By Mahir Can, Zhenheng Li, Benjamin Steinberg, Qiang Wang
This e-book features a number of fifteen articles and is devoted to the 60th birthdays of Lex Renner and Mohan Putcha, the pioneers of the sector of algebraic monoids.
Topics offered include:
structure and illustration conception of reductive algebraic monoids
monoid schemes and purposes of monoids
monoids concerning Lie theory
equivariant embeddings of algebraic groups
constructions and houses of monoids from algebraic combinatorics
endomorphism monoids brought about from vector bundles
Hodge–Newton decompositions of reductive monoids
A part of those articles are designed to function a self-contained advent to those themes, whereas the remainder contributions are study articles containing formerly unpublished effects, that are certain to develop into very influential for destiny paintings. between those, for instance, the real fresh paintings of Michel Brion and Lex Renner displaying that the algebraic semi teams are strongly π-regular.
Graduate scholars in addition to researchers operating within the fields of algebraic (semi)group concept, algebraic combinatorics and the speculation of algebraic crew embeddings will reap the benefits of this particular and vast compilation of a few basic ends up in (semi)group concept, algebraic staff embeddings and algebraic combinatorics merged less than the umbrella of algebraic monoids.
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The examine effects released during this ebook diversity from natural mathematical concept (semigroup thought, discrete arithmetic, and so forth. ) to theoretical machine technological know-how, particularly formal languages and automata. The papers deal with matters within the algebraic and combinatorial theories of semigroups, phrases and languages, the constitution idea of automata, the category conception of formal languages and codes, and purposes of those theories to numerous components, like quantum and molecular computing, coding conception, and cryptography.
This can be an advent to puzzling over basic arithmetic from a categorial perspective. The objective is to discover the implications of a brand new and basic perception concerning the nature of arithmetic. Foreword; notice to the reader; Preview; half I. the class of units: 1. units, maps, composition; half II.
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Additional resources for Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics
X; to / equals '. (i) There exists a unique morphism « W Y T ! x/; t / on X T . x/ on X T ). Proof. x; t / 7! x; t / 7! OX / D OY in view of Lemma 1. y; to /, where y 2 Y , and the assumption (iii) of that lemma holds with s being the inclusion of Y To in X To . Finally, the assumption (iv)0 of Remark 13 is satisfied, since T is connected. By that remark, we thus have g D g ı s ı f on X T . Hence there 46 M. Brion exists a unique morphism « W Y T ! y; t /. x/ on X T . t u Remark 14. The preceding result has a nice interpretation when X is projective.
In view of Theorem 5, every nontrivial algebraic semigroup law on an irreducible curve S is commutative; by Proposition 17 again, it follows that S has an idempotent F -point whenever S and are defined over F . 4 Rigidity In this subsection, we obtain two rigidity results (both possibly known, but for which we could not locate adequate references) and we apply them to the study of endomorphisms of complete varieties. Our first result is a scheme-theoretic version of a classical rigidity lemma for irreducible varieties (see [8, Lem.
D 0 ; a contradiction. Thus, we must have n D 2, and we obtain a nonconstant morphism D 2 W S ! A1 , where the semigroup law on A1 is the multiplication. The image of contains 0 and a nonempty open subset U of the unit group Gm . Then U U D Gm and hence is surjective. e/ D 1. Then e is the desired nonzero idempotent. t u Remark 10. One may also deduce the above theorem from the description of algebraic semigroup structures on abelian varieties (Proposition 21), when the irreducible curve S is assumed to be nonsingular and nonrational.