116 research outputs found
Local structure of algebraic monoids
We describe the local structure of an irreducible algebraic monoid at an
idempotent element . When is minimal, we show that is an induced
variety over the kernel (a homogeneous space) with fibre the two-sided
stabilizer (a connected affine monoid having a zero element and a dense
unit group). This yields the irreducibility of stabilizers and centralizers of
idempotents when is normal, and criteria for normality and smoothness of an
arbitrary . Also, we show that is an induced variety over an abelian
variety, with fiber a connected affine monoid having a dense unit group.Comment: Final version, minor changes, to appear in Moscow Mathematical
Journa
Conjugation in Semigroups
The action of any group on itself by conjugation and the corresponding
conjugacy relation play an important role in group theory. There have been
several attempts to extend the notion of conjugacy to semigroups. In this
paper, we present a new definition of conjugacy that can be applied to an
arbitrary semigroup and it does not reduce to the universal relation in
semigroups with a zero. We compare the new notion of conjugacy with existing
definitions, characterize the conjugacy in various semigroups of
transformations on a set, and count the number of conjugacy classes in these
semigroups when the set is infinite.Comment: 41 pages, 14 figure
The development version of the CHEVIE package of GAP3
I describe the current state of the development version of the CHEVIE
package, which deals with Coxeter groups, reductive algebraic groups, complex
reflection groups, Hecke algebras, braid monoids, etc... Examples are given,
showing the code to check some results of Lusztig.Comment: 24 pages. arXiv admin note: text overlap with arXiv:1003.492
Some results on embeddings of algebras, after de Bruijn and McKenzie
In 1957, N. G. de Bruijn showed that the symmetric group Sym(\Omega) on an
infinite set \Omega contains a free subgroup on 2^{card(\Omega)} generators,
and proved a more general statement, a sample consequence of which is that for
any group A of cardinality \leq card(\Omega), Sym(\Omega) contains a coproduct
of 2^{card(\Omega)} copies of A, not only in the variety of all groups, but in
any variety of groups to which A belongs. His key lemma is here generalized to
an arbitrary variety of algebras \bf{V}, and formulated as a statement about
functors Set --> \bf{V}. From this one easily obtains analogs of the results
stated above with "group" and Sym(\Omega) replaced by "monoid" and the monoid
Self(\Omega) of endomaps of \Omega, by "associative K-algebra" and the
K-algebra End_K(V) of endomorphisms of a K-vector-space V with basis \Omega,
and by "lattice" and the lattice Equiv(\Omega) of equivalence relations on
\Omega. It is also shown, extending another result from de Bruijn's 1957 paper,
that each of Sym(\Omega), Self(\Omega) and End_K (V) contains a coproduct of
2^{card(\Omega)} copies of itself.
That paper also gave an example of a group of cardinality 2^{card(\Omega)}
that was {\em not} embeddable in Sym(\Omega), and R. McKenzie subsequently
established a large class of such examples. Those results are shown to be
instances of a general property of the lattice of solution sets in Sym(\Omega)
of sets of equations with constants in Sym(\Omega). Again, similar results --
this time of varying strengths -- are obtained for Self(\Omega), End_K (V), and
Equiv(\Omega), and also for the monoid \Rel of binary relations on \Omega.
Many open questions and areas for further investigation are noted.Comment: 37 pages. Copy at http://math.berkeley.edu/~gbergman/papers is likely
to be updated more often than arXiv copy Revised version includes answers to
some questions left open in first version, references to results of Wehrung
answering some other questions, and some additional new result
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