342 research outputs found
Every group is the maximal subgroup of a naturally occurring free idempotent generated semigroup
Gray and Ruskuc have shown that any group G occurs as the maximal subgroup of
some free idempotent generated semigroup IG(E) on a biordered set of
idempotents E, thus resolving a long standing open question. Given the group G,
they make a careful choice for E and use a certain amount of well developed
machinery. Our aim here is to present a short and direct proof of the same
result, moreover by using a naturally occuring biordered set.
More specifically, for any free G-act F_n(G) of finite rank at least 3, we
have that G is a maximal subgroup of IG(E) where E is the biordered set of
idempotents of End F_n(G). Note that if G is finite then so is End F_n(G)
Subgroups of free idempotent generated semigroups: full linear monoid
We develop some new topological tools to study maximal subgroups of free
idempotent generated semigroups. As an application, we show that the rank 1
component of the free idempotent generated semigroup of the biordered set of a
full matrix monoid of n x n matrices, n>2$ over a division ring Q has maximal
subgroup isomorphic to the multiplicative subgroup of Q.Comment: We hope to use similar methods to study the higher rank component
On Maximal Subgroups of Free Idempotent Generated Semigroups
We prove the following results: (1) Every group is a maximal subgroup of some
free idempotent generated semigroup. (2) Every finitely presented group is a
maximal subgroup of some free idempotent generated semigroup arising from a
finite semigroup. (3) Every group is a maximal subgroup of some free regular
idempotent generated semigroup. (4) Every finite group is a maximal subgroup of
some free regular idempotent generated semigroup arising from a finite regular
semigroup.Comment: 27 page
Maximal subgroups of free idempotent generated semigroups over the full linear monoid
We show that the rank r component of the free idempotent generated semigroup
of the biordered set of the full linear monoid of n x n matrices over a
division ring Q has maximal subgroup isomorphic to the general linear group
GL_r(Q), where n and r are positive integers with r < n/3.Comment: 37 pages; Transactions of the American Mathematical Society (to
appear). arXiv admin note: text overlap with arXiv:1009.5683 by other author
On Semigroups with Lower Semimodular Lattice of Subsemigroups
The question of which semigroups have lower semimodular lattice of subsemigroups has been open since the early 1960s, when the corresponding question was answered for modularity and for upper semimodularity. We provide a characterization of such semigroups in the language of principal factors. Since it is easily seen (and has long been known) that semigroups for which Green\u27s relation J is trivial have this property, a description in such terms is natural. In the case of periodic semigroupsāa case that turns out to include all eventually regular semigroupsāthe characterization becomes quite explicit and yields interesting consequences. In the general case, it remains an open question whether there exists a simple, but not completely simple, semigroup with this property. Any such semigroup must at least be idempotent-free and D-trivial
Free idempotent generated semigroups and endomorphism monoids of free -acts
The study of the free idempotent generated semigroup over a
biordered set began with the seminal work of Nambooripad in the 1970s and
has seen a recent revival with a number of new approaches, both geometric and
combinatorial. Here we study in the case is the biordered
set of a wreath product , where is a group and
is the full transformation monoid on elements. This wreath
product is isomorphic to the endomorphism monoid of the free -act
on generators, and this provides us with a convenient approach.
We say that the rank of an element of is the minimal number of
(free) generators in its image. Let For
rather straightforward reasons it is known that if (respectively, ), then the maximal subgroup of
containing is free (respectively, trivial). We show that if
where , then the maximal
subgroup of containing is isomorphic to that in
and hence to , where is the
symmetric group on elements. We have previously shown this result in the
case ; however, for higher rank, a more sophisticated approach is needed.
Our current proof subsumes the case and thus provides another approach to
showing that any group occurs as the maximal subgroup of some .
On the other hand, varying again and taking to be trivial, we obtain an
alternative proof of the recent result of Gray and Ru\v{s}kuc for the biordered
set of idempotents of Comment: 35 page
Quivers of monoids with basic algebras
We compute the quiver of any monoid that has a basic algebra over an
algebraically closed field of characteristic zero. More generally, we reduce
the computation of the quiver over a splitting field of a class of monoids that
we term rectangular monoids (in the semigroup theory literature the class is
known as ) to representation theoretic computations for group
algebras of maximal subgroups. Hence in good characteristic for the maximal
subgroups, this gives an essentially complete computation. Since groups are
examples of rectangular monoids, we cannot hope to do better than this.
For the subclass of -trivial monoids, we also provide a semigroup
theoretic description of the projective indecomposables and compute the Cartan
matrix.Comment: Minor corrections and improvements to exposition were made. Some
theorem statements were simplified. Also we made a language change. Several
of our results are more naturally expressed using the language of Karoubi
envelopes and irreducible morphisms. There are no substantial changes in
actual result
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