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Since a,a1 are dyadic fractions and a > a 1 , a-a1 is a positive dyadic fraction. 6, (3b'eF1 )(3b"eG' )(b"-b' a'+b. Similarly, we can see that numbers of the form a+b" are cofinal on the right. By the cofinality theorem a+b = {a+b'}|{a+b n }. Note that a+b' and a+b" are dyadic fractions. Also since F' = {b 1 } has no maximum, neither does {a+b 1 }. Similarly AN INTRODUCTION TO THE THEORY OF SURREAL NUMBERS 36 {a+b11} has no minimum. 3, a+b is a real number. Now suppose neither a nor b is a dyadic fraction.

Now is a lower element for the same side of ac acx < 1. ac iff respectively i f f a c °a and c are on e G iff c °a > c. (This follows from the e a r l i e r statement regarding the map b -• b°a 1 *) Since c l°^l s a t i s f i e s the equation o the inequality c1 a1 > c (a-a 1 )c 1 + a x x = 1 and ax > 0 is equivalent to ( a - a ^ c ^ axc < 1. The left-hand side of this inequality is nothing but Hence the lower elements for elements are greater than that cl°a1 * c ac 1. are less than (Note that since so that the negation of ">" axc + a c ^ a^.

Furthermore, the canonical representation of a a is {a$: 3. Again, if H is a set of ordinals then { a ^ } ^ = a a where a is the least ordinal such that a > H. If H has a maximum 3 then a = 3+1. b. H. a is called the sequent of H and denoted by seq H in the literature. In summary, as far as the order properties are concerned, the identification is quite reasonable. Thus for convenience of notation we use a instead of a a . So the ordinals are the sequences which consist only of pluses. (Incidentally, note that our definition of cofinality [see p.