Download PDF by N. Bourbaki: Elements de Mathematique. Algebre. Chapitre 9

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By N. Bourbaki

ISBN-10: 3540353380

ISBN-13: 9783540353386

ISBN-10: 3540353399

ISBN-13: 9783540353393

Formes sesquilin?©aires et formes quadratiques.

Les ?‰l?©ments de math?©matique de Nicolas BOURBAKI ont pour objet une pr?©sentation rigoureuse, syst?©matique et sans pr?©requis des math?©matiques depuis leurs fondements.

Ce neuvi??me chapitre du Livre d Alg??bre, deuxi??me Livre du trait?©, est consacr?© aux formes quadratiques, symplectiques ou hermitiennes et aux groupes associ?©s.

Il contient ?©galement une word historique.

Ce quantity est une r?©impression de l ?©dition de 1959.

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Extra info for Elements de Mathematique. Algebre. Chapitre 9

Example text

Then it is isomorphie to a Rees matrix semigroup M = M(I ,G,J;P) over a group G with a sandwich matrix P. We can identify Sand M. Let (i ,a,j), (k,b ,n) E S be arbitrary elements. Then (i,a ,j)(k ,b ,n) = (i,a,j)2(i, (Pj,iapj,d-Ipj,kb,n) E (i ,a,j)2S which implies that S is a left Putcha semigroup. We can prove, in a similar way, that S is a right Putcha semigroup. li}) A sem igroup is an archimedean left and right Putcha semigroup containing at least one idempotent elem ent il and only il it is a retract ext ension 01 a completely simple sem igroup by a nil semigroup.

As the semigroups listed in the theorem are tl-bands, the theorem is proved. S. Putcha chara cterized semigroups which are decomposable into semilat tice of archimedean semig roups. He showed that a semigroup S is a semilattice of arehirneden semigroups if and only if, for every a, b E S, the assumption a E S1bS 1 implies an E S1 a 2 SI for some positive integer n . Semigroups with this condition are called Putcha semigroups. 1). It is proved that a semigroup is a simple left and right Putcha semigroup if and only if it is com ple tely sim ple.

Sin ee 8 is subdirectly irredueibl e, C{sd = ids for some SI E 8. It is easy to see that nsdsESC{s} = ids and so C{S 2} = ids for some s2 =/: SI. Henee SI and s2 are two different disjunctive elements of 8. 0 11 a se m igroup 8 with a zero has a non-zero disjunctive elemen t th en 8 has a core and eve r y disjunctiv e element of 8 is in th e core. 5 {[85}} Proof. Assume that a semigroup 8 with a zero has a non-zero disjunctive element k . Sinee r(k) = {s E 8: (Vx ,y E 8 1 ) xsy =/: k} is a C{k}-c1ass and an ideal of 8 , it follows that r(k) = {O} (beeause k is disjunctive).

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Elements de Mathematique. Algebre. Chapitre 9 by N. Bourbaki

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