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By Donald Knutson
Knutson D. Lambda-Rings and the illustration conception of the Symmetric staff (LNM0308, Springer, 2007)(ISBN 3540061843)(1s)_Mln_-o
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The sequence and let l l , 1 2 , . . , 1 q ( l l , 1 2 , . . , 1 q) is / also a p a r t i t i o n Its d i a g r a m diagonal. The is o b t a i n e d b y f l i p p i n g Thus ~ E ~(n), partition of ~, d e n o t e d ~ the d i a g r a m for ~ a l o n g its for ~ = ( 6 , 5 , 3 , 3 , 1 , I , i ) , set of all p a r t i t i o n s (For a t a b l e For of n, the c o n j u q a t e another common ~'= ( 7 , 4 , 4 , 2 , 2 , 1 ) . of a n u m b e r n is d e n o t e d of the size of H(n) for n=l,2 ..... 200, notation .
R w is c e r t a i n l y isomorphic a k-algebra, and under the hypothesis, is to l + R [ [ t ~ +. The Proof of the original R has k-operations, R is a k-ring theorem is n o w accomplished. and hence Y-operations, iff R is a Y-ring. A useful If and is torsion-free, restatement of the theorem is the following proposition. Proposition: Let R b e a t o r s i o n - f r e e Suppose there is a ring homomorphism. There and S b e any ring. is given a map of sets ~:S--~> l + R [ ~ t ~ +. Then is a ring h o m o m o r p h i s m k-structure ring, iff the c o m p o s i t e map S --~ I + R [ [ t ~ + - ~ R If S is a pre-k-ring, ~ preserves the iff the c o m p o s i t e map L~ does.
T h a t t h i s c a t e g o r y of v a r i e t i e s is a g o o d e x a m p l e of a c a t e g o r y w h i c h has symmetric but whose Grothendieck and H e n c e R is not a k - r i n g and no o t h e r d e f i ~ i t i o n however, _affine sums, ring products, is not over k and a k-ring. 54 W e n o w u s e the t h e o r e m to c o n s t r u c t k-rings. K(S) Let S b e c l a s s of a set, K a field of c h a r a c t e r i s t i c the set of all m a p s two m a p s a general is d e f i n e d as usual, identity.