5 research outputs found
An infinite natural sum
As far as algebraic properties are concerned, the usual addition on the class
of ordinal numbers is not really well behaved; for example, it is not
commutative, nor left cancellative etc. In a few cases, the natural Hessemberg
sum is a better alternative, since it shares most of the usual properties of
the addition on the naturals.
A countably infinite version of the natural sum has been used in a recent
paper by V\"a\"an\"anen and Wang, with applications to infinitary logics. We
provide an order theoretical characterization of this operation. We show that
this countable natural sum differs from the more usual infinite ordinal sum
only for an initial finite "head" and agrees on the remaining infinite "tail".
We show how to evaluate the countable natural sum just by computing a finite
natural sum. Various kinds of infinite mixed sums of ordinals are discussed.Comment: v3 added a remark connected with surreal number
Intermediate arithmetic operations on ordinal numbers
There are two well-known ways of doing arithmetic with ordinal numbers: the
"ordinary" addition, multiplication, and exponentiation, which are defined by
transfinite iteration; and the "natural" (or Hessenberg) addition and
multiplication (denoted and ), each satisfying its own set of
algebraic laws. In 1909, Jacobsthal considered a third, intermediate way of
multiplying ordinals (denoted ), defined by transfinite iteration of
natural addition, as well as the notion of exponentiation defined by
transfinite iteration of his multiplication, which we denote
. (Jacobsthal's multiplication was later rediscovered by
Conway.) Jacobsthal showed these operations too obeyed algebraic laws. In this
paper, we pick up where Jacobsthal left off by considering the notion of
exponentiation obtained by transfinitely iterating natural multiplication
instead; we will denote this . We show that
and that
;
note the use of Jacobsthal's multiplication in the latter. We also demonstrate
the impossibility of defining a "natural exponentiation" satisfying reasonable
algebraic laws.Comment: 18 pages, 3 table