849 research outputs found
Decidability and Independence of Conjugacy Problems in Finitely Presented Monoids
There have been several attempts to extend the notion of conjugacy from
groups to monoids. The aim of this paper is study the decidability and
independence of conjugacy problems for three of these notions (which we will
denote by , , and ) in certain classes of finitely
presented monoids. We will show that in the class of polycyclic monoids,
-conjugacy is "almost" transitive, is strictly included in
, and the - and -conjugacy problems are decidable with linear
compexity. For other classes of monoids, the situation is more complicated. We
show that there exists a monoid defined by a finite complete presentation
such that the -conjugacy problem for is undecidable, and that for
finitely presented monoids, the -conjugacy problem and the word problem are
independent, as are the -conjugacy and -conjugacy problems.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1503.0091
Conjugacy Classes of Renner Monoids
In this paper we describe conjugacy classes of a Renner monoid with unit
group , the Weyl group. We show that every element in is conjugate to an
element where and is an idempotent in a cross section
lattice. Denote by and the centralizer and stabilizer of in , respectively. Let act by conjugation on the set of left
cosets of in . We find that and () are
conjugate if and only if and are in the same orbit. As
consequences, there is a one-to-one correspondence between the conjugacy
classes of and the orbits of this action. We then obtain a formula for
calculating the number of conjugacy classes of , and describe in detail the
conjugacy classes of the Renner monoid of some -irreducible monoids.
We then generalize the Munn conjugacy on a rook monoid to any Renner monoid
and show that the Munn conjugacy coincides with the semigroup conjugacy, action
conjugacy, and character conjugacy. We also show that the number of
inequivalent irreducible representations of over an algebraically closed
field of characteristic zero equals the number of the Munn conjugacy classes in
.Comment: A reference ([13]) and Corollary 4.5 are added to show the connection
between the result in Theorem 4.4 of the previous version and the results in
[13]. A paragraph on page 12 is new to show that Theorem 4.4 can also be
deduced from the results in [13]. Two necessary concepts from [13] to
describe the connection are inserted in Section 2.
Inverse monoids and immersions of 2-complexes
It is well known that under mild conditions on a connected topological space
, connected covers of may be classified via conjugacy
classes of subgroups of the fundamental group of . In this paper,
we extend these results to the study of immersions into 2-dimensional
CW-complexes. An immersion between
CW-complexes is a cellular map such that each point has a
neighborhood that is mapped homeomorphically onto by . In order
to classify immersions into a 2-dimensional CW-complex , we need to
replace the fundamental group of by an appropriate inverse monoid.
We show how conjugacy classes of the closed inverse submonoids of this inverse
monoid may be used to classify connected immersions into the complex
Bijection between Conjugacy Classes and Irreducible Representations of Finite Inverse Semigroups
In this paper we show that the irreducible representations of a finite
inverse semigroup over an algebraically closed field are in bijection
with the conjugacy classes of if the characteristic of is zero or a
prime number that does not divide the order of any maximal subgroup of
Knuth-Bendix algorithm and the conjugacy problems in monoids
We present an algorithmic approach to the conjugacy problems in monoids,
using rewriting systems. We extend the classical theory of rewriting developed
by Knuth and Bendix to a rewriting that takes into account the cyclic
conjugates.Comment: This is a new version of the paper 'The conjugacy problems in monoids
and semigroups'. This version will appear in the journal 'Semigroup forum
Hopf monoids from class functions on unitriangular matrices
We build, from the collection of all groups of unitriangular matrices, Hopf
monoids in Joyal's category of species. Such structure is carried by the
collection of class function spaces on those groups, and also by the collection
of superclass function spaces, in the sense of Diaconis and Isaacs.
Superclasses of unitriangular matrices admit a simple description from which we
deduce a combinatorial model for the Hopf monoid of superclass functions, in
terms of the Hadamard product of the Hopf monoids of linear orders and of set
partitions. This implies a recent result relating the Hopf algebra of
superclass functions on unitriangular matrices to symmetric functions in
noncommuting variables. We determine the algebraic structure of the Hopf
monoid: it is a free monoid in species, with the canonical Hopf structure. As
an application, we derive certain estimates on the number of conjugacy classes
of unitriangular matrices.Comment: Final Version, 32 pages, accepted in "Algebra and Number Theory
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
On the Dual Canonical Monoids
We investigate the conjugacy decomposition, nilpotent variety, the Putcha
monoid, as well as the two-sided weak order on the dual canonical monoids
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