HHYPEROPERATIONS IN EXPONENTIAL FIELDS
JUAN DIEGO JARAMILLO SALAZARUNIVERSIDAD DEL VALLECOLOMBIA
Abstract.
New sequences of hyperoperations [1–5] are presented togetherwith their local algebraic properties. The commutative hyperoperations re-ported by Bennet [1] are introduced as a sequence of monoids; after identify-ing semirings along the sequence, the corresponding fields are constructed viainverse completion. Introduction
Hyperoperations are infinite sequences of binary operations extending recursivelythe definition of the ordinary addition and multiplication [1–5]. The earliest knownreport on hyperoperations is due to Bennet [1] in 1914, where commutative hyperop-erations are introduced starting from the multiplicative identity of the exponentialfunction: exp( a + b ) = exp( a ) × exp( b ). More than a decade later, building on apaper by Hilbert [2], Ackermann [3] introduces a sequence of hyperoperations thatextends the definition of the ordinary addition, multiplication and exponentiation.The latter sequence is here referred as regular hyperoperations, noting that theyare noncommutative beyond multiplication. A numerical system based on regularhyperoperations is introduced by Goodstein [4] in 1947, the complete hereditaryrepresentation of nonnegative integers. He also coined the names of tetration, pen-tation, etc. to denote the hyperoperations after exponentiation, and included the successor as the primitive hyperoperation.This study presents an algebraic characterization of Bennet’s sequence of com-mutative hyperoperations and a recursive procedure to create new sequences. Thegeneral goal is to provide a more solid ground to the study of hyperoperations.The content is relevant for the study of commutative fields. Possible applicationscould be found in computer arithmetic, where research remains active in the searchfor alternatives to the IEEE floating-point to overcome numeric overflow/underflow[9, 10]. In particular, it provides a formal background to proposals involving gener-alized exponentiation such as the Elias ω code [6] and the level-index codes [7, 8].It may also prove useful to the subfield of weighted automata, where semirings playa central role [11, 12].This paper is organized as follows: The section 2 reviews the original sequenceof commutative hyperoperations and proves that consecutive hyperoperations forma commutative semiring. The section is closed by constructing new sequences of Key words and phrases.
Hyperoperations, Tetration. a r X i v : . [ m a t h . A C ] D ec JUAN DIEGO JARAMILLO SALAZAR UNIVERSIDAD DEL VALLE COLOMBIA hyperoperations from the insight that each commutative hyperoperation is the ini-tial object of a new sequence of hyperopertations. In the section 3 an algorithmto represent nonnegative numbers using commutative hyperoperations is presented.In the section 4 the method of inverse completion [13] is used to construct a com-mutative field from each commutative semiring, closing with remarks on the localconstruction of the rationals and the reals along the sequence of semirings. Finally,in the section 5 the sequence of commutative hyperoperations is extended to nega-tive values of the index via inverse completion of the monoid of homomorphisms.2.
Monoids & semirings
Notation . Denote the set of natural numbers from Peano axioms as N = { , , , , ... } , together with its successor function s : N → N \ { } and its inversefunction s − . Definition 2.2.
The sequence of regular hyperoperations generated from the suc-cesor function in N , is the sequence of binary operations H n : ( N ) → N , for n ∈ N ,defined recursively as, H n ( a, b ) = s ( b ) , if n = 0; a, if n = 1 and b = 0;0 , if n = 2 and b = 0;1 , if n ≥ b = 0; H n − ( a, H n ( a, s − ( b ))) , otherwise . (2.3) Remark . The tuple ( N , H , H ) is a commutative semiring with additive identity0 and multiplicative identity 1. See the Prop. 6.1 in the Appendix. Definition 2.5.
Define the function E : N → N : E ( a ) = H ( w , a ), for a givenbase w ∈ N \ { , } . Denote by L its inverse function. Notation . For n ∈ N , denote n +1 N = E ( n N ) with N = N . Denote, n N = { n , n , n , . . . } such that for n +1 k ∈ n +1 N and n k ∈ n N , n +1 k = E ( n k ). Remark . The function E generates a descending filtration in N , meaning that n +1 N (cid:40) n N . Definition 2.8.
The sequence of commutative hyperoperations generated from theabelian monoid ( N , H ) is the sequence of binary operations F n : ( n − N ) → n − N for n = 2 , , , . . . , defined recursively as, F n ( a, b ) = (cid:26) E ( H ( L ( a ) , L ( b ))) , if n = 2; E ( F n − ( L ( a ) , L ( b ))) , otherwise . (2.9) Notation . Let a, b ∈ N . Denote, F n ( a, b ) = (cid:26) H ( a, b ) , if n = 0; H ( a, b ) , if n = 1 . (2.11) Remark . In Bennet’s original report [1] there is no discussion of the set in whichthe commutative hyperoperations are defined, and Euler’s exponential function isused in the recursive definition, instead of an exponential with an integer base, asin Def. 2.8.
YPEROPERATIONS IN EXPONENTIAL FIELDS 3
Remark . The motivation for Def. 2.8 can be expressed informally as follows.One is familiar with two operations in the natural numbers: ordinary addition andmultiplication. Let a, b and w be natural numbers with w other than zero andone. For convenience, denote iterated exponentiation as ( w ∧ a = w a ): w ∧ w ∧ a = w ∧ ( w ∧ a ) , (2.14a) w ∧ w ∧ w ∧ a = w ∧ ( w ∧ ( w ∧ a )) , (2.14b)...For numbers of the form w ∧ a and w ∧ b , one can define the following operation: F ( w ∧ a, w ∧ b ) = w ∧ ( a × b ) . (2.15)It is tempting to also define, F (cid:48) ( w ∧ a, w ∧ b ) = w ∧ ( a + b ) , (2.16)but it is already defined, being equivalent to w ∧ a × w ∧ b . One can continuewith the definition of new operations recursively as, F ( w ∧ w ∧ a, w ∧ w ∧ b ) = w ∧ w ∧ ( a × b ) , (2.17)...The operations F (cid:48) n ≥ defined in analogy to Eq. 2.16 can always be expressed through F n − . Remark . Let a, b ∈ N . The multiplicative identity of the exponential functiontakes the form, F ( a, b ) = E ( F ( L ( a ) , L ( b ))) . (2.19)See the Propostion 6.22 in the Appendix. Remark . The sequence of 3-tuples [( n N , F n , F n +1 )] n ∈ N is a sequence of commu-tative semirings with additive identity n n
1, respectively.The function E is the generator of the homomorphisms in the sequence. See theTheorem 6.25 in the Appendix. As a corollary, the sequences of tuples [( n N , F n )] n ∈ N and [( n N , F n +1 )] n ∈ N are both sequences of commutative monoids, where ( n N , F n ) is asubmonoid of ( n − N , F n ). Remark . Since n n +1 n
0. They can begenerated through tetration as, H ( w , −
1) = 0 , (2.22a) H ( w ,
0) = 1 , (2.22b) H ( w ,
1) = w , (2.22c) H ( w ,
2) = w w , (2.22d)...In the expression H ( a, n ), the argument a is known as the base and the argument n is known as the height or level . For completness in Eq. 2.22 the domain of tetration JUAN DIEGO JARAMILLO SALAZAR UNIVERSIDAD DEL VALLE COLOMBIA has been extended to allow for negative integers in the height, despite in Def. 2.2it is restricted to nonnegative integers.The rest of this section generalize the former hyperoperations. Any commutativemonoid ( n N , F n ) becomes a primitive to construct hyperoperations. Notation . Denote, H n ( a, b ) = a [0 : n ] b and F n ( a, b ) = a [ n : 1] b . Definition 2.24.
Define the successor function in n N as the function s n : n N → n N \ { n } : s n ( a ) = n n : 1] a , and denote its inverse function as s − n . Definition 2.25.
The sequence of regular hyperoperations generated from thesuccesor in n N is the sequence of binary operations [ n : m ] : ( n N ) → n N , for m ∈ N defined recursively as, a [ n : m ] b = s n ( b ) , if m = 0; a, if m = 1 and b = n n , if m = 2 and b = n n , if m ≥ b = n a [ n : m − a [ n : m ] s − n ( b )) , otherwise . (2.26) Notation . Henceforth assume the hierarchy of operations: a [ n : m ] b [ n + 1 : m ] c = a [ n : m ]( b [ n + 1 : m ] c ) (2.28)and a [ n : m ] b [ n : m + 1] c = a [ n : m ]( b [ n : m + 1] c ) , (2.29) a [ n : m + 1] b [ n : m ] c = ( a [ n : m + 1] b )[ n : m ] c . (2.30) Proposition 2.31.
Let a, b ∈ n N , then a [ n : 2] b = a [ n + 1 : 1] b. (2.32) Proof.
Let a, b ∈ n N . Without loss of generality, let b = n k , where k ∈ N . From theDef. 2.25 one obtains that, a [ n : 2] b = a [ n : 1] a [ n : 1] . . . [ n : 1] a (cid:124) (cid:123)(cid:122) (cid:125) k copies of a . (2.33)To prove the right hand side is equal to a [ n + 1 : 1] b one can proceed by induction.Case n = 0 follows from Def. 2.2 since the binary operations [0 : 1] and [1 : 1] standfor H and H , respectively. For n ≥
1, assume, c [ n : 1] d = c [ n − c [ n − . . . [ n − c (cid:124) (cid:123)(cid:122) (cid:125) k copies of c . (2.34) YPEROPERATIONS IN EXPONENTIAL FIELDS 5 for any c, d ∈ n − N and c = n − k . Then, for a, b ∈ n N (cid:40) n − N and b = n k , one obtainsthat, a [ n + 1 : 1] b = E ( L ( a ) [ n : 1] n − k ) (2.35)= E ( L ( a ) [ n − L ( a ) [ n − . . . [ n − L ( a ) (cid:124) (cid:123)(cid:122) (cid:125) k copies of L ( a ) ) (2.36)= a [ n : 1] a [ n : 1] . . . [ n : 1] a (cid:124) (cid:123)(cid:122) (cid:125) k copies of a . (2.37) (cid:4) Proposition 2.38.
The tuple ( n N , [ n : 1] , [ n : 2]) is a commutative semiring withadditive identity n and multiplicative identity n .Proof. It can be proven in analogy to Theorem 6.1 in the Appendix. Alternatively,from Prop. 6.25 in account of Prop. 2.31. (cid:4)
The regular hyperoperations, [ n : 1] and [ n : 2] are commutative. The nextproposition shows how commutativity is broken for the regular hyperoperation [ n :3], which plays the role of exponentiation in the semiring ( n N , [ n : 1] , [ n : 2]). This isdone by relating [ n : 3] with the commutative hyperoperation [ n + 2 : 1], in analogyto Prop. 2.31. Proposition 2.39.
Let a ∈ n +1 N and b ∈ n N , then, a [ n : 3] b = a [ n + 2 : 1] E ( b ) . (2.40) Proof.
Without loss of generality, let b = n k , where k ∈ N . From Def. 2.25 andProp. 2.31 one obtains that, a [ n : 3] b = a [ n : 2] a [ n : 2] . . . [ n : 2] a (cid:124) (cid:123)(cid:122) (cid:125) k copies of a (2.41)= a [ n + 1 : 1] a [ n + 1 : 1] . . . [ n + 1 : 1] a (cid:124) (cid:123)(cid:122) (cid:125) k copies of a (2.42)= a [ n + 1 : 2] E ( b ) (2.43)= a [ n + 2 : 1] E ( b ) . (2.44) (cid:4) Notation . Let the variable n denote elements in N . For a given n ∈ N ,let the elements in ( n , N ), denoted with the variable n , be equipped with theoperation, ( n , n ) + l = ( n , n + l ) , (2.46)for any n ∈ N and l ∈ Z , such that n + l ≥ Definition 2.47.
Define the function E n : n N → n +1 N : E n ( a ) = w [ n : 3] a , fora given base w ∈ n N \ { n , n } . Denote by L n its inverse function. JUAN DIEGO JARAMILLO SALAZAR UNIVERSIDAD DEL VALLE COLOMBIA
Definition 2.48.
The sequence of commutative hyperoperations generated fromthe abelian monoid ( n N , [ n : 2]) is the sequence of binary operations [ n : 1] : ( n − N ) → n − N for n = 2 , , , . . . , defined recursively as, a [ n : 1] b = (cid:26) E n ( L n ( a ) [ n : 2] L n ( b )) , if , n = 2; E n ( L n ( a ) [ n − L n ( b )) , otherwise . (2.49) Notation . Let a, b ∈ n N . Denote, a [ n : 1] b = (cid:26) a [ n : 1] b, if , n = 0; a [ n : 2] b, if , n = 1 . (2.51) Notation . In general, for a given n k − ∈ ( n k − , N ), let the elements in ( n k − , N )be equipped with the operation,( n k − , n k ) + l = ( n k − , n k + l ) . (2.53)for any n k ∈ N and l ∈ Z , such that n k + l ≥
0. For completeness, let n = 0. Definition 2.54.
Define the successor function in n k N as the function s n k : n k N → n k N \ { n k } : s n k ( a ) = n k n k : 1] a , and denote its inverse function as s − n k . Definition 2.55.
The sequence of regular hyperoperations generated from thesuccesor function in n k N is the sequence of binary operations [ n k : m ] : ( n k N ) → n k N ,for m ∈ N defined recursively as, a [ n k : m ] b = s n k ( b ) , if m = 0; a, if m = 1 and b = n k n k , if m = 2 and b = n k n k , if m ≥ b = n k a [ n k : m − a [ n k : m ] s − n k ( b )) , otherwise . (2.56) Remark . Note that the hyperoperation ( n k N , [ n k : N ]) requires the choice of asequence: w , w , w , . . . , w k − , where w r ∈ n r N \ { n r , n r } , for r = 0 , , , . . . , k − Definition 2.58.
For k ∈ N \{ } , define the function E n k − : n k N → n k +1 N : E n k − ( a ) = w k − [ n k − : 3] a , for a given base w k − ∈ n k − N \ { n k − , n k − } . Denote its inversefunction as L n k − . Definition 2.59.
The sequence of commutative hyperoperations generated fromthe abelian monoid ( n k − N , [ n k − : 2]) is the sequence of binary operations[ n k : 1] : ( n k N ) → n k N , for n k ∈ ( n k − , N ), defined recursively as, a [ n k : 1] b = (cid:26) E n k − ( L n k − ( a ) [ n k − : 2] L n k − ( b )) , if n k = 2; E n k − ( L n k − ( a ) [ n k − L n k − ( b )) , otherwise . (2.60) Notation . Let a, b ∈ n k − N . Denote, a [ n k : 1] b = (cid:26) a [ n k : 1] b, if n k = 0; a [ n k : 2] b, if n k = 1 . (2.62) YPEROPERATIONS IN EXPONENTIAL FIELDS 7 n n n n n . . .. . .. . . ... ... Figure 1.
The diagram shows five sequences of commutative hy-peroperations. In particular, the sequence { [ n : 1] } n ∈ N , is ob-tained after the choices: n = 2 , n = 1 , n = 1 and n = 3. Remark . The homomorphism from ( n N , [ n : 1] , [ n + 1 : 1]) to ( n k N , [ n k :1] , [ n k + 1 : 1]) takes the form, E n k n k − ◦ . . . ◦ E n n ◦ E n , (2.64)where ◦ stands for function composition and E n j n j − stands for apply n j -times thefunction E n j − .The present proposal to generate new sequences of commutative hyperoperationscan be visualized as a binary tree diagram, see Fig. 1. Remark . The sequence of 3-tuples [( n k N , [ n k : 1] , [ n k + 1 : 1])] n k ∈ N is a sequenceof commutative semirings with additive identity n k n k E n k − is the generator of the homomorphisms in thesequence. The proof is analogous to the Theorem 6.25 in the Appendix.3. Representation of natural numbers
A recursive representation of nonnegative integers using commutative hyperop-erations can be obtained from its power series expansion. It is well known that anyinteger a ∈ N can be represented uniquely as a power series base w in the form, a = d + d w + d w + · · · + d m w m , (3.1)where the coefficients d k ∈ { , , , . . . , s − ( w ) } are the digits and the maximumexponent m ∈ N is such that d k>m = 0. This power series can be written recursivelyas, a k = F ( F ( d k , k ) , a k +1 ) , (3.2) JUAN DIEGO JARAMILLO SALAZAR UNIVERSIDAD DEL VALLE COLOMBIA where the exponents k ∈ N , a = d and a m = a . The term k is the scale of thedigit d k . Applying the function E n to both sides of Eq. 3.2 one arrives at the moregeneral statement that any element b ∈ n N can be represented as a power series inthe semiring ( n N , F n , F n +1 ), taking the form, b k = F n ( F n +1 ( e k , n +1 k ) , b k +1 ) , (3.3)where the coefficients e k ∈ { n , n , n , . . . , s − n ( n w ) } are the ‘digits’, the term n +1 k isthe ‘scale’ of the digit e k ; among the ‘exponents’ k ∈ N there is a maximum q suchthat e k>q = n b = e and b q = b .In Eq. 3.2 the maximum exponent m is finite, but arbitrarily large. One canwrite all exponents k as a power series base w , proceeding iteratively. This isknown as the hereditary representation of a natural number and can be expressedusing hyperoperations: Since the scale k belongs to N , it can be represented as apower series in the semiring ( N , F , F ). The subsequent ‘scales’ belong to N andcan be expanded in the semiring ( N , F , F ). The procedure repeats iteratively untilall ‘scales’ are expanded. For example, the expansion of 266 base w = 3 is,266 = 2 × + 1 × + 2 × + 1 × . (3.4)Let a = 266 ∈ N , then, a = , (3.5a) a = F ( F ( , , a ) , (3.5b) a = F ( F ( , , a ) , (3.5c) a = a , (3.5d) a = a , (3.5e) a = F ( F ( , , a ) = a. (3.5f)According to the hereditary representation, the scale a , being greater than w ), must be expanded. Note that, = 3 × +1 × . Let, b = ∈ N ,then, b = , (3.6a) b = F ( F ( , , b ) = b. (3.6b)There is no further iteration. The expansion of 266 base w = 2 is,266 = 2 + 2 + 2 . (3.7) YPEROPERATIONS IN EXPONENTIAL FIELDS 9
Let a = 266 ∈ N , then, a = , (3.8a) a = F ( F ( , , a ) , (3.8b) a = a , (3.8c) a = F ( F ( , , a ) , (3.8d) a = a , (3.8e)... a = a , (3.8f) a = F ( F ( , , a ) = a. (3.8g)The scale a must be expanded. Let b = ∈ N , then, b = , (3.9a) b = F ( F ( , , b ) = b. (3.9b)There is no further iteration in a . The scale a must be expanded. Note that, = 2 . Let c = ∈ N , then, c = , (3.10a) c = c , (3.10b) c = c , (3.10c) c = F ( F ( , , c ) = c. (3.10d)The scale c must be expanded. Note that, = 2 . Let c (cid:48) = ∈ N ,then, c (cid:48) = , (3.11a) c (cid:48) = F ( F ( , , c (cid:48) ) = c (cid:48) . (3.11b)There is no further iteration. 4. Groups & fields
In App. 6.1 is the standard procedure of inverse completion on a commutativesemigroup. When applied to the additive monoid of integers, it produces the addi-tive group of integers. If applied to the multiplicative monoid of non-zero integers,it generates the multiplicative group of positive rationals. In general, after this pro-cedure, any commutative, cancellable semigroup results in an abelian group [13].
Notation . Denote as ( n Z , F n ) for n ∈ N , the inverse completion of the commu-tative monoid ( n N , F n ), and denote by n T its inverse operation , i.e., for any a ∈ n Z : F n ( n T a, a ) = F n ( a, n T a ) = n . (4.2) Proposition 4.3.
The sequence of abelian groups [( n Z , F n )] n ∈ N is compatible withthe domain extension of the exponential function E , in the sense that for a ∈ n Z : E ( n T a ) = n +1 T E ( a ) . (4.4) Proof.
Extending the definition of F n +1 in 2.8 from n +1 N to n +1 Z , one obtains that, F n +1 ( E ( n T a ) , E ( a )) = E ( F n ( L ( E ( n T a )) , L ( E ( a )))) (4.5a)= E ( F n ( n T a, a )) (4.5b)= E ( n
0) = n +1 . (4.5c) (cid:4) Remark . The tuple ( n Z , F n , F n +1 ) for n ∈ N , is an integral domain. See theProp. 6.38 in the Appendix. Remark . Let a ∈ ( n Z , F n , F n +1 ). The reader can prove the following propertiesof the additive inverse: n T ( n T a ) = a, (4.8a) n T F n ( a, b ) = F n ( n T a, n T b ) , (4.8b) F n +1 ( n T a, b ) = F n +1 ( a, n T b ) = n T F n +1 ( a, b ) , (4.8c) F n +1 ( n T a, n T b ) = F n +1 ( a, b ) . (4.8d)One can use these properties to reduce any computation in ( n Z , F n ) to a computationin ( n N , F n ) up to the overall action of the additive inverse n T . Remark . From the properties above, one obtains that, n T a = F n +1 ( n T n , a ) . (4.10) Notation . For every integral domain, there is a field of quotients [13]. Denote as( n Q , F n , F n +1 ) for n ∈ N , the field of quotients of the integral domain ( n Z , F n , F n +1 ). Remark . By construction:(1) The identity element in ( n Q , F n +1 ) is n n Z \ { n } , F n +1 ) has an inverse element in ( n Q , F n +1 ). Abus-ing of notation, denote the inverse operator as n +1 T .(3) Every element of ( n Q , F n +1 ) can be written as F n +1 ( a, n +1 T b ) where a ∈ n Z and b ∈ n Z \ { n } . Remark . Let n ∈ N . Note that the element n n +1 Z (cid:83) { n } , F n +1 ) and is the infimum of n +1 Z as a subset of the ordered field n Q . YPEROPERATIONS IN EXPONENTIAL FIELDS 11
Remark . The absolute value in the field ( n Q , F n , F n +1 ) is defined as, | a | n = n T a, if a < n n , if a = n a, if a > n . (4.15) Remark . There are several ways to construct the real numbers ( R ) from therational numbers ( Q ), as ordered fields. Analogously, one constructs n R from n Q .In particular, the definition of F n and F n +1 in n R is analogue to the definition ofordinary addition ( F ) and multiplication ( F ) in the reals ( R ). Remark . Denote n R + = { x ∈ n R : x > n } . Note that n +1 R = n R + , after all n +1 R = E ( n R ). While the tuple ( n +1 R , F n +1 , F n +2 ) forms a field, the tuple ( n +1 R (cid:83) { n } , F n , F n +1 )forms a semiring. For n = 0, the latter is simply the semiring of nonnegative reals,also known as the probability semiring in the context of weighted automata [12]. Remark . Some observations regarding the classification of real numbers alongthe sequence of fields: • Irrational numbers in ( n R , F n , F n +1 ), can become rational in ( n +1 R , F n +1 , F n +2 ),e.g., if w = 2, the number √ R , F , F ) while it is rationalin ( R , F , F ), it can be written as T
2. This entails that the quadratic,algebraic equation x − R , F , F ) can be written in ( R , F , F ) asthe linear algebraic equation, x [2 : 1] T . (4.19) • Rational numbers in ( n R , F n , F n +1 ), can become irrational in ( n +1 R , F n +1 , F n +2 ),e.g., if w = 2, the number R , F , F ) while it is irrationalin ( R , F , F ). The proof of the latter is analogous to the proof that log (3)is irrational in ( R , F , F ). • Trascendental numbers in ( n R , F n , F n +1 ) can become algebraic and irrationalin ( n +1 R , F n +1 , F n +2 ), e.g., if w = 2, the Gelfond-Schneider constant 2 √ is trascendental in ( R , F , F ) while algebraic and irrational in ( R , F , F ).Note that x ± = 2 ±√ are the solutions to, x [2 : 1] x [1 : 1] T , (4.20)which is a quadratic, algebraic equation in ( R , F , F ). Note that the fun-damental theorem of algebra for ( n R , F n , F n +1 ) is proven in analogy to theone in ( R , F , F ). Remark . For every polynomial f in ( n R , F n , F n +1 ) there is a polynomial g in( n +1 R , F n +1 , F n +2 ), related by the conjugation, g ( x ) = E ◦ f ◦ L ( x ) . (4.22) Notation . Let a k be an infinite sequence of elements in n R . Denote the corre-sponding infinite series as,[ n : 1] ∞ k =1 a k = a [ n : 1] a [ n : 1] a [ n : 1] · · · (4.24)Denote the factorial in n N as, n k ! = n k [ n : 2] s − n ( n k ) [ n : 2] s − n ( n k ) [ n : 2] · · · [ n : 2] n n : 2] n , (4.25)where s − rn stands for apply r -times the function s − n .Remark . Euler’s exponential function in n R , denoted exp n , takes the form,exp n ( x ) = [ n : 1] ∞ k =1 x [ n : 3] n k [ n : 2] n +1 T ( n k !) , (4.27)using the hierarchy of operations from Eq. 2.30. It is related by conjugation withthe ordinary exponential function as: exp n ( x ) = E n ◦ exp ◦ L n ( x ).After the construction of n R from n N , the following proposition offers a character-ization of n N from the perspective of n R . Proposition 4.28.
The set n N \ { n } is a natural way to count iterations in themonoid ( n R , F n ) in the sense that for a ∈ n R : F n +1 ( n , a ) = a, (4.29a) F n +1 ( n , a ) = F n ( a, a ) , (4.29b) ... F n +1 ( n k, a ) = F n ( a, F n ( a, . . . F n ( a, F n ( a, a )) . . . )) (cid:124) (cid:123)(cid:122) (cid:125) k copies of a . (4.29c) Proof.
By induction. It is easy to prove that, F ( k, a ) = F ( a, F ( a, . . . F ( a, F ( a, a )) . . . )) (cid:124) (cid:123)(cid:122) (cid:125) k copies of a . (4.30)For n ≥
2, assume, F n ( n − k , a ) = F n − ( a, F n − ( a, . . . F n − ( a, F n − ( a, a )) . . . )) (cid:124) (cid:123)(cid:122) (cid:125) k copies of a , (4.31)then, F n +1 ( n k, a ) = E ( F n ( n − k , L ( a ))) (4.32)= E ( F n − ( L ( a ) , . . . F n − ( L ( a ) , F n − ( L ( a ) , L ( a ))) . . . ) (cid:124) (cid:123)(cid:122) (cid:125) k copies of a ) (4.33)= F n ( a, F n ( a, . . . F n ( a, F n ( a, a )) . . . )) (cid:124) (cid:123)(cid:122) (cid:125) k copies of a . (4.34)For an alternative proof, note that Def. 2.25 and Prop. 2.31 still hold if one allowsthe first entry ( a ) to be in n R and the second entry ( b ) to remain in n N . Then, n k [ n + 1 : 1] a = a [ n + 1 : 1] n k = a [ n : 2] n k . (4.35) (cid:4) YPEROPERATIONS IN EXPONENTIAL FIELDS 13 Commutative hyperoperations with negative index
With inverse completion one can extend the sequence of commutative semirings[( n N , F n , F n +1 )] n ∈ N to negative values of the index n .Start from the commutative monoid ( M, ◦ ), induced by the composition of theexponential function E with domain in N . The elements of M take the form, E , (5.1a) E = E, (5.1b) E = E ◦ E, (5.1c) E = E ◦ E ◦ E, (5.1d)...For a generic element x ∈ N one obtains that, E n +1 ( x ) = E ( E n ( x )) and E ( x ) = x. (5.2)The successor function in M is given by composition with E . The monoidal oper-ation is given by, E m ◦ E n = E m + n , (5.3)where m, n ∈ N . The explicit steps of inverse completion can be found in theAppendix 6.1, after replacing ( S, ∗ ) and ( T (cid:48) , ⊕ (cid:48) ) by ( M, ◦ ) and ( G, ◦ ), respectively.The elements in G take the form E n , where n ∈ Z . Consistent with the definitionof the logarithm L set, E ±| n | = L ∓| n | . (5.4)Changing the element x ∈ N in Eq. 5.2 simply changes the representation of themonoid and of its subsequent group. The generator of these representations is givenby precomposition with the successor function s of N , i.e., s ∗ : G → s ∗ G : s ∗ E n = E n ◦ s, (5.5)for all n ∈ Z . The successor operator s ∗ is assumed to be distributive on thecomposition of exponential functions, i.e., s ∗ E m ◦ s ∗ E n = s ∗ E m + n . (5.6)A dual representation of the sequence of monoids is given by, n N ∗ = { E n , s ∗ E n , ( s ∗ ) E n , ( s ∗ ) E n , . . . } . (5.7)The corresponding regular and commutative hyperations can be reconstructed inanalogy to Defs. 2.2 and 2.8, starting from the successor operator s ∗ . The evalu-ation of n N ∗ at n N for n ∈ Z . In particular, the sequence of commutativehyperoperations from Def. 2.8 is extended to binary operations F n : ( n − N ) → n − N for n = 0 , − , − , . . . , defined recursively as, F n ( a, b ) = L ( F n +1 ( E ( a ) , E ( b ))) . (5.8) Note that the definition of F above, is compatible with Eq. 2.11 when restrictedto N . The reader is invited to prove that the descending filtration n N (cid:40) n +1 N and thesemiring structure ( n N , F n , F n +1 ) remain valid for n ∈ Z . Remark . The elements of − N \ { − } take the form, − L ( , − L ( , − L ( , . . . (5.10)The element − − N , F − , F ) is a subalgebra ofthe so called log semiring [11], notable beacause in the limit w → + ∞ it becomesa subalgebra of the max tropical semiring , where − −∞ . Theprevious discussion relied on the condition w ∈ N \{ , } which implies w < + ∞ .In the limit w → + ∞ , the property N (cid:40) − N is no longer valid. Instead onecan use the sequence { n M } n ∈ Z whose elements are characterized by the relation n M = n N (cid:83) n +1 M . The latter fulfills the property n +1 M (cid:40) n M and as soon as w becomesfinite one obtains that n M = n N . To verify that ( − M , F − , F ) is a subalgebra of themax tropical semiring, consider a, b ∈ M (cid:40) − M . Then, F − ( a, b ) = lim w → + ∞ L ( F ( E ( a ) , E ( b ))) . (5.11)Without loss of generality, assume a ≤ b . Note that,lim w → + ∞ F ( E ( a ) , E ( b )) = lim w → + ∞ w a + w b (5.12a)= lim w → + ∞ w b ( w a − b + 1) (5.12b)= lim w → + ∞ w b . (5.12c)Therefore, F − ( a, b ) = max { a, b } . (5.13)In particular, F − ( a, a ) = a , which makes it an idempotent semiring. In fact, thesubset I = M (cid:83) { − } is an ideal of the semiring ( − M , F − ) as it fulfills the followingproperties [14]:(1) ( I, F − ) is a submonoid of ( − M , F − ).(2) If a ∈ − M and b ∈ I , then F ( a, b ) ∈ I .(3) I (cid:40) − M . 6. Appendix
Proposition 6.1.
The tuple ( N , H , H ) is a commutative semiring.Proof. First, one proves that ( N , H ) is a monoid, then prove that ( N , H ) is amonoid and H is distributive on H . Let a, b, c ∈ N . YPEROPERATIONS IN EXPONENTIAL FIELDS 15 (1)
Closure of H in N . Note H ( a,
0) = a ∈ N . For the remaining cases, H ( a, b ) = H ( a, H ( a, b − H ( a, H ( a, H ( a, b − H ( a, H ( a, . . . H ( a, H ( a, . . . )) (cid:124) (cid:123)(cid:122) (cid:125) b copies of H (6.2c)= H ( a, H ( a, . . . H ( a, H ( a, a )) . . . )) (cid:124) (cid:123)(cid:122) (cid:125) b copies of H ∈ N . (6.2d)(2) Identity element of H in N . From Def. 2.2 one obtains that, H ( a,
0) = 0 , (6.3)and, H (0 , a ) = H (0 , H (0 , . . . H (0 , H (0 , . . . )) (cid:124) (cid:123)(cid:122) (cid:125) a copies of H (6.4a)= H (0 , H (0 , . . . H (0 , H (0 , . . . )) (cid:124) (cid:123)(cid:122) (cid:125) a − H (6.4b)= H (0 , H (0 , . . . H (0 , H (0 , . . . )) (cid:124) (cid:123)(cid:122) (cid:125) a − H (6.4c)= H (0 , H (0 , . . . H (0 , H (0 , . . . )) (cid:124) (cid:123)(cid:122) (cid:125) a − H (6.4d)...= H (0 , a −
1) (6.4e)= a. (6.4f)(3) Commutativity of H in N . Let a < b , then, H ( a, b ) = H ( a, H ( a, . . . H ( a, H ( a, a )) . . . )) (cid:124) (cid:123)(cid:122) (cid:125) b copies of H (6.5a)= H ( a, H ( a, . . . H ( a, H ( a, H (1 , a )) . . . ))) (cid:124) (cid:123)(cid:122) (cid:125) b − H (6.5b)= H ( a, H ( a, . . . H ( a, H (1 , H (1 , a ))) . . . )) (cid:124) (cid:123)(cid:122) (cid:125) b − H (6.5c)...= H ( a, H ( a, . . . H ( a, H ( a, H ( a, b − . . . )) (cid:124) (cid:123)(cid:122) (cid:125) a +1 copies of H (6.5d)= H ( a, H ( a, . . . H ( a, H ( a, b )) . . . )) (cid:124) (cid:123)(cid:122) (cid:125) a copies of H (6.5e)= H ( b, a ) . (6.5f) (4) Associativty of H in N . H ( a, H ( b, c )) = H ( a, H ( a, . . . H ( a, H ( a, H ( a, a ))) . . . )) (cid:124) (cid:123)(cid:122) (cid:125) H ( b,c ) copies of H (6.6a)= H ( a, H ( a, . . . H ( a, H ( a, H (1 , a )) . . . )) (cid:124) (cid:123)(cid:122) (cid:125) H ( b,c ) − a (6.6b)...= H ( a, H ( a, . . . H ( a, H ( a, H ( a, b ) − . . . )) (cid:124) (cid:123)(cid:122) (cid:125) c +1 copies of H (6.6c)= H ( a, H ( a, . . . H ( a, H ( a, H ( a, b ))) . . . )) (cid:124) (cid:123)(cid:122) (cid:125) c copies of H (6.6d)= H ( H ( a, b ) , c ) . (6.6e)This concludes the first part of the proof, that ( N , H ) is an abelian monoid.The proofs that ( N , H ) is a monoid and H is distributive in H are thefollowing:(5) Closure of H in N . H ( a, b ) = H ( a, H ( . . . H ( a, H ( a, a )) . . . )) (cid:124) (cid:123)(cid:122) (cid:125) b − H ∈ N . (6.7)(6) Identity element of H in N . H ( a,
1) = H ( a, H ( a, H ( a,
0) (6.8b)= a, (6.8c)and, H (1 , a ) = H (1 , H ( . . . H (1 , H (1 , . . . )) (cid:124) (cid:123)(cid:122) (cid:125) a − H (6.9a)= H (1 , H ( . . . H (1 , H (1 , . . . )) (cid:124) (cid:123)(cid:122) (cid:125) a − H (6.9b)= H (1 , H ( . . . H (1 , H (1 , . . . )) (cid:124) (cid:123)(cid:122) (cid:125) a − H (6.9c)...= H (1 , a −
1) (6.9d)= a. (6.9e) YPEROPERATIONS IN EXPONENTIAL FIELDS 17 (7) H is distributive on H , in N . Proving the case H ( b, c ) = 0 is straight-forward since it implies b = c = 0. For the remaining cases: H ( a, H ( b, c )) = H ( a, H ( a, . . . H ( a, H ( a, a )) . . . )) (cid:124) (cid:123)(cid:122) (cid:125) H ( b,c ) − H (6.10a)= H ( a, H ( a, . . . H ( a, H ( a, . . . )) (cid:124) (cid:123)(cid:122) (cid:125) H ( b,c ) − H (6.10b)= H ( a, H ( a, . . . H ( a, H ( a, . . . )) (cid:124) (cid:123)(cid:122) (cid:125) H ( b,c ) − H (6.10c)...= H ( a, H ( a, . . . H ( a, H ( a, H ( a, c ))) . . . )) (cid:124) (cid:123)(cid:122) (cid:125) b copies of H (6.10d)= H ( a, H ( a, . . . H ( a, H ( H ( a, a ) , H ( a, c ))) . . . )) (cid:124) (cid:123)(cid:122) (cid:125) b copies of H (6.10e)= H ( a, H ( a, . . . H ( a, H ( H ( a, , H ( a, c ))) . . . )) (cid:124) (cid:123)(cid:122) (cid:125) b − H (6.10f)= H ( a, . . . H ( a, H ( H ( a, H ( a, , H ( a, c ))) . . . )) (cid:124) (cid:123)(cid:122) (cid:125) b − H (6.10g)= H ( a, . . . H ( a, H ( H ( a, H ( a, , H ( a, c ))) . . . )) (cid:124) (cid:123)(cid:122) (cid:125) b − H (6.10h)...= H ( H ( a, H ( a, b − , H ( a, c )) (6.10i)= H ( H ( a, b ) , H ( a, c )) . (6.10j)To prove H ( H ( b, c ) , a ) = H ( H ( b, a ) , H ( c, a )), one can proceed by in-duction. Note that, H ( H ( b, c ) ,
1) = H (1 , H ( b, c )) (6.11a)= H ( H (1 , b ) , H (1 , c )) (6.11b)= H ( H ( b, , H ( c, . (6.11c)Assume H ( H ( b, c ) , a ) = H ( H ( b, a ) , H ( c, a )), then H ( H ( b, c ) , H ( a, H ( H ( H ( b, c ) , a ) , H ( H ( b, c ) , H ( H ( H ( b, c ) , a ) , H ( b, c )) (6.12b)= H ( H ( H ( b, a ) , H ( c, a )) , H ( b, c )) (6.12c)= H ( H ( H ( b, a ) , b ) , H ( H ( c, a ) , c )) (6.12d)= H ( H ( H ( b, a ) , H ( b, , H ( H ( c, a ) , H ( c, H ( H ( b, H ( a, , H ( c, H ( a, . (6.12f)(8) Commutativity of H in N . By induction. It is known that H ( a,
1) = H (1 , a ). Assume H ( a, b ) = H ( b, a ), then, H ( a, H (1 , b )) = H ( H ( a, , H ( a, b )) (6.13a)= H ( H (1 , a ) , H ( b, a )) (6.13b)= H ( H (1 , b ) , a ) . (6.13c) (9) Associativty of H in N . By induction. Note that, H ( H ( a, b ) ,
0) = 0 (6.14)= H ( a,
0) (6.15)= H ( a, H ( b, . (6.16)Assume H ( H ( a, b ) , c ) = H ( a, H ( b, c )), then H ( H ( a, b ) , H (1 , c )) = H ( H ( H ( a, b ) , , H ( H ( a, b ) , c )) (6.17)= H ( H ( a, b ) , H ( H ( a, b ) , c )) (6.18)= H ( H ( a, b ) , H ( a, H ( b, c ))) (6.19)= H ( a, H ( b, H ( b, c ))) (6.20)= H ( a, H ( b, H (1 , c ))) . (6.21) (cid:4) Proposition 6.22.
Let w ∈ N \ { , } and a, b ∈ N , then, H ( w , H ( L ( a ) , L ( b ))) = H ( a, b ) . (6.23) Proof. H ( w , H ( L ( a ) , L ( b ))) (6.24a)= H ( w , H ( w , . . . H ( w , H ( w , w )) . . . )) (cid:124) (cid:123)(cid:122) (cid:125) H ( L ( a ) ,L ( b )) − H (6.24b)= H ( w , H ( w , . . . H ( w , H ( w , . . . )) (cid:124) (cid:123)(cid:122) (cid:125) H ( L ( a ) ,L ( b )) − H (6.24c)= H ( w , H ( w , . . . H ( w , H ( w , . . . )) (cid:124) (cid:123)(cid:122) (cid:125) H ( L ( a ) ,L ( b )) − H (6.24d)...= H ( w , H ( w , . . . H ( w , H ( w , L ( a ))) . . . )) (cid:124) (cid:123)(cid:122) (cid:125) L ( b ) copies of H (6.24e)= H ( w , H ( w , . . . H ( w , H ( a, w )) . . . )) (cid:124) (cid:123)(cid:122) (cid:125) L ( b ) copies of H (6.24f)= H ( w , H ( w , . . . H ( w , H ( a, H ( w , . . . )) (cid:124) (cid:123)(cid:122) (cid:125) L ( b ) − H (6.24g)= H ( w , H ( w , . . . H ( w , H ( a, H ( w , . . . )) (cid:124) (cid:123)(cid:122) (cid:125) L ( b ) − H (6.24h)...= H ( w , H ( a, H ( w , L ( b ) − H ( a, H ( w , L ( b ))) (6.24j)= H ( a, b ) . (6.24k) (cid:4) Proposition 6.25.
The tuple ( n N , F n , F n +1 ) for n ∈ N , is a commutative semiring.Proof. By induction. From Theorem 6.1 the tuple ( N , H , H ) is a commutativesemiring and from Notation 2.10, ( N , H , H ) = ( N , F , F ). Assume the tuple ( n − N YPEROPERATIONS IN EXPONENTIAL FIELDS 19 , F n − , F n ) is a commutative semiring, with additive identity n − n − a, b, c ∈ n N , then,(1) Closure of F n in n N . F n ( a, b ) = E ( F n − ( L ( a ) , L ( b ))) ∈ E ( n − N ) = n N . (6.26)(2) Identity element of F n in n N . F n ( a, n
0) = E ( F n − ( L ( a ) , n − E ( L ( a )) (6.27b)= a. (6.27c)(3) Commutativity of F n in n N . F n ( a, b ) = E ( F n − ( L ( a ) , L ( b ))) (6.28a)= E ( F n − ( L ( b ) , L ( a ))) (6.28b)= F n ( b, a ) . (6.28c)(4) Associativity of F n in n N . F n ( a, F n ( b, c )) = E ( F n − ( L ( a ) , L ( F n ( b, c )))) (6.29a)= E ( F n − ( L ( a ) , L ( E ( F n − ( L ( b ) , L ( c )))))) (6.29b)= E ( F n − ( L ( a ) , F n − ( L ( b ) , L ( c )))) (6.29c)= E ( F n − ( F n − ( L ( a ) , L ( b )) , L ( c ))) (6.29d)= F n ( E ( F n − ( L ( a ) , L ( b ))) , E ( L ( c ))) (6.29e)= F n ( F n ( E ( L ( a )) , E ( L ( b ))) , c ) (6.29f)= F n ( F n ( a, b ) , c ) . (6.29g)This concludes the first part of the proof, that ( n N , F n ) is an abelian monoid.The proofs that ( n N , F n +1 ) is a monoid and F n +1 is distributive in F n arethe following:(5) Closure of F n +1 in n N . F n +1 ( a, b ) = E ( F n ( L ( a ) , L ( b ))) ∈ E ( n − N ) = n N . (6.30)(6) Identity element of F n +1 in n N . F n +1 ( a, n
1) = E ( F n ( L ( a ) , n − E ( L ( a )) (6.31b)= a. (6.31c) (7) F n +1 is distributive on F n , in n N . F n +1 ( a, F n ( b, c )) = E ( F n ( L ( a ) , L ( F n ( b, c ))))= E ( F n ( L ( a ) , F n − ( L ( b ) , L ( c ))))= E ( F n − ( F n ( L ( a ) , L ( b )) , F n ( L ( a ) , L ( c ))))= F n ( F n +1 ( a, b ) , F n +1 ( a, c )) . (8) Commutativity of F n +1 in n N . F n +1 ( a, b ) = E ( F n ( L ( a ) , L ( b ))) (6.32a)= E ( F n ( L ( b ) , L ( a ))) (6.32b)= F n +1 ( b, a ) . (6.32c)(9) Associativity of F n +1 in n N . F n +1 ( a, F n +1 ( b, c )) = E ( F n ( L ( a ) , L ( F n +1 ( b, c )))) (6.33a)= E ( F n ( L ( a ) , L ( E ( F n ( L ( b ) , L ( c )))))) (6.33b)= E ( F n ( L ( a ) , F n ( L ( b ) , L ( c )))) (6.33c)= E ( F n ( F n ( L ( a ) , L ( b )) , L ( c ))) (6.33d)= F n +1 ( E ( F n ( L ( a ) , L ( b ))) , E ( L ( c ))) (6.33e)= F n +1 ( F n +1 ( E ( L ( a )) , E ( L ( b ))) , c ) (6.33f)= F n +1 ( F n +1 ( a, b ) , c ) . (6.33g) (cid:4) Lemma 6.34.
The tuple ( n Z , F n , F n +1 ) for n ∈ N , is a commutative ring.Proof. By construction, the tuple ( n Z , F n ) is an abelian group. It remains to provethat the tuple ( n Z , F n +1 ) is a commutative monoid. Let a, b, c ∈ n N , then,(1) Closure of F n +1 in n Z . Since n Z = n N (cid:83) ( n T n N ), all operations fall into threecases:(a) F n +1 ( a, b ) ∈ n N . It follows from the Theorem 6.25.(b) F n +1 ( n T a, b ) ∈ n T n N . It follows from the Property 4.8c and the Theorem6.25.(c) F n +1 ( n T a, n T b ) ∈ n N . It follows from the Property 4.8d and the Theo-rem 6.25.(2) Identity element of F n +1 in n Z . From the Property 4.8c and the Theorem6.25 one obtains that, F n +1 ( n T a, n
1) = n T F n +1 ( n , a ) (6.35a)= n T a. (6.35b)Similarly one proves, F n +1 ( n , n T a ) = n T a . The remaining cases follow di-rectly from the Theorem 6.25.(3)
Commutativity of F n +1 in n Z . Since n Z = n N (cid:83) ( n T n N ), all operations fall intothree cases: YPEROPERATIONS IN EXPONENTIAL FIELDS 21 (a) F n +1 ( a, b ) = F n +1 ( b, a ). It follows from the Theorem 6.25.(b) F n +1 ( n T a, b ) = F n +1 ( b, n T a ). It follows from the Property 4.8c andthe Theorem 6.25.(c) F n +1 ( n T a, n T b ) = F n +1 ( n T b, n T a ). It follows from the Property 4.8dand the Theorem 6.25.(4)
Associativity of F n +1 in n N . One must prove associativity for all eight cases: F n +1 ( a, F n +1 ( b, c )) = F n +1 ( a, b ) , c ) , (6.36a) F n +1 ( n T a, F n +1 ( b, c )) = F n +1 ( n T a, b ) , c ) , (6.36b) F n +1 ( a, F n +1 ( n T b, c )) = F n +1 ( a, n T b ) , c ) , (6.36c) F n +1 ( a, F n +1 ( b, n T c )) = F n +1 ( a, b ) , n T c ) , (6.36d) F n +1 ( n T a, F n +1 ( n T b, c )) = F n +1 ( n T a, n T b ) , c ) , (6.36e) F n +1 ( n T a, F n +1 ( b, n T c )) = F n +1 ( n T a, b ) , n T c ) , (6.36f) F n +1 ( a, F n +1 ( n T b, n T c )) = F n +1 ( a, n T b ) , n T c ) , (6.36g) F n +1 ( n T a, F n +1 ( n T b, n T c )) = F n +1 ( n T a, n T b ) , n T c ) . (6.36h)The relation 6.36a follows from the Theorem 6.25. The relation 6.36b fol-lows from the Property 4.8c and the Theorem 6.25: F n +1 ( n T a, F n +1 ( b, c )) = n T F n +1 ( a, F n +1 ( b, c )) (6.37a)= n T F n +1 ( F n +1 ( a, b ) , c ) (6.37b)= F n +1 ( n T F n +1 ( a, b ) , c ) (6.37c)= F n +1 ( F n +1 ( n T a, b ) , c ) . (6.37d)The reader is invited to prove the remaining cases using the Properties 4.8and the Theorem 6.25. (cid:4) Theorem 6.38.
The tuple ( n Z , F n , F n +1 ) for n ∈ N , is an integral domain.Proof. By Lemma 6.34, the tuple ( n Z , F n , F n +1 ) is a (non-zero) commutative ring.What remains is to prove that ( n Z , F n , F n +1 ) has no zero divisors, i.e., that theproduct F n +1 is closed in n Z \ { n } . The proof is analogous to the proof that thering of integers has no zero divisors. (cid:4) Inverse Completion:Definition 6.39.
Let ( S, ∗ ) be an algebraic structure. An element a ∈ ( S, ∗ ) is cancellable if and only if: ∀ a, b ∈ S : x ∗ a = x ∗ b = ⇒ a = b, (6.40a) ∀ a, b ∈ S : a ∗ x = b ∗ x = ⇒ a = b. (6.40b) Let ( S, ∗ ) be a commutative semigroup which has cancellable elements. Let( C, ∗ C ) ⊆ ( S, ∗ ) be the subsemigroup of cancellable elements of ( S, ∗ ), where ∗ C denotes the restriction of ∗ to C .Consider the external product ( S × C, ⊕ ), i.e., ∀ ( x, y ) , ( u, v ) ∈ S × C : ( x, y ) ⊕ ( u, v ) = ( x ∗ u, y ∗ C v ) . (6.41) Remark . Let x, y ∈ S and a, b ∈ C . The cross-relation (cid:2) defined as,( x ∗ a, a ) (cid:2) ( y ∗ b, b ) ⇐⇒ x = y, (6.43)is a congruence relation on ( S × C, ⊕ ). Remark . Let [[( x, y )]] (cid:2) be the equivalent class of ( x, y ) under (cid:2) , then,[[( x ∗ a, y ∗ a )]] (cid:2) = [[( x, y )]] (cid:2) . (6.45) Remark . Consider the quotient structure,( T (cid:48) , ⊕ (cid:48) ) = (cid:18) S × C (cid:2) , ⊕ (cid:2) (cid:19) , (6.47)where ⊕ (cid:2) is the operation induced on ( S × C ) / (cid:2) by ⊕ . For any c ∈ C , the identityelement is [[( c, c )]] (cid:2) . If all elements of S are cancellable ( C = S ), then the tuple( T (cid:48) , ⊕ (cid:48) ) is an abelian group. References [1] Albert A. Bennet,
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