and by the property that the elements form an Abelian group with respect to addition. Our postulates are equivalent to this as theorem 5.1 shows,and moreover are completely independent as will be shown. Still another definition demands that if a and bare in I, a - b t I as well aa Postulates I and II. This, of course, is also implied by our definition, since it is a property of Abelian groups. (Part II) 6.1 Theorem. 25 The Postulates I, II, I I I for an ideal are completely independent. -. Character Ring (+ + +) (+ + -) Additions a+b= Mul tipllcation: ab ~ Integers a+b ab Linear set of elements a= ~i+a2J (a1+b1 )i + (a2+b2)j (a1+a2)b1i + (a1+a2)b2j b =b1i+b2j over the rational integers Ideal Even integers All elements of form (a1 i+a2 j)i = (a1+a.2)i. = ij j is no t in the ideal ; I so Post.III is false • Same as (+ + -) (a1+a2 )i + (b1+b2 )j • ;~ ~ b +b )i + a (b +b )j 1 2 2 1 2 (- + +) Integers a+b ab Zero c:Llld all integers n such that f nl > 5 (- - +) Saine as (+ + -) Same as (+ - +) Same as(+ - +) All elements aj where a i s an i nteger such t hat (-¼ - +) i.ll elements of f orm i(a1 i + a 2 j) I al> 5 or a :: 0 (- + -) Same as (+ + -) Same as (+ + -) Same as (+ + -) All elements aj where a i s an integer such that Jal> 5 or a= 0 (+ - -) Rationals a.+ b ab Rationals r such that 2r is an integer (- - -) Integers a+b ab Positive even integers 26 PART I;!:;J: DIVISION SYSTEMS O,O Definition. Suppose that to ea.ch element x of a class K there corresponds a unique subclass Px ~ K. K will be called a division system if the Postulates I through V below hold for every element of K. 0.1 Definition. The unique subclass corresponding to xis called the principal ideal of x, 0.2 Definiti9n. Pi If there exists an Eilement i in K such that =K, i is called a unit ,21. !• 0.3 Postulate I. a E: Pa· 0.4 Postulate II. If a 0.5 Every subset S ~ K is contained in at Po!iitulate III. E: Pb, then Pa~ Pb. least one and at most a finite number of principal ideals. 0.6 For every a f: K there is an element b not Po§tulate IY. in Pa such that Papb F Pa• 0.7 Postulate V. The (logical) product of two principal ideals is a principal ideal. 0.8 Theorem. The Postulates I through V are consistent and independent. Proof: By following examplea: 27 (Part III) Principal ideal of x, fx Character Class K {+ + + + +} Positive integers. All multiples of x by a positive integer. (+ + + + -) Positive integers with 6 omitted. All multiples of x by a positive integer other than 6. (Eg. PzP3 is not a principal ideal.) (+ + + - +) Positive integers, All integers y >' x. (+ + - + +) Integers> l x and all its multiples by an integer > l. (Eg. no principal ideal contains the pair, (2,3).) (+ + - + +)* 1 and all fractions with even denominators in lowest terms. x and all its multiples by a :fraction with even denominator in lowest terms. {Every element belongs to an infinite number of ideals.) (+ - + + +) x = (Xi,x2); P consists of all couples (yl'y2) such that either y1 is a multiple of x1 or y 2 is a multiple of x2 where 1 ,( yl ~ 100; 1 ~ y ~ 100. 2 Integer couples (a,b) l ~ a i 100; l ~ b ~ 100. (- + + + +) Positive integers. 0.9 Examples of Divisiou Systenua• that given under(+++++). P1 = K; Px for x > 1 consists of all integers y > x. The simplest example is Postulate I expresses the fact that divisibility is a reflexive relation and II that it is transitive. III insures the existence of a unit, and moreover adds the restriction *Two examples a.re given., one in which a subset Smay belong to no principal ideal but at most belongs to a finite number; and a seeond in which Sis contained in at least one and not necessarily in a finite number of principal ideals. (Part III) 28 that every integer has at most a finite number of divisors. IV rules out trivial systems such as totally ordered classes, which satisfy all the other postulates. Postulate V states that among the common multiples of a set there is always a least, i.e., one that divides all the rest. Another example is all the integers positive and negative but not zero. This illustrates the more general situation that equivalent elements, as the equivalence relation will be defined in 1.0, are not equal but merely associate. of 1.0, For instance by the definition l@-1 and in general n@-n, since both n and -n determine the same set of multiples. The positive Gaussian integers form a division system with t.he principal idea.ls defined as usual. Equivalence again means dif- fering 011ly by a unit factor, eg., 2@ 2i since i is a unit. The integers of an algebraic number field in general, however, do not form a division system. Thus in the quadratic integers a + b C5, where a and bare rational integers, t.he common divisors of 9 and 3 - 6 /:5 a.re !l, ±3, ±(2 - /:s) no one of which is divisible by all the others; so there exists no G.C.D. 29 (Part III) 1.0 Definition. We define a.n equality in K as follows: a= b if and only if Pa= Pb 1.1 Remark. The correspondence between classes of equivalent elements and principal ideals is now one to one. 1.2 Theorem. a= a. 1.3 Theorem. If a= b, b = a. 1.4 Theorem. If a= band b =c, then a= c. 1.5 Remark. Equality as defined in 1.0 is an equivalence rela tion. 1.6 Definition. 1. 7 Theorem. 1.8 Theorem. Proof: a Ia (by Postulate I). If a I b and b I e, then a I c. Since a I b, bf Pa. o t Pb~ Pa so that c E Pa• 1.9 Tb;eorem. If b E Pa' a i s said to divide b, written a ; b. By Postulate II, P'b ~ Pa• Since b I c, Therefore a\ c. If alb and b/a, then a= b. Pr_gof': By Postulate II, since alb, b c Pa and Pb( Pa• Similarly Pa~ Pb. By 1.0, then, a= b, since Pa= Pb. 1.10 Definition. called a multiple 2£ J!• If ab, a is called a divisor of .2, and bis 30 (Part III) 1.11 Theorem. Prgof) There exists a unit in K. By Postulate III, K is the principal ideal of some element 1, which is then a unit by def'ini tion. 1.12 Theorem. Proof: il There exists only one unit i in K. Suppose there were two units i 1 and iz. =i 2 by l.O. 1 .. 13 Then K = Pi = P1 1 2 Theorem.. The unit di vides every element of K. Every element of K has a finite number of 1.14 Theorem. divisors. Proof: a a. Let a be any element of K. It has at least one divisor, since It has at most a finite number, since if xta, a f Px and by Postulate III a belongs to a finite number of principal ideals Px• Let a. 1- i. 1.15 Theorem. Then there exists an element b such that a does not divide band b does not divide a. Proof: Direct from Postulate IV. 2.1 Defini tio9;. An element which is a multiple of every element of a set Sis a coIDillOn multiple of S. 2.2 Theorem. The product of a finite number of principal ideals is a principal ideal. Proqf: The theorem is trivial for one principal ideal, and for two it i s a direct consequence of Postulate V. Assume it true for then principal 31 (Part III) ideals. Then their product Pa= Palp a is a principal ideal. Now consider the 2 • •• n + 1 principal ideals This is a principal ideal by Postulate V and t he induction is complete. 2.3 element. Theorem. Every principal ideal contains at l east one For, if Px is any principal ideal, x ~ Px by Postulate I. 2.4 Remark. The null class is not a principal ideal. 2. 5 Theorem. .Every finite set has at least one common multiple. Let S be a set of elements a 1 , a 2 , ••• , 8n• The product P = Pa Pa ... P8n 1 2 of t heir principal ideals is a principal ideal by 2.2. It is non-empty by 2.3 • .Every element of Pis a common multiple of S by definitions 1.10 and 2.1. 2.6 Definition. A common multiple of a set S which divides ever-/ other common multiple is a least common multinle (L.C. M.) of S. 2.7 Theorem. Every finite set has a least common multiple in K. Proof: Let S ~ K be any finite set of elements a1 , a 2, •.• 8n• Let Pm= Pa Pa ••• P~. Pm is a pri ncipal ideal by 2. 2. Consider the 1 1 element m of which Pm is t.he principal ideal. m is a multi ple of each of t he elements a1 t S, since mt Pm and Pm is the class of all common multiples of s. Any other commo11 multiple m1 is thu s also in Pm• (Part III) 32 Therefore m' is a multiple of m or m divides m1 , and is the required. L.C.M. by .2.6. 2.8 Progft Theorem. Every finite set has a unique L.C.M. It has at least one by the previous theorem. J.l Definition. If there were two, An element which divides every member of a set Sis a common divisor of S. 3.2 Theorem. Every set S ~ K has at least one and at most a finite number of common divisors. Progfl By Postulate III Sis contained in at least one and at most a finite number of principal ideals Pa, Pa, ••• , Pa_• 1 2 Then the elements -xi a1 , ••. , an are common divisors of s, since each one divides every element of its principal ideal by 1.6. If there were any other common divisor a, S ~ Pa; so that Pa must be a.DWng the P¾. and a among the a 1 . 3.3 Definition. A common divisor of a set S which is a multiple of every ot,'ier common dh'"isor of S is the greatest common divi§or (G.c.D.) of s. 3.4 Theorem. Every set S ~ K possesses a greatest corllllon divisor in K. PrqQf: S possesses at least one common divisor and a.t most a finite number by 3.2. If Dis the class of all common divisors of s, then D (Part III) is finite and non-empty. by 2.8. Therefore it will possess a unique L.C.M., d, We shall show that dis a G.C.D. of s. element of s. Let ai be an arbitrary Every element of D divides a1; so 81. is a common multiple of D by 1.10 and 2.1. Then d I ELj, since the L.C.M. of a set divides all the other co.rrurr-0n multiples by 2.6. dis a common divisor of s. d' € D. .33 Since this is true for ro:ry ai, Let d' be any other common divisor. Then But d is a multiple of every element of D by 2.6 and 2.1. d' I d by 1.10. Therefore d is a G,.C.D. of S from 3.3. 3.5 Theorem. Every set S ~ K possesses a unique G.C.D. in K. PtQQ,f: It has e.t least o:ne by the previous t heorem. If there were two, d1 and d2 , d1 [ d 2 and d2 [ d1 ; so dl = d 2 by 1.9. 3~6 Theorem. The G.c.n. of a set Sis the L.C.M. of the common divisors of s. (Tlds is a direct consequence of the way the element d was constructed in the proof of theorem 3.4.) 3.7 Remark. It is worthy of note that while only finite sets possess an L.C.M., all sets possess a G.C.D. just as is the case in 8.ri t hmetic.