1,284 research outputs found
Monoids of modules and arithmetic of direct-sum decompositions
Let be a (possibly noncommutative) ring and let be a class
of finitely generated (right) -modules which is closed under finite direct
sums, direct summands, and isomorphisms. Then the set
of isomorphism classes of modules is a commutative semigroup with operation
induced by the direct sum. This semigroup encodes all possible information
about direct sum decompositions of modules in . If the endomorphism
ring of each module in is semilocal, then is a Krull monoid. Although this fact was observed nearly a decade ago, the
focus of study thus far has been on ring- and module-theoretic conditions
enforcing that is Krull. If
is Krull, its arithmetic depends only on the class group of and the set of classes containing prime divisors. In this paper
we provide the first systematic treatment to study the direct-sum
decompositions of modules using methods from Factorization Theory of Krull
monoids. We do this when is the class of finitely generated
torsion-free modules over certain one- and two-dimensional commutative
Noetherian local rings.Comment: Pacific Journal of Mathematics, to appea
Strong forms of self-duality for Hopf monoids in species
A vector species is a functor from the category of finite sets with
bijections to vector spaces (over a fixed field); informally, one can view this
as a sequence of -modules. A Hopf monoid (in the category of vector
species) consists of a vector species with unit, counit, product, and coproduct
morphisms satisfying several compatibility conditions, analogous to a graded
Hopf algebra. A vector species has a basis if and only if it is given by a
sequence of -modules which are permutation representations. We say that a
Hopf monoid is freely self-dual if it is connected and finite-dimensional, and
if it has a basis in which the structure constants of its product and coproduct
coincide. Such Hopf monoids are self-dual in the usual sense, and we show that
they are furthermore both commutative and cocommutative. We prove more specific
classification theorems for freely self-dual Hopf monoids whose products
(respectively, coproducts) are linearized in the sense that they preserve the
basis; we call such Hopf monoids strongly self-dual (respectively, linearly
self-dual). In particular, we show that every strongly self-dual Hopf monoid
has a basis isomorphic to some species of block-labeled set partitions, on
which the product acts as the disjoint union. In turn, every linearly self-dual
Hopf monoid has a basis isomorphic to the species of maps to a fixed set, on
which the coproduct acts as restriction. It follows that every linearly
self-dual Hopf monoid is strongly self-dual. Our final results concern
connected Hopf monoids which are finite-dimensional, commutative, and
cocommutative. We prove that such a Hopf monoid has a basis in which its
product and coproduct are both linearized if and only if it is strongly
self-dual with respect to a basis equipped with a certain partial order,
generalizing the refinement partial order on set partitions.Comment: 42 pages; v2: a few typographical errors corrected and references
updated; v3: discussion in Sections 3.1 and 3.2 slightly revised, Theorem A
corrected to include hypothesis about ambient field, final versio
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
Zappa-Sz\'ep products of Garside monoids
A monoid is the internal Zappa-Sz\'ep product of two submonoids, if every
element of admits a unique factorisation as the product of one element of
each of the submonoids in a given order. This definition yields actions of the
submonoids on each other, which we show to be structure preserving.
We prove that is a Garside monoid if and only if both of the submonoids
are Garside monoids. In this case, these factors are parabolic submonoids of
and the Garside structure of can be described in terms of the Garside
structures of the factors. We give explicit isomorphisms between the lattice
structures of and the product of the lattice structures on the factors that
respect the Garside normal forms. In particular, we obtain explicit natural
bijections between the normal form language of and the product of the
normal form languages of its factors.Comment: Published versio
Invariant means on Boolean inverse monoids
The classical theory of invariant means, which plays an important role in the
theory of paradoxical decompositions, is based upon what are usually termed
`pseudogroups'. Such pseudogroups are in fact concrete examples of the Boolean
inverse monoids which give rise to etale topological groupoids under
non-commutative Stone duality. We accordingly initiate the theory of invariant
means on arbitrary Boolean inverse monoids. Our main theorem is a
characterization of when a Boolean inverse monoid admits an invariant mean.
This generalizes the classical Tarski alternative proved, for example, by de la
Harpe and Skandalis, but using different methods
Binomial Ideals and Congruences on Nn
Producción CientíficaA congruence on Nn is an equivalence relation on Nn that is compatible with the additive structure. If k is a field, and I is a binomial ideal in k[X1,…,Xn] (that is, an ideal generated by polynomials with at most two terms), then I induces a congruence on Nn by declaring u and v to be equivalent if there is a linear combination with nonzero coefficients of Xu and Xv that belongs to I. While every congruence on Nn arises this way, this is not a one-to-one correspondence, as many binomial ideals may induce the same congruence. Nevertheless, the link between a binomial ideal and its corresponding congruence is strong, and one may think of congruences as the underlying combinatorial structures of binomial ideals. In the current literature, the theories of binomial ideals and congruences on Nn are developed separately. The aim of this survey paper is to provide a detailed parallel exposition, that provides algebraic intuition for the combinatorial analysis of congruences. For the elaboration of this survey paper, we followed mainly (Kahle and Miller Algebra Number Theory 8(6):1297–1364, 2014) with an eye on Eisenbud and Sturmfels (Duke Math J 84(1):1–45, 1996) and Ojeda and Piedra Sánchez (J Symbolic Comput 30(4):383–400, 2000).National Science Foundation (grant DMS-1500832)Ministerio de Economía, Industria y Competitividad (project MTM2015-65764-C3-1)Junta de Extremadura (grupo de investigación FQM-024
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