Non-associative (or structured, or arborescent) numbers

on März 12, 2011 in General

The perhaps most basic concept of mathematics are the natural numbers and the act of counting. For example the assertion of Kronecker “Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.” (“The integers are made by God, everything else is human made.”) stresses the important role of the natural/the integer numbers.

Counting is closely related with the concept of sets and their cardinality. Two sets have by definition the same number of elements if there is a one-one assignment of their elements. Sets have this strange property that one can arbitrarily rearrange their elements without changing the set, which is reflected in the associativity and the commutativity of the union-operation.

In computer science however however it is rather cumbersome to implement this unsortedness of the elements of a set. Here the underlying structure are ordered lists.

In a view from universal algebra and theoretical computer science one is given operations and constants (which can be seen as 0-ary operations) and they generate all the terms that can be made from these operations. For example if we have a constant/terminal element 1 and a binary operation + we can form the terms, 1, 1+1, (1+1)+1, 1+(1+1), (1+1)+(1+1), etc.

In the natural numbers all the grouping/bracketing falls flat and in the end it is just the number of 1′s regardless of the order or bracketing, e.g. 1+(1+1) = (1+1)+1 = 1+1+1.
From the standpoint of universal algebra one has to impose the associative law on the terms made up by + and 1 (which implies the commutative law) to beat the term structure flat.

If we however keep the bracketing structure one can still define multiplication (beside the inherent addition) in the default way (by left-distributivity), which is again associative, and we arrive at the concept of structured numbers.
When writing my Ph.D. thesis [Tra07] — which basically deals with exponentiation and the higher operations ladder — I was not aware of the existence of other literature about the topic. Just later slowly it turned out that there was some sparse activity from the 1950 to 1970 about that topic which however seems to remain unpursued, I think due to the difficulty of the matter.

To make the topic more explicit we call an algebraic structure non-associative arithmetic if it has a constant 1, an addition + and a multiplication *, which satisfy:
1 * x = x and (x+y)*z = x*z+y*z and which is generated from 1. The multiplication is then automatically associative.

In this article I want to summarize the findings by giving a commented literature list. I hope to attract people interested in that matter which in turn may contribute literature I didn’t find.

The logarithmetic – as introduced by Etherington – of a groupoid is defined as the set of exponents T (groupings/bracketings) of elements such that x^T=y^T for all x, y. Addition is defined by x^{S+T}=x^Sx^T and multiplication by x^{ST}=(x^S)^T.

[Bru55] uses associative right neoring as description and shows division by the center.

[Fra55] introduces symmetric law: (A+B)+(C+D)=(A+C)+(B+D).
[Eth59] calls it entropic law.

[Eva57] The additive entropic/symmetric law implies commutativity of the multiplication. By induction:
(A+B)(C+D) = (A+B)C + (A+B)D = C(A+B) + D(A+B)
= (CA+CB) + (DA+DB) = (CA+DA) + (CB+DB)
= A(C+D) + B(C+D)
= (C+D)(A+B)
S is N divided by the entropic law.
This then gives a commutative neoring. Embeddability into neofield?

[Eva57] embeds the free cyclic (generated by 1 element) groupoid N into a loop I, making addition invertible on both sides. Shows that still prime number decomposition is valid.

Questions about embeddability arise:
Embeddability of N into a structure with division
Embeddability of I into a structure with division (right skew neofield?)
Embeddability of S into a structure with division

[Min59] proves number theoretic theorems for the free cyclic logarithmetic:
(1) if PQ=QP then P and Q are powers of the same element
(2) if X^m P = Y^n or P X^m  = Y^n where P is prime and m,n>1 then X and Y are powers of P.
(3) X^p + Y^q = Z^r with p,q,r>1 then X=2^k, Y=2^m, Z=2^n with kp=mq=nr-1.

[Bun57] shows cancellation of addition and (induced) multiplication of the free commutative groupoid.

[Tra07] introduces left-commutative law: a+(b+c) = b+(a+c) and constructs the free division structure of N.
constructs the free division structure of the left-commutative N.

Literature (with links to the abstract in Zentralblatt)

[Eth39] Etherington, I. M. H. (1939). On non-associative combinations. Proc. R. Soc. Edinb., 59, 153–162.

[Eth48] Etherington, I. M. H. (1948). Non-associative arithmetics. Proc. R. Soc. Edinb., Sect. A, 62, 442–453.

[Eth55] Etherington, I. M. H. (1955). Theory of indices for non-associative algebra. Proc. R. Soc. Edinb., Sect. A, 64, 150–160.

[Rob49] Robinson, A. (1949). On non-associative systems. Proc. Edinb. Math. Soc., II. Ser., 8, 111–118.

[Bru55] Bruck, R. H. (1955). Analogues of the ring of rational integers. Proc. Am. Math. Soc., 6, 50–58.

[Fri55] Frink, O. (1955). Symmetric and self-distributive systems. Amer. Math. Monthly, 62, 697–707.

[Eva51] Evans, T. (1951). On multiplicative systems defined by generators and relations. I. Normal form theorems. Proc. Camb. Philos. Soc., 47, 637–649.

[Eva53] Evans, T. (1953). On multiplicative systems defined by generators and relations. II. Monogenic loops. Proc. Camb. Philos. Soc., 49, pp 579-589 doi:10.1017/S0305004100028772

[Eva65] Evans, T. (1956). Some remarks on a paper by R. H. Bruck. Proc. Am. Math. Soc., 7, 211–220.

[Eva57] Evans, T. (1957). Nonassociative number theory. Am. Math. Mon., 64, 299–309.

[Min57] Minc, H. (1957). Index polynomials and bifurcating root-trees. Proc. R. Soc. Edinb., Sect. A, 64, 319–341.

[Min59] Minc, H. (1959). Theorems on nonassociative number theory. Amer. Math. Monthly, 66, 486–488.

[Bol67] Bollman, D. (1967). Formal nonassociative number theory. Notre Dame J. Formal Logic, 8, 9–16.

[Bol73] Bollman, D., & Laplaza, M. (1973). A set-theoretic model for nonassociative number theory. Notre Dame J. Formal Logic, 14, 107–110.

[Bun76] Bunder, M. W. (1976). Commutative non-associative number theory. Proc. Edinburgh Math. Soc. (2), 20(2), 133–136.

[Blo98] Blondel, V. D. (1998). Structured numbers: Properties of a hierarchy of operations on binary trees. Acta Informatica, 35(1), 1–15.

[Duc98] Duchon, P. (1998). Right-cancellability of a family of operations on binary trees. Discrete Mathematics and Theoretical Computer Science, 2(1), 27–33.

[Tra07] Trappmann, H. (2007). Arborescent numbers: Higher arithmetic operations and division trees. Ph.D. thesis, University of Potsdam, Potsdam.

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