428 research outputs found
Representation Theory of Finite Semigroups, Semigroup Radicals and Formal Language Theory
In this paper we characterize the congruence associated to the direct sum of
all irreducible representations of a finite semigroup over an arbitrary field,
generalizing results of Rhodes for the field of complex numbers. Applications
are given to obtain many new results, as well as easier proofs of several
results in the literature, involving: triangularizability of finite semigroups;
which semigroups have (split) basic semigroup algebras, two-sided semidirect
product decompositions of finite monoids; unambiguous products of rational
languages; products of rational languages with counter; and \v{C}ern\'y's
conjecture for an important class of automata
Schreier split extensions of preordered monoids
Properties of preordered monoids are investigated and important subclasses of
such structures are studied. The corresponding full subcategories of the
category of preordered monoids are functorially related between them as well as
with the categories of preordered sets and monoids. Schreier split extensions
are described in the full subcategory of preordered monoids whose preorder is
determined by the corresponding positive cone
Commuting upper triangular binary morphisms
A morphism from the free monoid into itself is called upper
triangular if the matrix of is upper triangular. We characterize all upper
triangular binary morphisms and such that .Comment: 14 page
Quantale Modules and their Operators, with Applications
The central topic of this work is the categories of modules over unital
quantales. The main categorical properties are established and a special class
of operators, called Q-module transforms, is defined. Such operators - that
turn out to be precisely the homomorphisms between free objects in those
categories - find concrete applications in two different branches of image
processing, namely fuzzy image compression and mathematical morphology
The Algebra of Binary Search Trees
We introduce a monoid structure on the set of binary search trees, by a
process very similar to the construction of the plactic monoid, the
Robinson-Schensted insertion being replaced by the binary search tree
insertion. This leads to a new construction of the algebra of Planar Binary
Trees of Loday-Ronco, defining it in the same way as Non-Commutative Symmetric
Functions and Free Symmetric Functions. We briefly explain how the main known
properties of the Loday-Ronco algebra can be described and proved with this
combinatorial point of view, and then discuss it from a representation
theoretical point of view, which in turns leads to new combinatorial properties
of binary trees.Comment: 49 page
Factorizations of Elements in Noncommutative Rings: A Survey
We survey results on factorizations of non zero-divisors into atoms
(irreducible elements) in noncommutative rings. The point of view in this
survey is motivated by the commutative theory of non-unique factorizations.
Topics covered include unique factorization up to order and similarity, 2-firs,
and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and
Jordan and generalizations thereof. We recall arithmetical invariants for the
study of non-unique factorizations, and give transfer results for arithmetical
invariants in matrix rings, rings of triangular matrices, and classical maximal
orders as well as classical hereditary orders in central simple algebras over
global fields.Comment: 50 pages, comments welcom
Factorization theory: From commutative to noncommutative settings
We study the non-uniqueness of factorizations of non zero-divisors into atoms
(irreducibles) in noncommutative rings. To do so, we extend concepts from the
commutative theory of non-unique factorizations to a noncommutative setting.
Several notions of factorizations as well as distances between them are
introduced. In addition, arithmetical invariants characterizing the
non-uniqueness of factorizations such as the catenary degree, the
-invariant, and the tame degree, are extended from commutative to
noncommutative settings. We introduce the concept of a cancellative semigroup
being permutably factorial, and characterize this property by means of
corresponding catenary and tame degrees. Also, we give necessary and sufficient
conditions for there to be a weak transfer homomorphism from a cancellative
semigroup to its reduced abelianization. Applying the abstract machinery we
develop, we determine various catenary degrees for classical maximal orders in
central simple algebras over global fields by using a natural transfer
homomorphism to a monoid of zero-sum sequences over a ray class group. We also
determine catenary degrees and the permutable tame degree for the semigroup of
non zero-divisors of the ring of upper triangular matrices over a
commutative domain using a weak transfer homomorphism to a commutative
semigroup.Comment: 45 page
Solution sets for equations over free groups are EDT0L languages
© World Scientific Publishing Company. We show that, given an equation over a finitely generated free group, the set of all solutions in reduced words forms an effectively constructible EDT0L language. In particular, the set of all solutions in reduced words is an indexed language in the sense of Aho. The language characterization we give, as well as further questions about the existence or finiteness of solutions, follow from our explicit construction of a finite directed graph which encodes all the solutions. Our result incorporates the recently invented recompression technique of Jez, and a new way to integrate solutions of linear Diophantine equations into the process. As a byproduct of our techniques, we improve the complexity from quadratic nondeterministic space in previous works to NSPACE(n log n) here
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