By F. Borceux, et al.,

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**Extra resources for Algebra in a Local Topos With Applns to Ring Theory**

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For any symmetry set R, we let B(R) = BCR) ^ where R is the ~ - c l o s u r e of R. the r o o t s y s t e m of R. CLASSICALSYMMETRY SETS AND THEIR CLASSIFICATION 7- DEFINITION. ~ We call B(R) such that R* = Hom A classical symmetry set is a symmetry set R (R,~) separates R. THEOREM. 2 free is a classical rootset. Proof. 3, R a n d R are i s o m o r p h i c . sical rootset is c l e a r since B(R) contains R + ~ R. 3 for all a E ~ - ~ and the a* Proof. [] One d i r e c t i o n free, follows that R is b o u n d e d .

S(b) b E S, It Then s ~ 1 at a in S is a r e f l e c t i o n , s normal, = aib ras = if it is a-normal. u at a in S has = b for all all (respectively symmetry mal for it p e r m u t e s PROPOSITION. every R. A bijection a-orbits a symmetry for that and ras(aib) [] (respectively 2. Symmetries PROPOSITION. and on S ~ R. (-r ~ i ~ q). = SsCb)(a) at a in S is n o r m a l we = a-is(b) if it n o r m a l i z e s on Sb(a) if j = a~(b). at a in S o f p e r i o d b E S. E R - 1) evident. Let a E R a n d S c • if S(Sb(a)) is a - n o r m a l (a,b < i < q.

R' is a mapping f: R ~ R' such that f(ab) = f(a)f(b) that f - 1 E unipotents T 2 are in the same orbit in T 2 under V if A h o m o m o r p h i s m from a groupset R to a groupset DEFINITION. f: R + R' in the set HOMOMORPHISMSAND ISOMORPHISMS 3. that is, a'*(b') = a*(b) and ra,(b') Caftan = ra(b)' E R - i. following that S2 N R ~ 3,3 THEOREM. d e n o t e d s(b) = b', Then suppose tt' are n o r m a l , tt' i [] theorem T 2 of r e f l e c t i o n s a in S: such u = v normal. The for all Proposition under Conversely, u at a in S.