The Foundations of General Arithmetic Citation Ward, Morgan (1928) The Foundations of General Arithmetic. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/XDSZ-KF26. https://resolver.caltech.edu/CaltechETD:etd-03042005-135853 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) 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 1928 Record Number: CaltechETD:etd-03042005-135853 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-03042005-135853 DOI: 10.7907/XDSZ-KF26 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 863 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 07 Mar 2005 Last Modified: 03 Oct 2019 23:03 Thesis Files Preview PDF (Ward_m_1928.pdf) - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

THE FOUNDATIONS OF GENERAL ARITHMETIC Thesis By Morgan Ward. In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy. California Institute of Technology Pasadena, California 1928 Section 1. Section 2. Section 3. Section 4e Section 5.. Section l. Section 2. Section 3. Section l. Section 2. Section 3. Section 4. Section i. Section 2. Section 3. Section 4 Section 5. Section 6. Section 7. Section 8. CONTENTS Part I. Preliminary Notions from Formal Logic. Introduction. Ons to One Correspondence. Functions. Transformations Counting Functions of Integerse Integral Functions of One Variable. Integral Functions of Two Variables. Operational Functions. Part III, Semi-Groups. Introduction. Glosed Systems. Closed Associative Systems. The Identity. Part IV. The Operation of Division. Inverse Operations. Division in General. Units. Integrality. Types of Semi-Groups. Equivalence. Divisors and Multiples, Conclusion and Summary. Page agar P ta @ 26 30 37 41 50 54 57 60 68 — THE FOUNDATIONSOF GENERAL ARITHMETIC PART I. PRELIMINARY NOTIONS FROM FORMAL LOGIC. SECTION 1. INTRODUCTION. We assume as known the whole machinery of formal logic, such as the notion of Ks, relation, propositional function, correspondence, formal squivalence, counting, and the like. We shall use this machinery to investigate a kind of class called a "collection" whose elements cone sist of "entities" which in a given collection are sither all "objects" or all “marks,” and certain special types of propositional functions associated with tha collection. The reason for introducing these terms is to avoid confusing the general use of the words "class" “elament" in our reasoning about col- lections and entities with the particular collections and entities themselves. Loosely speaking, by “marks we mean bare symbols which are distinguishable from one anothsr, but which have no direct connatation. By an "object" we mean something which can be denoted by a "mark." Ths distinction between "mark" and “object" is somewhat vague; to state it intelligibly would be to solve one of tha major problems of epistemology. our ultimate aim is to lay the foundations for a precise defin~ ition of an “arithmetic" analagous to the postulational definition of an abstract group. For the present, by an "arithmetic" we mean any system where in . . 1. All operations possible can be carried out in a finite number of steps. 2e Division is not always a possible operation. 3e Unique factorization into primes is always a possible operation. SECTION 2: ONE TO ONE CORRESPONDENCE. PRELIMINARY: In this section, we describe one of the most fundamental ideas of logic, that of the one to one correspondence between two classes. NOPATION: Consider two 00112 Tions of entities C, and Cy, which have the same cardinal number, so that they can be put into one to one correspondence with each other. We shall denote the fact that Cc , oad C, are in one to one correspondence by writing (ai) = C, Cu “which is read * LC, corresponds to C, ." Let of, f denote the particular general element of CG and Az ‘the particular general element of Ca 3 then (202) Ay 7 An isa propos itional function. If m1, 9 Wg 3 M7, 5 Fz, Genote cor= responding entities in the two collections, then the class of true pro~ | positions V7, <I Wp, A, <7 Fay - are the values of the propositional function (2.2). THEOREM 2.1: Fundamental Properties of © 7 « . The relation co is symmetric, reflexive, and transitive. PROOF: Clear from formal logic. THEOREM 2.2: The Five Types of Correspondence. Ali correspondences (201) can be separated into five types accord- ine as Cc, and Co contain marks or objects, and are the sams or different. These types are given by the following table: Elements of Type de 26 Be he 5e objects objects objects marks marks Elements of objects ob jects marks marks marks PROOF: Clear from formal logic. DEFINITION 201: Labelling. Relation between C, and Ca / sams different different same different A correspondence of type 3 is called “labelling,® or "naming," and the mrks are called the "labels" or ‘names"* of their corresponding objects. Each mark is said to denote its corresponding object. ' SECTION 3: FUNCTIONS. PRELIMINARY: We here describe two kinds of propositional function which we shall need subsequently. DEFINIETON 3.1: Functions of One Variable. Let 7” denote the particular general element of a collection of antities, C | - Suppose we have a rule which assigns to each value" of A 3 that is, to each entity of C , a wnique second entity of C aenoted by mA + We then say 7Z/ is @ one-valued function of A defined over C 3 briefly, a function of 7 .« NOTATION: (3.1) ¥Y = F(a) entity entity In yerticular, if the rule assigns to the chigeet the ebseos g » we write (302) an Fi (7 / XH is called the independent, and Z/ the dependent variable of the function, 2 is said to be the value of 7/ determined by the value g of a@ « It need not be distinct from 7 . DEFINITION 3.2: Functions of Two Variables. Let zc and +7 £denote particular general elements of a col- lection of entities C « Suppose we have a mie wiich assigns to each pair of "values" of ¢4¢ and 7 ¢ that is, to each pair of entities of (© , a unique third entity of (Co denoted by yw . We then say 4 is a one-valued function of ¢¢ and 7“ defined over (© , briefly a function of cw and 7 . NOTA ION: (88) We FA, v) In particular, ie the rule assigns to the Seats 9 and of C tm Sopeet’ , we write | (34) a= FCG 4) “x ani}? are called the independent, and A the dependent variables of the function. /z is said to be the value of +” deter- mined by the values G and 4 of Zc and 7“ . These values “need not be distinct, but in general FUG J and Fg) ara distinct. REMARK: (1.1) and (1.3) are propositional functions, (1.2) and (1.4) values of these propositional functions. REMARK: These two types of function are extremely general. Ths following fart ioular Kind of function is highly important in wnat follows. THEOREM 3.1: Connection between Operations and Functions. Suppose C is a collection of objects denoted by 777, 77, . and 0 en operation by which we can combine any two objects and which satisfies the following condition, called the postulate of closure: "If 77 , 77 «denote any tw elements of ( , then the result of combining the objects denoted by 77 and 77 by means of 2 is a unigue object of C aenotea by Va ott Then the object denoted by ft is a function of the objects denoted by 77 and 77 PROOF: Clear. REMARK: We may express ‘the hypotheses of Theorem 3-1 in the following form: (305) CC / m, 7 * (306) Wie We (a7 CSF (3.8) ft is unique. The emelusion then reads, "the conditions (3-5) = (3.8) imply f= Flan) DEFINITION 3.3: Equality of Two Functions. Two functions are said to be equal when and only when they have the same number of independent variables and are identical for all values of these variables. wotarts: «F(A = 6H), FLAY) = O4Y ad C/m is read t C oontains 77 ," or more accurately, " ‘C contains the entity denoted by 7, «" SECTION 4: TRANSFORMATIONS. PRELIMINARY: We here develop another fudamentel idea of forml logic | which we shall constantly employ. THEOREM 4.1: Isomorphism. Let Cc, and (CC, be tw collections of entities between which there exists a definite one to ons correspondence symbolized by (401) CY, C7 CL and assume that there exists a function Z,= F(A, 4) setined over Cc, as in Df. 326 Then wa may define a function Z,- G (Hx) over C, by means of and the relations 2,0 2,, ZW, HOM implied by (4.1) in accordance with Df. 3.2. PROOF: Clear from forml logic. THEOREM 402: The function GOK, H/ of Th. 41 depends upon two factors (4) the function PC% YB) in C, (14) the way in which the correspondence (4.1) has been set up; that is, the values of the propositional fumction 7G 67 x + PROOF: Clear from formal logic. DEFINITION 4.1: System. The collection C, and the function 7 (7 7/) are said to form a system. voration: L£C,, F(A, Y// ‘DEFINITION 4.2: Isomorphic Sys tems. ) | The systems [ C, , FCA AIL end [lCza, GCA, Y/] are said to be isomorphic. NOTATION: [C,, FCA, BIT L Ca. CURL GI] This is read " LC,, FCK, AI JS is isomorphic with [O,, COL 4/] 0m . THEOREM 4.3: Chief Properties of Isomorphism. The relation of isomorphism between systems defined in Df. 4.2 is reflexive, symmetria, ani transitive. PROOF: Clear from formal logic. REMARK 1: We now introduce a concept of great importance in all that is to follow. DEFINITION 4.3: Formal Transformation. The correspondence (3.1) is said to define a formal transformation 7 of the system LCy, al he 7 into /C,, CCH, x). NOTA ION: S(G)—> Cr 7 (%)— mK TOF (4, g)) —> CCH A) The class of all possible correspondences (3.1) between Cc and Cy 4s said to define the class / 7 # of all formal transfor~ mations 7 .« REMARK 2: The general ain of this paper may now be concisely formulated. It is to discover and study the invariants of the class A 7f of formal transformations; that is, to investigate those properties of Fe 0 M7 and OO (4, %/) Which are independent of the factor (ii) in Theorem 4ebe 10 In view of Theorem 2.2, we may say that our aim is to study those ‘properties of a system — LC, F(4#x7/ waich are independent of ‘the names we give the entities of C . | REMARE 3: We shall now give three examples of such properties. DEFINITION 4.4: Symmetry. Let LC, FCAWY / be any system. If 7CUAY) = FY A forall valuss of 7 ami 7 , FCK ww) is said to be symmetric in A and 7 , or briefly, to have symmetry, and the system is said to be commutative. THEOREM 4.4: Syumetry is an invariant of the class / 7} of formal transformations. PROF: If CC, 676 » then by (262) OM, FOR, 2672, Sines Ay Ht) = FCB) + 4, T)= 2a. and TPF CHA!) | > CCK ABI * 22 VT CF CA,%)) —> COBH) = Ze CCM, w%/) + CY, HS » so that GK Yw/ is also symmetric. DEFINITION 4.4: Primitivity. tet £ C, FIXy/ | be any system, and let 4 and ws be two values of cA - « Then if the functions of the single variable wv are equal (see Df. 3.3) when and only when A= 2 for every pair of values oA » € of SA >» the function FCAY il is said to be primitive with respect to Z/ . . Primitivity with respsct to 7 is similarly defined. A funetion which is primitive with respect to both independent variables is said. to be primitive" wi thout any qualifying phrase. THEOREM 4.5: Primitivity is an invariant of the class / 7} of formal transformations ° PROOF: Clear from proof of The 4.4. DEFINITION 4.5: Associativity. Let £[£C, Flay] be any system, and let 4 , €@ , 7 denote any entities of C e Suppose PLAC) = 77, FCC a/)= 4 where. 77 and -2 denote known entities of CO. The functim (7% 7/ is said to be associative if — F fm, 2) = FVKAS for all sets of three entities h ’ L » 7 in C ° THEOREM 4.6: Associativity is an invariant of the class /7} of formal transformations. PROOF: Clear from proof of The 44. DEFINITION 4.26: Semi~group. A system consisting of a collection C of entities and a primitive associative function /(% wy defined over C is said to form a semi~group. If the satities of (C. are marks, the semi-group is said to be abstract. — oration: £0, PC% YL] “REMARK: A large part of this paper is devoted to developing the properties of abstract semi-groups. Part III.) (Seo the remark at the beginning of 12 13 SECTION 5. COUNTING. PRELININARISS: In view of the remarks under Df. 4.3, and Th. 262, it is sufficient to confine our attention to collections of marks. We assume henceforth that the marks in any collection C are all distinct and denumerable. We shall continue to refer to such special collections of marks as "collections" in accordance with the following definition. DEFINITION 5.1: Collection of Marks. A class whose elements consist of mrks which are all distinct and denumerable is called a collection of marks. NOTATION: We shall now restrict C to denote such aclass. We shall use other letters as the occasion arises for collections of objects. ig Pg ST denote marks of C » then we write was’ if S am SS denote the sam mark, and wes’ ig S and 5° denote different mrks. DEFINITION 5.2: Set of Marks. If we count the marks of a collection C » we obtain a set of marks, C ° NOTATION: We shall denote the set of mrks CG Ly Co 5, nu, It follows from Df. 5.1, and the definition of counting, that if we regard Sy, Sn, Gr es as denoting marks, T= when and only when ¢ =// . Ofcourse yy, 4, G5, ... Gee marks are distinguishable. 14 DEFINITION 5.3: Order of a Set. ‘There is alastmrk Sy of G& or no last mrk according as the cardinal number of (CC is the positive integer “/ or Q, . We shall say in these two cases © is of order / or order 2 . REMARK: ‘The word "type" is usually used instead of the word “order in Df. 5.5, but we wish to reserve "types" for more important purposes. THEOREM 5.1: Counting is a special type of formal transformation. PROOF: In Df. 441, 2 take for C), the collection of mrks C’ and for C, the collection of objects consisting of the numbers 7% 2, 3, ... The theorem is now clear from the definition of counting. REMARK: Befora wa can take advmtage of this extremely important theoren, it will be necessary to develop the properties of functions of positive integral variables. This development cannot be done by specializing section 3, because the functions of saction 3 do not correspond to the most general types of positive integral function. PART Il. FUNCTIONS OF INTZGERS. SECTION 1. INTEGRAL FUNCTIONS. DEFINITION 1.1: “Range. The set of N numbers 2,3,.. ., N. is called a finite range of order N ° The set of all thea numbers /,2a,3%,. . . etc. is called an infinite range of order ww . NOTATION: A range whose order is unspecified is denoted by f,2%3%.- DEFINITION 1.2: Equality of Range se fwo ranges are said to be equal when and only when they have the same order. REMARK: There is only one range of order uw) . DEFINITION 1.5: The Positive Integral Variable. The particular general menber A of the range /, 2, J, is said to be a positive integral variable. DEFINITION 1.4: Integral Funct ione Suppose we have a rule vhereby we can associate with each value of the positive integral variable 7A’ a definite integer 4/= W(A/ ‘then Y(H/ is said to be an integral function of 7 XY 1s called the independent, and 7/ the dependent variable of the function. 15 16 REMARK: The various values of 7/ need not be distinct. NOBAt 10M: “We shall denote YA by 4% «+ We shail use thruout tha paper ay, Fy, >. 6) ba, Mp. to denote integral functions. DEFINITION 1.42 Order of a Functions The order of a function is defined as the order of the range of its independent variable. DEFINITION 1.5: Matrix of a Function. Phe ordered set of integers is called the matrix of the function 7 « Notation: jf Hi || (¢24%5--- 7) DEFINITION 1.6: Equality of Functions. Two integral functions 7 = YA) , Z= Z(A/ are said to be equal when and only when they are of the same order am their matrices are squale | THEOREM 1e1: Derived Functions. Let Wf (A/ be a given integral function. We can define a unique function TF of the positive integral variable +/ by the following rule: ; Y= 4 Ur* Be & Hey, 17 BH Ht uty, or Yr tGRY SH LUALLY, amd WotY “HP LALAY wh 6-H) and sO One Moreover the numbers 7, , Yo ' hoy. > are all distinct. PROOF: Clear from Df. 1.4, and rule. DEFINITION 1-7: Derived Function. The function wy, which is known as soonas 4, is known is called the derived function of Va « Note that the ranges of YY and 2% need not be equal. THEOREM 1.2: ‘Types of Integral yunctions. All fanctions Wa fall into threes distinct types given in the table below according as the ranges of 7 and Y= are finite or infinite. Qable Type Range x W 1 F F 2 I FE 3 I I PROOF: Clear. DEFINITION 1.8: Range of Dependent Variable. The set of numbers | ° 18 is called ths range of the dependent variable oy e DEFINITION 1.9: Primitive Function. | The function Z/ = VY (a) is said to be primitive if Yor Yj | Ween and only when ¢ =/ for all numbers 2) Js belonging to the ranga of A . THEOREM 1-3: Ths necessary and sufficient condition that a function be primitive is that it should be equal to its derivative. 20 SECTION 2. INTEGRAL FUNCTIONS OF TWO VARIABLES. “DEPINIDION 2.1: Integral Function. Let 4 , 7 denote two independent positive integral variables. — | | Suppose we have a rule which assigns to each pair of val ues of 4é and VY a definite integer Zz Zz (4, Y/ . Men Z is called an integral function of 4¢ and *% . NOTATION: We dnote Z(4%7/ by Zyy « DEFINITION 2.2: Matrix of a Function. The array of integers 24, , 2 Z3 f 4as Z~,5 / a Ze 2 232, A 23, is called the matrix of the function Zz e woration: // Z<; |] (é,j242,3,...) DEFINITION 2.5: Equality of Punctions. | Two functions are said to be equal when and only when their matrices are equal. THEOREM 2.1: Derived Function. By reading the matrix I] Zej /[ by diagonals, we can define @ function Zz { Ww) : of the positive integral variable /+/ . The - first few values of this function are 4, = Su. Ze- Zig. 23% Fars Zy = 23. Zeger Fza2,.-- OME AA (Behe. eZ, & a eo nes 2 . et PROOF: Clear. "DEFINITION 2.4: The derivative Z Cw) of Z (/4/ is called the derived function of ZY, vw), | THEOREM 2.2: Types of Integral Functions, Al functions Z (¥uY, Vv) fall into seven types given by the table below according as the ranges of 4c , * and WW are finite or infinite. _ Table Type Range “uw a ev 1 F a a 2 I F F 3 I F L 4 FF Lt 5 F I L $ A I F ? L L DEFINITION 2.5: Range of Dependent Variable. The numbers ¢ 4 ‘ 4, . 22 , 2; + . , 4 are called the range of the dependent variable Z ., DEFINITION 2.6; Primitive Function. It is apparent that Zer where A is a fixed value of the range of ¢¢ , is an integral function of the variable ~~ . a2 If ZL ~ is a primitive function of 7 for all values of A inthe range of cc , the function Z(4/ is said to be primitive with respect to “ . Similarly, if 2, ¢ is a primitive function of 4c for all values of 2% inthe range of ¢ , Zltt,v/ is said to be primitive with respect to 2¢ ° Afanction 2/4, V/ which is primitive with respect to both ite independent variablea is said to be primitive, without any qualifying phrase. REMARK: It should be carefully noted that the values of a primitive function of two variables are not necessarily all distinct; for example, let the ranges of ¢< and / £«dbe of order 7 and ZZ, given by / 2, 3 } Se, Then Zya = 4277 433 , but nevertheless —~Z (zu v/ is primitive. THEOREM 2.3: Types of Primitive Functions. All primitive functions 2Z2/4,,/ fall into four types accord~ ing as the ranges of wlk , 7 and p/ ~ are finite or infinite, where // igs the range of the independent variable of the derived , function 2 (w/ . These types are given by the following table. fable Type Range | i YoW 1 ZL F Ff 2 F fF F 3 lL £ T 4 F F F° DEFINITION 2.7: Symmetric Function. re Zo j) > 20,7 for all values of ¢ and / in the ranges of cc and . then Z is said to be a symmetric function. ao GRORION 3. OPERATIONAL FUNCTIONS. DEFINITION Sel: Operational Functione . The integral function Z(u,v~/ is said to de an operational function provided l. The renges of c¢ and ~* are identical. 2. The range of c2 and 7 contains the range of the derived function Z THEOREM Sel: ‘Types of Operational Functions. All operational functions Z(4 V/ fall into five types according as their orders and ranges are equal or not, given by the fol- lowing table: abies Order of Order of Range of Range of Relation between Typ 3 van ~*~ zZ° “and Zz ranges 1 NV vV F F equal 2 N MeN F F unequal 3 Ww WV L F unequal 4 aw wW L r unequal 5 w ew IL x equal PROOF: Clear. | REMARK: Operational functions are an example of the functions of Part I, Section 3, Df. 3.2. THEOREM 3.2: No function of type 5, The 3.1 is primitive. ~ PROOF: Clear. THEOREM 3.3: A necessary condition that a function 2/4 %*/ be symme tric is that a and “*" have the same range. All symmetric functions are of types 1, 6 or 7 under The 2.2. PROOF: Clear. | DEFINITION 5o2: Associative Function. Let Z (4 v~/ be an operational function, and let f , L,™ be any three numbers in the range of 4 and Ww 6 tat | 2f(40/= % ; Z/(Lom/=s Then % , " are in the range of ¢¢ and 7 , since Z(u,~/ is operational (Df 3.1). If now ZlAm) = 24S) for all sets of numbers A , € , 77 in the range of 4e and w >» z(u,v/ is said to be an associative functions DEFINITION 3.5: Functional. A primitive associative operational function will be called a functional and will be said to generate a semi-group. DEFINITION 3.4: Types of Functionals. 4 functional of type 1 or 5 Th. 3.1 will be called orthoid, a functional of type 2 or 4 holoid. PART III. SEMI~-GROUPS. SECTION le INTRODUCTION. Thea postulates for a semi-group given by Dickson are as follows.* "Given a function @od& of two arguments, ad a set of elements, ws say that the elements fom a semi~group with respect to © when the following postulates hold: (1} For every two elements @ amd ¢& of th set, aed is uniquely determined as an element of the sete (2) (aed jeC = aeGeoe) whenever @, £ ,¢ and all the determinations of aot , ec , (ach Joe ama ae(Gee) occur in the sete | (3) If @ , w# , HA’ occur in the set, and if there are equal determinations in the set of @oW , aor’ , then We= A’. (4) If there are equal determinations of you , Aca ~, then HW: , Ar THEOREM 1.1: The definition of Dickson and the definition 4.5 of Part I, Section 4 are identical. PROOF: Glear from Part I. REMARK: The only advance we have made so far is the following: Since we have shown in Part I, Section 4 that the defining properties of a semi~ group are invariant under thse set { 7H of formal transformations, it * won Semi~groups and the General Isomorphism between Infinite Groups," " Bransactions, Vol. 6, 1925, pp. 205-208. av follows that if we assume that the elements of our semigroup are denum- erable, it is sufficient to study either the ordered set of marks G. SK, Sn, Sy, and an operation ¢ satisfying Dickson's postulates, or the ranga of numbers L. /, 2, 3G, and a primitive, associative operational function, isomorphic (Df. 4.2) with GS and ° to deduce all the properties of denumerable semi~ . groups. Before embarking upon this program, it is convenient to define the term “semi=group" to man denumerable semi-group, and to restate Dickson's postulates as follows: DEFINITION 1.1:. The Semi-Group. A set of mrks Ge KS, and a function Sy ? Sy is said to form a semi-group if the fol- lowing postulates hold. POSTULATE I. Closure. If J, and %_ are any two elements of eG » oy is a uniquely determined element of G ° POSTULATE Il. Associativity. Tf Yoo» GY > oe are any three elements of CS ’ (Sno Sg)°Se = Sn 2 C42 Ge) POSTULATE III. Gancellativity. are any three elements of G ’ If 1 Go kg then 28 (ab YF ee = Ber SI, = Ve (b) lf Seed, = VeS2, Sy = Se DEFINITION 1.2: Left and Right Semi-groups. | _ A system LG: o/ satisfying Postulates I, II, Ili(a) is called a left semi~group. Asystem [Go] satisfying Postulates I, Il, I11(b) is called a right semi=group. DEFIN FP 10N 165: Closed and Closed Associative Systems. A system [G, o/ . consisting of a set of marks ( anda function So dy which satisfies Pos. I is said to be a closed system. a system [6:07 satisfying both Pos. I and Pos. II is said to be a closed associative system. NOTATION: Closed system [ GC, o /z] Closed associative system [G, ¢ / za] Left semi~group [G, o JZ. zz, TE ca) | Right semi-group [G, o/Z, ZL, Ltt 7 Semi~group [Gy o /Z, ZZ, zz 7 REMARK: Note that these systems are arranged in order of decreasing gensrality. It follows thet the properties of [ G,-/Z] and [ G o of Zz Zz/ developed in the next two sections are also properties of semi~groups. DEFINITION 1.4: Commtativity. It is necessary to consider later systems which satisfy a fourth 29 _ postulate, which we shall call the postulate of commutativity, viz:— POSTULATE IV. Gonmutativity. If Uw + SL are any two elemuts of G » then eye eos | any systen £ Gy oJ satisfying this postulate is said to be commutative. The extension of the notation under Df. 1.2 is obvious. DEFINIZION 1.5: Abelian Semi-group. The system L6G /¢f/ZL£, zz Zz] is called an Abelian semigroup. 30 SECTION 2. CLOSED SYSTEMS. . PRELIMINARY: Consider the system [ GC, o /. 'Z_/ so that we have a set of narks 42 Ja, Sv; a and a function Uy ¢ Vy satisfying Postulate I. DEFINITION 2.1: Sequences. Let Jn denote an integral function of 7 whose order is not greater than the order of (G , and denote the mark S, LY uy + Then the ordered set of marks 4 , is called a sequence of mrks of G » or, briefly, a sequence. The ranges of 77 is called the range of the sequence. Notation: 9 (2) 9 “4,, 0 (201) (Ly) LU, a ys he, DEFINITION Bek: Equality of Sequences. @wo sequences - (4, ). th, 4,4, . --. Ha, . - (7 Ze ée as ’ are said to be equal when and orfly when they have the same rangs, and “L, = Vy for all numbers -2 in their ranges noration: 9 (24, ) = (%/ | DEFINITION 2.3: The Displacement Symbol. Let @ be anumber. Denote the operation of replacing in (2.1) the mark <<, by the mark W“,,, by a.- (2.2) Et, = keg (a 4% 2/7 31 a . Then is called a displacement symbol.* ‘DEFINITION 264: The Symbol Z .« If we allow @ toassum the value © in (22), we have a E. Me = He a so that £. leaves any mark of (21) unchanged. og We denote £ by 2 4, so that Ea, = Lhd, = Hy DEFINITION 2.5: Multiplication of Displacement Symbols. a let 3 £ ° a be two displacement symbols, where it is understood henceforth that a, & > eo e By (2-2) ce t a _ ota E_ CE lt, J = E Kipt = Kap be * E “a, os a dé Cra Fo, CE are said to be multiplied to form the product F_ . THEOREM 2el: Multiplication of displacement symbols is associative and commutative. PROOF: Clear from Dfse 203, 205. DEFINITION 266: Powers of a Displacenent Symbol. wan 2 The displacensnt symbol (7,2 +0) is called the th a yw power of £ .. a Fad THEOREM 202: Any displacemnt symbol E is the power of the fundamental displacement symbol ££. defined by _—/ Aree * Stefansson: Interpolation, Chap. 1, Baltimore, 1927. We tacitly assume is lass than or equal to the order of tha SEQUENCOs ** again, we assume 4% +Zr+a < the order of the sequence. 32 PROOF: Clear from Dfs. Zed, Zed, Zebe “THEOREM 205: Assume that the order of (Gana of In in Df. 201 is aw . If after a certain value 2% of -< , the set of mrks | a fe 2, Jap . . . . “l,7,, 4 in the sequence ( au, ) repeats itself indefinitely in the sequence, there will be a least value -< ‘ ft for which this is the case. PROOF: | Clear. REMARK: If the order of Ty is , we consider the set of marks 7 where “,, = Sy m7 mm 7 meet NM C4/k nw ££ NM) and apply Theorem 2.3 to it. DEFINITION 2.7: With the hypotheses of The 2.5, the 4< marks A, lbs 47 uf » 4%, + 2. , &e are said to form a cyole of period ,< , and the sequence ( <,/ is said to be recurring with period fe 8 NOTATION: We write ( LLy y, Ue for a recurring sequence of yriod fos | THEOREM 2-4: A recurring sequence is completely determined by its cycle, and to every recurring sequence there corresponds a finite ordered set of marks of ‘EG ° PROOF: Clear from Df. 2-7. | THEOREM 2.5: If (4, / reowsfor <2: <z° , /¢ isa divisor of 14° PROOF: Clear from Df. 207. DEFINITION 2.8: Displacement of a Gyole- From (263) 1t follows that if ( iy, 4. is a recurring sequence 33 of riod K, a ft Fo uw, = £'u,= lo Qe lic met (/J<«€ C¥ yt) Hence £[(u,) = : (be . ae, Ger; mG 1 Kee, Aye, a, , Hy ha NoMPioN: = (ej, = (tty, 2 Ne (“iva 4, is called a displacement of (i, Du ° THEOREM 206: The operations z 7 a a ae = ’ fe associated with thea recurring sequence ( 2a yi of period fe form a cyclic group of which A is the generator and EY the identity. PROOF: Clear from the definition of a cyclic group. NOTATION: We shall denote a cyclic group of order /< by LCGf, ° We note for future reference the following important properties of a cyclic group. Properties a ic Group.* 1. Every element of a LOE, is soma power of a single element of period /4 , and there is one and only one cyclic group whose order is an arbitrary natural number /& « Furthermore, if fe isa prime 7 , the only group of order 7 is the [CG/, > and there is one and only one cyclic group of infinite order. 2. The LE 6], contains one and only one sub-group (also eyclie) whose order is a given divisor of fe 9 80 that the determination of all the factors of j- sives all the sub-groups of the /C E4,, . . Finite Groups, Miller, Blichfeldt and Dickson, pp. 54-57, New York, 1916. 3 If ke ', the sub-group of order 4 contains glk ) generators, nadif £ generates /[CE/,. ’ lam “generates - lCéJ,, when and only wen “(8 4 /=/. 4. The group of cyclic substitutions on -- letters, the group of least positive residues modulo #~. with the operation addition, and th the group of the ff ~ voots of unity with the operation of mltiplica- tion are simply isomorphic with the /C G4, * DEFINITION 2.9: The Cyclic Group of a Cycle. The cyclic group [C6], E/E E,. 2. EX is called the cyclic group of displacements associated with the cycle DEFINITION 2.10: Equivalent Cycles. Let (42) ’ VY le be two cycles of the sams period “~ . then ( 7, j, 18 said to be equivalent to (4, when and only Me when it isa displacemnt of (4 Lye . Notation: (14 ye <7 ( ka) THEOREM 2.7: The class of all cycles equivalent to (“4 J, forms the set of cyclic substitutions on the sc marks “,, Hs, as, . 2, C& PROOF: Clear from Dfs. 2010, 2¢8 and properties of the Lee}, . [HEOREM 208: The relation C7 of Dfe 2.10 is symmetric, transitive and reflexive. PROOF: Clear from Dfs. 228, 210. 35 REMARK: We have developed the section above as the simplest example of - the "equivalence relation" which appears continually in general _ arithmetic. THEOREM 209: Assume G contains precisely .// marks ST, 3, %, . -, Sw and let . ae ™, 777 a Wy ft = 7) : 77. . - be the factorization of » , where 77, J, , _ TS are distinot primes numbers. Then G contains precisely # ¢ fe. pe gyoles of period /“ . PROOF: Let 4 (a) be the number of sequences (4/2 ) which recur for ~1:= W and MK) the number which have a period of exactly 4 « Then by Th. 2.5, 1p) = a WV (a) But clearly if ’ is the mmber of mrksof G , fF Al(H) = WM The result stated now follows by Dedekind's inversion theorem*, the summations extending over the combinations /, 2, ...,¥ ata time of the distinct prime factors 77, 72, ° °° 7s of # « THEOREM 2.10: There are V(/z) distinct sets of equivalent cycles of period uw in G PROOF: Clear from The 2.8, Df. 210. | . Dickson's History, Vole 1, pe 441, eq. (2)- 36 DEFINITION 2-11: Product of Recurring Sequences. | Let ( f , (4% )) be two recurring sequences of periods 4c and XY yaspectively. The sequence (2? % J is called the product of the sequences (42) and (A) . notation: 9 (4,0% =) = (4, /)e(%) REMARK: It is scarcely necessary to add 4,07, denotes a definite mark of @ determined by KHnr= Un , Ke = &, Myo vy = ths ¢SZe THEOREM 2.11: With the hypotheses of Df. 211, (u,¢7%%/ is also a recurring sequence, whose order is the least common multiple of fe and y., PROOF: Clear. REMARK: This theorem is fundamental in the general theory of recurring SOrIGS. | REMARK: Any finite sequence 4, Uy, ay, . 2, He, 2 2. by can be written as a recurring sequence, by letting Mim = Si = My mE A at N (SKAEN) so that Df. 2-11 gives us a method for defining the product of any two sequences. 37 SECTION 3. CLOSED ASSOCIATIVE SYSTEMS. | PRELIMINARY: Consider the system /. GCG, eo /Z,HZ/ 0 tat w have asetof marks | | G - Sy, Sn, So5 and a function Sy ody satisfying Postulates I and II. DEFINITION 3.1: Power of an Element. | If <S is any element of CS » ~2 any positive integer, the symbol WS defined by tre A (3-1) S = So S / - OS. pi is called the -Z power of Se pa THEOREM Sel: G contains the -2 power of any element in it. PROOF: Clear from Df. Sel, Pose I and induction. THEOREM 3.2: If cf is any elementof © , (302) Pore e Poet pt PROOF: ort Pose sos’ dy (301) Ass une see = sot Then sos. pe (obs) by Df. Sel = estos by The 3e1, Pos. IL = (rée) oS | by hype 2 sth 5 by Df. 301. Hence by induction : (A). sos *. uty ¢ a aA ‘ _ vere yess ers 5 Oy (3,2) (t=42,3 .. . SF _ she t. sete t** | (#& then | (B) tt oh 54a) = Se (ros*) - ts Jar® = (var? jor = SAS ESY = Soltkost) = (vost Jor or (c) pate 5 test me ptrk gp treet and yp tts _ [ser JeS _ wt (stos! - wt sh! _s t,t or etree! ; pet, ,4 bee Hence by induction . “2 ¢ Ja a pote 38 fixed ) by Df. Sel py (A) for 27: by Th. 3.1, Pos. II by (A) -by Pos. I1 by hype by Pose II by (A) and Df. 3.1 by hype by Pos. II by (B) by (Cc) 39 DEFINITION 3.2: One-dimensional Spread. A sequence . where . | Ty 4 (3.3) 4, = G,t ¥ and S is an element of (GS is called a one-dimensional spread, of which J isa generator. NOTATION: /9/ THEOREM BBs. Any one-dimensional spread is an instance of a system which is closed, associative, and commutative; (3.4) 42,0 4 = ho ty = 4 (4,5 27,4,5%--. 7) PROOF: Clear from Th. 3.1, 35.2, Die 302. THEOREM 3.4: If a one-dimensional spread contains an infinite number of different elements, they aré all distinct. PROOF: Let /u/ be a one-dimensional spread, so that JS a4 in the sequence 4, ) . If the 4 3 are not all distinct, there exist positive integers ¢ and §/ sach that (5) 4 = 4eyy Then hye 44 = fey aes ( Oo €& 2 €y-t y ep teny = Acar 77 = og, 4, ay- > by (362), (33), (304), (3.5) and induction, so that Gay) consists of at most <4y+/ different slements ty, 42, Fe... 45,4, 44 ae ‘ @- 24 g-ys gy orsre "ee y mrs ey REMARK: Although the general theory of spreads of any number of dimen- ‘ sions whose generators belong to a closed associative system, a semi- group, or a commtstive semi-group, is a subject of extreme interest and importance, it will not be developed here, as it would take us too far from our original aim of defining an arithmetic. Such a definition is, indeed, an indispensible preliminary to the theory of spreads itself. Suffice it to say that in the theory of spreads we have thse Key to the whole problem of ideals and ideal nwnbers in any conceivable , arithmetic, in the wider sense. AL SECTION 4, THE IDENTITY. DEFINITION 4.1: Identity of a Left Semi-group. Let [@, e /Z, Zw, Te as] be a left semi~group. If ( contains an elemant ¢ such that for at least one elemnt S of @ (401) Secs SF, Clr. then ¢ iscalled an identityof G . THEOREM 401: If the identity ¢ in (41) exists, 4 where UT is any element of G e PROOF: JSos’= sos’ . SOC = iF" | by hyp. (Fe tjes's sos’ | Se loos) = Jes’ by substitution py Pos. II , ( Loss by Pos. IlI(a) THEOREM 442: If SS’ is any element of (S commtative with the element S of (461), so that Ses = ST ‘oS then 4 . ¢ S ¢ge4 «- S PROOF: Soeas a by Hyp. Se (Se ej = ST aS CSosjel=ses py Pos. II Ves Jee = ver by hyp. ry foo - vos’ Se (Sat) = 4? by Pos. II / Sol = SI by Pos. Iil(a) THEOREM 4.3: If the element c of (4.1) exists, it is unique. ‘PROOF: Suppose there were a second element / such that ‘REMARK: It seems impossible to infer from (401) that S oz = 5‘ for every element s’ of G without the use of Pos. I1I(b). To prove this conjecture it would be necessary to construct an actual example, which I have not yet been able to do. THEOREM 4.4: Theorems 4el, 402, 403 remain true if the in (4.1) is replaced by the < of (4el). PROOF: Clears | REMARK: The corresponding theorems and definition for a right semi~group are aprarente THEOREM 405: Let L@ le /L, EL, the 7 be a semi-group. Then if (S contains anelement cz such that coe -¢ » then for any element S of G ’ | Set - Cefet S and, moreover, ¢ is unique. PROOF: Clear from The 401 ~ 464 and the remarks above. DEFINITION: 442: If the ¢ of Theorem 4.5 exists, it is called the identity of the somi-group. MEORM 4.6: Let / G2 / 2,2, Dla] be a left semi- groups If GE | contains an element <¢ such that (4.1) holds for every element S of G 5 then for every JS Sot = Cosi and ¢ is wmigue. PROOF: Clear from Th. 41 we , REMARK: The corresponding theorem for a right semi~group is apparente Hereafter, we shall state all definitions and theorems for left semi~ | groups alone; the corresponding definitions and theorems for right semi~ groups ars | easily inferred. REMARK: The substance of these theorems on the identity are given in Dickson's paper, but thay are proved only for a semi=-group, on much stronger assumptions. REMARK: Although it is unnecessary to postulate the existence of an identity element in a system in order for it to be an “arithmetic” in the narrow sense of the definition given at the close of Part I, Section 1*, nevertheless instances of systems in which an identity exists are of frequent occurrence, ani the abstract theory of such systems is of cosiderable interest. We accordingly frame the following postulate: POSTULATE Ye The set. CG contains an element < such that (a) for every element JS in the set, Jot =z s (bd) for every element 5» in the sat, | Ceos- SF (ec) gogee (THE OREM 4.7: The properties expressed in Pos. V are invariants of the class v T St of formal transfomations defined in Part I, Section 4. ‘PROOF: Clear. * Por example, the set of all numbers of the form ashe! vhere XxX » £ » YY Frum over all the positive integers form an arithmetic in the sense of Part I, Section 1, but the set does not contain an identity. PART IV. THE OPSRATION OF DIVISION. SECTION 1. INVERSE OPERATIONS. DEFINITION 1.1: Table of Entry of a System. Let. L GC, e JZ / be any closed system. The matrix I] ¢S Wi (Of -hus oy where we write for each Jf ¢ Jj its wmique determination in (© is called the table of entry of the system. REMARK: The table of entry is simply isomorphic with the matrix |/Z,; / of the owrational function Z(A y/ associated with JS,oty . DEFINITION 1.2: Inversion Problems. . tet / G, ¢/ =] ve any closed system. In the function (1) SZ = Spe Sy we regard Jy as known when 4, and Jy are given. Let us suppose instead now that ~/, is known; there are three conceivable Cas6Se (4) Given S » to find XZ, ama Sy . (43) Given YZ and Jy , to find Sy . (444) Given S, am Vy , to find Sy . These three cases ars said to constitute tha inversion problem for the function (1). Obviously (i) includes (ii) md (iii). THEOREM lel: Types of Solution. There ara six types of solution conceivable for an inversion pro blem, according as we can find an infinite number, a finite number, or no Sy ‘8 and Sy ‘s to satisfy (1) for a given 5_ «+ ‘These types = ~are given by the following table: Bab le Type _ Number of Number of | : Sy Sy 1 T I 2 IT iF 3 FE L 4 F 5 none ~~ 6 — | none PROOF: Clear. THEOREM 1e2: The necessary and sufficient condition that all possibile solutions of the inverse problem be of type 4 in Th. 1el, is that any particular element S of CS appear in the table of entry of the system at most a finite number of times. PROOF: Clears THEOREM 1.3: Phe necessary and sufficient condition that the inverse problem always have a solution is that every clement J of G ap= pear in the table of entry of the system at least once. PROOF: Gleare THEOREM Leds Solutions of Types 1, 2, 3 cam ocour only when the set is of order (O . PROOF: Otherwise the table of entry is finite. THEOREM 1.5: A sufficient condition that the problem of inversion always be solvable in a finite number of steps is that G be of finite order. ‘PROOF: Clear. | ‘THEOREM 1.6: If the inversion problem in L Gy o/Z/ nas the property that all its solutions are of types 4, The 1.2, this property is an invar= jant of the sot / 7} of forml transformtions of (6 defined in Part I, Section 4 for G of order VY or order w « PROOF: Clear. THEOREM 1.7: If (G is of order «) , a necessary and sufficient condition that all the solutions of the inversion problem for be of type IV is that in the operational function w= Z2(4% 4 / = Flav) associated with the system, (1.1) Za > as 7 oo and Zp we as V9 Moreover vhen this condition is satisfied, tha inversion problem can always be solved for any given value A of Z ina finite number of stepse PROOF: Let A. be any fixed value of 2Z , and consider the solutions o¢ (4) b-~ Fl(4 Vv) Assume first Z ~~? 2 as lt > Then we can find two fixed numbers Z and. ZC such that (14) ze WA for all values of < le 3 ise, for an infinite number of values of 4¢ . Hence if we consider the solutions of j2 F(4V/),a:Flar/, .. , Z+ Flav there is at Least ons value 4 ( /<#tes Z ) such that (i) has an infinite number of solutions cc , so that the inversion problem for this 4 _ is not of type 4. Similarly, it is necessary that Z-> L& as V—D>xX .« 47 Now assum that the conditions (1.1) are satisfied, and let us cons ider the solution of the inversion problem (1) for 7:4 ,a fixed nuniber. | | From (1.1), we can find a fixed number NV: MR) such that when > AY ’ ro WV ’ z> k& . It follows that all possible solutions of (i) lie in the matrix formed of the first 1/ rows and 1” colwms of WZ He + 80 that they are necessarily finite in number and can be found in a sinite number of steps. EXAMPLE: Consider this instance of [Ge / =] Hd the set of numbers / ,y 2», SF , Coa and ° ordinary maltiplication. Then Zs ao and the condition (1.1) is ful- filled. It is not difficult to maka the extension to algebraic integers and multiplication, unit factors disregarded, by thinking of them ranged in order of increasing NOIMS « . REMARK: It should be carefully noted that if (6 is of order ww , it is not sufficient that the solutions of the inversion problem be all finite for us to find them in a finite number of steps without the use of Th. 1.7; for if #- Flav) has no solutions, we must be able to assign limits to cc and 7 beyond which we need not look for. solutions, in order to be able to solve the inversion problem in this special case in a finite number of steps; but this is precisely what (1.1) gives use 48 SECTION 2. DIVISION. “PRELIMINARIES: wet G= L£ G5 og Jz, JIL, TIL fas, ga ] denote henceforth a left semi-group with elements | Ff ,, G, , a, , and an identity element Z . Weouse GF 4 GF ,. occasionally for particular general elements of G . The notation G/9,9) 94 mans (7 contains the elements 9 , G and 7 DEFINITION 2.1: Left Divisors. If for two particular elements Gs? Ih of CG » there exists 4 Gy in CG such that* (201) Gee Gy = Gp C/ Ga we say that <7 is a left divisor of _7f , or that it left divides Ue © Gy is called the “quotient” obtained by left division of Ge by Lb »* Or when there is no ambiguity possible, the "quotient." omits: 9G, : WC%) | & VEZ DEFINITION 2.2: Right Divisors. If for two particular elements GF ? Gp of G , there exists a Ly in G _ such that . Sy? Fh? Lk we say that G, is aright divisor of G «+ NOTATION: Gy - Lh, CH / * Compara Section 1, Df. 1.2. 49 DEFINITION 2e3: Divisor. If at least one of the two statements Gy - L, (Ge ) ’ Dy 2 Lh (Ge) is true, we say that , is a divisor of Gk» or that it divides G, « norton: 9,- W(x) » oe |G ‘ DEFINITION 20h Multiples. Under the conditions of Df. 2el, (202) Gg is called @ right (left) multiple of Ly, » and is said to right multiply. (left ml- tiply) OG, « NOTATION: on: pel 4%) (= 1, (%/ ) If at least one of the two statemnts FG = “1, (G%) >» Sh* Ve (Gy ) is true, we say G, isamultipleof gy 3 or that) 8 o, multiplies G, « NOTATION: Ge: 71%) THEOREM 201: The Ix of Df. 2-1 is unique. prom: GO / 94, Gx, Ge, Gp 6 Fy Ih. by hype Assume C/G," such that 9% «9 - gy Then ~The °Gy = Fs ¢ Pe 50 SECTION 3. UNITS. PRELIMINARY: The results of this section are due in the main to Dickson, who, however, proved them for a semi-group only. THEOREM 3.1: ‘ie for any element G, of G there exists an element FS & such that | (301) Gye Gt | (C/L vy Pos. Va) / then SA 44> 4 Furthermore, if such a Lf exists, it is uniques. PROOF: GC / Shr Gh, G4 Wp L by hype Gr * T4 (25° 4) +G,2 4 “Gy Gte (eG) - Ipod by Pos. II and III, Th.4.6 HU? Gp= A by Pos. IiI(a) Sk is wnique. For suppose there were a second elemnt Lp ‘ such that =F og,” = g . Then Le Ge = Gh eg" Gh = Gn. by Poss IlI(a). DEFINITION 3.1: Inverse. | Tha element G4 of (3.1) if it exists is called the inverse of dy . | THSORSM 3.2: (4) If Gy is the inverse of G, , G, is the inverse of .J . (ii) i is its owm inverse. PROOF: Clear from Th. 3.1, Df. 3.1, and Th. 4.5, Part III. 51 THEOREM 3.3: Every left divisor of 2 is aright divisor of Z and conversely. “PROOF: Let D4, * ZL/ (4) ’ Pes He (4) Lj,’ Py = L and Ly eG * Zz Cl Gxfy by Dfe 201,202 Gr Gp, 4 ami KH 1 by The Bel Ly _w/, (1j/ ma 4 ~Y (4) by Df. 202,261 We may henceforth speak simply of the divisors of 7 without ambiguity. DEFINITION 5.2: The Units. The class of divisors of -£ in G is denoted by Le. E,€,,... ,6, Any member of it, Cc, » is called a unit. THEOREM BeAt E. is not a null-class. eg: PROOF: ZL ZL, 2/2, Z isa unit. THEOREM 3.5: The class / of mits of CG form a group with respect to oe . PROOF: (41) If ©, , €, are any two elements of ££. , Ge, is uniquely determined as an slement of a For C,¢€y is uniquely determined as an element of GC , sine G /C, , & by Dfs 302, by Poss Ie Moreover, /_ / €,7€, | » for by Die Se2, 2el, The Bel Curcéyo A » Eye eee EZ / €y, Yy (E46 & ley és) > Led st Ene (byt (Eyc&) ) - Z- by Pos. II ECE, (fae €y/ Es 2 by Pos. II (E27 (fx Pt Vf ee rd . by Pos. Iz oy f Ee (E06 € a ~f ae Cre € ba (Ey <y) / 5 by The Bed (Greta fey) = 2 hy Far ge, [ 2 EE £06, by Df. 2.3, Bee (here eA j ) - - (44) It Ey » G 5 €% are eny three elements of x | (E, 0 &/? e = Ens (Epo éz/) | for G/ Ey, €&, Eg and Pos. II holds. tau) £. conuteins an elemant Ey sigh that for way elemenlE, 6,0 Ey s E, 2 Ep, = €2z tor & / 4; Gl Er, AE, = Enger ® 7% oy The 406, Fert Til. . (iv) Por every é, there is en €, such thet C42 Gr = Cre €,s Ea from Df. 3.2, 203, The Sal. But (i) = (iv) are the postlates for an abstract group. THLOREM 3.6; The produet of amy number of units is a wit. PROOP: Clear from Ths 3.5. CM S.7: Tf Gp Dg isauit, Gy md G4 are PROOF: Let G4 o Gy = E, 5 and let Ey be tha inverse of | G, » Then (G42 De J? €2z = €, ° €, = Z by Ne Bab Of - oo - a. , | (Bo°. 4 |? Fx G4 (9g 7&2 / be ZL W a Gye TI(4), 94-Mlt) — EL #4. 94 G,o9,, .-. Gr are all wits. PROOF: Clear from Pos. II and Th. 3.7. 53 by Pos. Il by Df. 2.3 is a wnit, SECTION 4. INTEGRALITY. PERLDONARY: tet G=- LGje/F, 2, mee, Fe] | | It may happen that Z. eoincides with G - It follows then that | G is a group by Th. 3.5. But in a group, the problem of inversion of Section l is trivial; for THR OREM 4el: If Oe . Ge are any two elements of a group G then 9G, left divides 9, amd 9, right divides G, . PROOF: Let OG, “ be the inverse of go, + Then (Ga? Ge)" 9" 4 °G = 46 But also (Ga ° Ga J eGy - Ga? (Ga 2%) Hence fa = L/, C4 ) \ and similarly Ga > Oe (He) It follows that in a group, every elemnt divides every other element, which otates ons of our fundamental conditions for an arithmetic (Part I, Section 1}. fo insure that our system shall not be a group, we frame the following postulates POSTULATE VI. Integrality. . The set G contains two elemnts Sy, and Vg such that (a) Ugo Sy = SL is satisfied for no elenent vy o G . (bd) Sy? Sy = Sh is satisfied for no elemat ty of G - 55 Assuming G contains the element ¢ of Posmlate V, the system contains an element Jy such that fe) Spe Sy = e is satisfied for no elensnt Sy of G . (a) Sy e SZ = c is satisfied for no element Sy of GB e DEFINITION 4.1: Intesral Left Semi-Group. The system / G ; eo /f/Z, IE, TE la), Bila, | is called an integral left semi-group. NOTATION: We shall continue to use CG amd 9 for this system. THEOREM 402: GC contains an infinite number of distinct elements. PROOF: Consider the elemat (Gp, of Pos. Vila). If Gy, a, on? pes _ are all distinct, the theorem is proved. Assume they are not all distinct. Then there exist integers “2 and Ww such that rs A A” I4 = Gf C 4,52 1) a re Now -, 7 “2 74°94 > Th 294 ~ % °CIp~¢(%y eg, ) SA > Gs (Gy ogy ) contradicting Pos. Vila). THEOREM 403: If Pos. Vi(a) is true in a left semi-group, then VI(b}, (c), (ad) are also all true, if Zhe Semt-pravuh tonF ain L PROOF: Clear. PHEOREM 4.4: If Pos. VI(a) is false, the left semi~group is a group. PROOF: Clear. 56 2 as . THEOREM 45: If |G is not a unit, G.@GsG ss... are ‘all distinct. PROOF: - Needless to say, this theorem assumes the system contains at least-one mit 7 . ; s Sr 2 If G =_G (SAY 5 54 GF 74> GF OF Z-> 9 LG? LI Azt, GF JL AD) so that G is a unite THEOREM 4.6: If G/z , G_ contains no element 7, such that FG°Fo = ft > forany Gg in C72 , PROOF: If GC / Go: LZ, G, * Go et Suppose for some 4 xx S°Se* Go "F* Fa ‘Then , G° Go- Go F= GerC#, Gt REMARK: If C is the class of positive integers and ¢ is mil- tiplication, Ie is zero. 57 SECTION 5. TYPES OF SEMI-GROUPS. ‘PRELIMINARY: We are now in a position to classify the possible types of semi-groups. The first obvious classification is into those which contain an identity element and those which do not. Consider now the system G-/G@,¢/£,£4,ilaw, Ze, wae] DEFINITION 5.1: The Three Classes of Elements in C . CLASS 1. Contains all elements GS such that we can find a positive integer -< for which 4 -G is called a root of mity. (This class is not null, since Let) s CLASS 2. Contains 211 elements _G such that (52) G¢ / x the 9 of (5.2) is called a unit. Class 1 is a sub-class of Class 2. CLASS 3. Contains all elements for which neither (51) nor (562) hold. These ¢lements are called integral elements. (This class-is-not-null-.by Posy-¥ite}-encthe--theorems Of Seetion 4). THEOREM 5.1: Classes 1, 2, 3 are closed under °¢ and , a belongs to the same class as JS , and conversely. PROOF: Clear. THEOREM 5.2: A sufficient condition for c to belong to Class 1 is that fo G ’ = 9°? £, 44, PROOF: Clear. THEOREM 5.3: The powers of an integral element are all distinct. PROOF: Clear. 58 DEFINITION 5.2: Unitary (Semi-) Groups. _: semi-group which contains only elements of Class I, that is, such that the inverse of any element in it may be expressed as positive power of that element, defines a type of group which includes finite discrete groups as a special case. Such groups are called unitary. THEOREM 5.4: There are an indefinite number of infinite discrete groups which are unitary. PROOF: Let 77 run through any infinite sequence of numbers which are all relatively primes to each other; for example, A Ay “4 ad * 4 a gO FTG TE gz where CA; J) is any sequence of pos itive numbers whatsoever, so that Ay _ 74 F1= fe : Ju tha ¢ prime. m — ; 2Ire a vias - 2 ad The totality of elements Yo Si SuJa, + xe form just such a group, and moreover, no two groups of this types are simply isomorphic. DEFINITION 5.3: Rational (Semi-) Groups. ri semi-group which contains no elements of Class 3 and at least one element of Class 2 is called a rational semi-group. A rational group G necessarily includes at least one root of wnity; namely, the identity element L | - If this is the only wnitary element unit, G is called pure; otherwise mixed. 69 THEOREM 5.5: Every rational group contains an infinite number of elements. PROOF: Clear. EXAMPLE: Tha pos itive and negative powers of 2 | » With the opration of ordinary multiplication, form a pure rational group; the positive and negative powers of + 2 and of —/ with the same opsration form a mixed rational group. DEFINITION 5.4: Integral Semi-group*. A semi~group CG which contains at least one integral elemnt G is called an integral semi—group. If it contains no elements of Class 1 and Class 2, it is called pure. Those elements of Class 1 and Class 2 contained in C are called its units; their detailed study is the subject matter of the theory of discrete groups. Ths detailed study of integral semi-groups is one of the chief objects of tha theory of general arithmetic. * Compare Section 4. mm etsy aN TA Te AY Ey rare SECTION 6. BOUIVALZSG. QOL INARY: Im this section we develop the relations between an integral | left semi-growep GCG =. LG, e/Zz, are far, TE tar, Yh) | and ite units £. 7 « din tee proofs, wa have omitted the reasons for tha simpler stecbs DRPINIQION 6.1: The Generel Equivalence Relation. Leb €, , | €, be any two unite and Ly » Pe any wwo glements of GC a G4, 18 said to be equivalent to when. and only when (601) Le? E = ey ae a MOLADIONs Tt Gk ‘“TESOREM 6.1: The relation ©? of equivalence is (i) symmetric, (ii) transitive, (i111) reflexive. 2 Des & UL. a PRooP: (4) Suppose Gt Gk so that by Df. Gel, If o Ey = Cy + Ge tet Ey and €,, be the inverses of <¢, md €, respectively. The n Cy eC Gee bay (Eee Eu OCG) CE, Cw?’ Gaz Ge ?-Gy 4 ° © ye = Ey “Gf, Sh. ley a by DP. Gals Gh (41) Suppose Ge +? G4, Ge <? Phe « Then vy Df. Gel, But S,lEe, Ere Gu are units Therafors Gf Ge by Df. 6a. | (ait) Z isauwunit, Geek = Leon QHOOREM 64.2: (4) ALL waits are equivalent. (11) If an clement ¢ of G& is equivalent to @ wnit, it eqals a unite PROOF: (4) Let E, » & be two units, €,. , €&y their inverses. Shen ey co? Es by Die bel. (41) Suppose GY <7 € « Mhen by DF. 6o1 Ge En»r~ E,0€ - a unit. vt G is @ unite by The 3%, DaPINITION 6.2: left and Richt Doeulvalence. If in (6.1), €,- 4. > 4 is said to be left equivalent $0 | GL 2 NODA? LOM: Gs LO Gk if in (6.1), C,e dt » gg is said to be right aguivalent to U4 . SESORSH 6.3: he relation fen of left equivalence is symmetric, trensitiva end raflexive. 62 PROOF: Clear from proof of Th. 6.1 on taking Cy £ in (i) and Es > €e> £ im (44). . PHOREM 604: (i) All units are left equivalent. (ii) If an elem nt | G of C is left equivalent to a unit, it equals a unit. PROOF: Clear. | THEO RAM 6.53 The necessary and sufficient condition that two elements Ge + Ge of G be left equivalent is that they left divide each other. PROOF: If Gy, L° Ge, Gs? €° GFR. Gp > Ly Ge) But if G4, “2 Gk, Gi G2 G#k by The 6.3. Gt 7 4 CH) Ife G4 - WD, C%, / and G - I (Gj » then Gh° Ge - Fh» Gh Gy G4 Gh Gy Gy ~ Gp Gy = GL Lh, CGxe°Gy! ~ Got are units, DEFINITION 6.3: Conjugate Equivalance. If in Df. 6-1, C, 7 Ey » Gg is said to be conjugate equivalent to FY in the first sense. NOTATION: Gh << Gk If in Df. 661 €,o Eg = ; G+ is said to be conjugate squivalent, in the second sense. NOTATION: Gp ©7 Gk | If in Di. 6&1, €,= E, ,and Gp, oe, 2 & » is said to be conjugate equivalent in the third sense. 63 NOPAT TON: Gp = Le | REMARK: Conjugate equivalence dn the third sense is the simplest generalization. conceivable of ordinary equality. It corresponds to integers - having the same absolute value in ordinary arithmetic. NOTATION: We shall use Gy CF GK to denote that Gy is conjugate to G4 in at least one of the senses above. THEORSM 606: The relation -to- of conjugate equivalence is symmetric, _ transitive, and reflexive. PROOF: Clears PHSOREM 6.7: (°., If YG is conjugate equivalent to a unit, _Y isa unite PROOF: Clear. REMARK: There is no analogy with Th. 6.2 (4) in general; it is obvious we are here encroaching upon one of the most important ideas of group theory, that of the transform of one element by another. For if Gy’ Fx= “aL and €, is the inverse of <=, , Gh €42° G40 Fa" If SH o Ex = es “¢O¢ SL = €, ° G4 cé, , If | . REMARK: All these relations of equivalence share a remarkable property: we camot infer Sy °F "C7 G6 F,' unless _ the units of C are commutative with all the elements of G . met Again, if _GY¥ / Fh. Tp "Fe, Ip Fm ‘thon we camot infer _G / Im without assuming the wits of G are commutative with all the elements of G . We thus have two properties of the integers (and for that matter of the alesbraic nunbers) wnich have to be sacrificed if we give up the commtative law. SECTION 7. DIVISORS AND MULTIPLES. . ; , ‘ Lepl . PRELIMINARY: We are working throughout this section in an integral, semi- group G <éontaining a unit, G- [Gy o/L, 2, M4, Bia, gicas_| THEOREM 701: If gG is any element of G 5 (i) 9 left divides 7 , (ii) Z left divides GF . (iii) If 7 left divides a unit, 7 isaunit. (iv) If 9-9, Ileftdivides G , 94 isa wit. (v) Every unit left divides every other wit. PROOF: Clear. THEOREM 7.2: If Dp left divides Ge and yg left divides G,, + G4 rett divides 9, . mor gy D1, (gg). 94> Uh (gn) OD IpePyr> Gh 1 GU Gy> Gm CL ¥x: Fy (9-4¢ G4) ¢4y > (Ly /*Im Gt, = LL CGm) THEOREM 7.3: If St left divides Gp, » all the left divisors of 4 are left divisors of Gg - PROOF: Clear from Th. 7.2. THEOREM 7.4: If Gk ‘has a finite number of left divisors, and oy left divides Gy , and is not left equivalent to oy Sf, nas fewer left divisors than Gy ° PROFS fp FL) ( Gy) »for it oy -L) (%/ since Dy > ZL, (Ge /, If, L£°. Gy by The 6.5, contrary 66 ' to hypothesis. But G¢¥z * LL, (94 ) by The Vel G4 has one less divisor than. x7} » by The 705-5 | THEOREM 7.5: If Gp eft diviaes Ge 5 Gy lett divides Sh ° Gm ° PROOF: SA - L/, a7; /, Gh bo Li, (G 2G, J by hyp. and Df. 2elL Gt ~ Ly (9 °Gm_) by The 702. THEORSM 7.6: If Gy, left divides on, » Geg¢y left divides Ge 4 and conversely. PROOF: Gps L, C% J by hype Sy °Gy ~ Gh by Df. 2.1 GS ¢ (94° Ge) = G94 FP? Gp, ~ Ly (9G) Conversely, if OG 1 Gy - LL, CF °F) S* $4°9, = 9° C/ Fr by Df. 2.1 S4A° GF, = Sk G4= LCG / THEOREM 7.7: If G is any element of G (i) |G right mltiplies G- ; (44) _G right multiplies 7 » (i114) if a unit right maltiplies G+ GY isa unit, (iv) if G right maltiplies G4 Gy : GL is aunit, (v) every unit right multiplies every other unite | PROOF: Clear from Df. 2.4. 67 THEOREM 7.8: If G, right mltiplies 7 and gy right multiplies Gan ’ Ge ight multiplies _7,,, o. PROOF: Clear from Die 204, The Ted THEOREM 7.9: If _S4 right miltiplies Sk » @ll the right multiples of G4 are right miltiples of It . PROOF: See The 7ed- THEOREM 7.10: If Sk rignt miltipliss G4 : Le °F om right miltiplies Gk ° PROOF: See fhe 765. THEOREM 7.11: If 4, right multiplies YA + 5° £& right multiplies G*GH and conversely PROOF: See The 746. REMARK: Note the strict analogy between the theoremson divisors and multiples. 68 SECTION 8. CONCLUSION AND SUMMARY. ‘We have now reacnad a logical stopping place. In Part I, we have subjected the concept of a class and a binary operation to a careful scrutiny, and shown that for the purposes of general arithmetic, where all classes are denumerable, we can assume the class ordered without any loss of generality, or replaced by tina mumbers one, two, tnres, , We have thus "arithmetized" gensral arithnetic in Kenecker's sense. In Part II, we have sketched the theory of arithmetized general arithmetic, without, however, any attempt at a systematic treatment. In Part III, we nave developed some of the properties of an abstract semi-group, a concapt indispensable both in group theory and arithmetic. Finally, in Part IV, w have given a fairly complete analysis of the concept of an inverse oyration in a semi-group, using the language of ordinary arithmetic. The linking together of the idsas in Part II and Part III remains a desiderata. Probably the most important consequence of our work is that we now know the problem of defining “arithmetio" in the narrow sense of ths introduction has already bean solved, provided wa assume our "multiplication" is commutative. For in view of the theory of formal transformations developed in Part I, the complete holoid realm of Julius Kosnig*® is an abstract arithmatic, provided we interpret his "Griisse" as marks" and his "Verknupfungen" as “functions't in the sense of our papere In cone lusion, I should like to thank Professor E. 7. Bell for criticism and advice in writing this paper. * see Algebraische Gréssom, Chap. I