The formal theory of hopf algebras Part I: Hopf monoids in a monoidal category

Abstract

The category Hopf ? of Hopf monoids in a symmetric monoidal category ?, assumed to be locally finitely presentable as a category, is analyzed with respect to its categorical properties. Assuming that the functors “tensor squaring” and “tensor cubing” on ? preserve directed colimits one has the following results: (1) If, in ?, extremal epimorphisms are stable under tensor squaring, then Hopf C is locally presentable, coreflective in the category of bimonoids in ? and comonadic over the category of monoids in C. (2) If, in ?, extremal monomorphisms are stable under tensor squaring, then Hopf ? is locally presentable as well, reflective in the category of bimonoids in C and monadic over the category of comonoids in ?.     

A Note on Almost GCD Monoids

A commutative cancellative monoid H (with 0 adjoined) is called an almost GCD (AGCD) monoid if for x,y in H, there exists a natural number n = n(x,y) so that xⁿ and yⁿ have an LCM, that is, xⁿH \cap yⁿH is principal. We relate AGCD monoids to the recently introduced inside factorial monoids (there is a subset Q of H so that the submonoid F of H generated by Q and the units of H is factorial and some power of each element of H is in F). For example, we show that an inside factorial monoid H is an AGCD monoid if and only if the elements of Q are primary in H, or equivalently, H is weakly Krull, distinct elements of Q are v-coprime in H, or the radical of each element of Q is a maximal t-ideal of H. Conditions are given for an AGCD monoid to be inside factorial and the results are put in the context of integral domains.     

Sets of Transformations as Criteria of Diversity within Sets of Objects

A set of transformations acting within a domain may serve as a standard in judging the diversity within any subset of that domain. We distinguish two possible outcomes of such a comparison (upper vs. lower bound to diversity), highlight some basic properties of both concepts, and illustrate them with examples. In the concluding section we indicate reasons of importance of such a “transformational approach to diversity”, by referring to the process of “transformation induction” and to problems in vision psychophysics.     

On Some Types of Rigid Sets in Banach Spaces

One of the properties characterizing Euclidean spaces says - roughly speaking- that their unit sphere has nice invariant properties. More precisely, a finite dimensional normed space has an Euclidean norm if and only if the group of isometries acts transitively on its unit sphere (the norm is “transitive”); such property of the sphere is also called “rigidity”. More recently, another notion of “rigidity” for compact sets, connected with “isometric sequences”, received some attention. Infinite rigid sets are diametral; moreover, under suitable assumptions on the space, they are also contained in the boundary of a sphere. These notions are connected with many problems, in different areas. Here we discuss and compare these two notions of rigid set, trying to indicate new relations among them and with some other properties of sets. Several examples complete the paper.     

The radical-annihilator monoid of a ring

Kuratowski’s closure-complement problem gives rise to a monoid generated by the closure and complement operations. Consideration of this monoid yielded an interesting classification of topological spaces, and subsequent decades saw further exploration using other set operations. This article is an exploration of a natural analogue in ring theory: a monoid produced by “radical” and “annihilator” maps on the set of ideals of a ring. We succeed in characterizing semiprime rings and commutative dual rings by their radical-annihilator monoids, and we determine the monoids for commutative local zero-dimensional (in the sense of Krull dimension) rings.     

Constructions and presentations for monoids

Presentations are found for the wreath product of two monoids, the Schützenberger product of two monoids, the Bruck-Reilly extension of a monoid, strong semilattices of monoids and Rees matrix semigroups of monoids.     

Measurable versions of the LS category on laminations

We give two new versions of the LS category for the set-up of measurable laminations defined by Bermúdez. Both of these versions must be considered as “tangential categories”. The first one, simply called (LS) category, is the direct analogue for measurable laminations of the tangential category of (topological) laminations introduced by Colman Vale and Macías Virgós. For the measurable lamination that underlies any lamination, our measurable tangential category is a lower bound of the tangential category. The second version, called the Λ-category, depends on the choice of a transverse invariant measure Λ. We show that both of these “tangential categories” satisfy appropriate versions of some well known properties of the classical category: the homotopy invariance, existence of a dimensional upper bound, a cohomological lower bound (cup length), and an upper bound given by the critical points of a smooth function. Also, we show possible applications of these invariants to variational problems.     

On Characterization of Monoids by Properties of Generators

Kilp and Knauer in(Comm. Algebra, 1992, 20(7), 1841–1856) gave characterizations of monoids when all generators in category of right S-acts(S is a monoid) satisfy properties such as freeness, projectivity, strong flatness, Condition(P), principal weak flatness, principal weak injectivity, weak injectivity, injectivity, divisibility, strong faithfulness and torsion freeness.Sedaghtjoo in(Semigroup Forum, 2013, 87: 653–662) characterized monoids by some other properties of generators including weak flatness, Condition(E) and regularity. To our knowledge,the problem has not been studied for properties mentioned above of(finitely generated, cyclic,monocyclic, Rees factor) right acts. In this article we answer the question corresponding to these properties and also f g-weak injectivity.     

Craig interpolation for semilinear substructural logics

The Craig interpolation property is investigated for substructural logics whose algebraic semantics are varieties of semilinear (subdirect products of linearly ordered) pointed commutative residuated lattices. It is shown that Craig interpolation fails for certain classes of these logics with weakening if the corresponding algebras are not idempotent. A complete characterization is then given of axiomatic extensions of the “R‐mingle with unit” logic (corresponding to varieties of Sugihara monoids) that have the Craig interpolation property. This latter characterization is obtained using a model‐theoretic quantifier elimination strategy to determine the varieties of Sugihara monoids admitting the amalgamation property.     

Transformations of discrete closure systems

Discrete systems such as sets, monoids, groups are familiar categories. The internal structure of the latter two is defined by an algebraic operator. In this paper we concentrate on discrete systems that are characterized by unary operators; these include choice operators σ, encountered in economics and social theory, and closure operators φ, encountered in discrete geometry and data mining. Because, for many arbitrary operators α, it is easy to induce a closure structure on the base set, closure operators play a central role in discrete systems. Our primary interest is in functions f that map power sets 2 ^U into power sets 2 ^U′, which are called transformations. Functions over continuous domains are usually characterized in terms of open sets. When the domains are discrete, closed sets seem more appropriate. In particular, we consider monotone transformations which are “continuous”, or “closed”. These can be used to establish criteria for asserting that “the closure of a transformed image under f is equal to the transformed image of the closure”. Finally, we show that the categories MCont and MClo of closure systems with morphisms given by the monotone continuous transformations and monotone closed transformations respectively have concrete direct products. And the supercategory Clo of MClo whose morphisms are just the closed transformations is shown to be cartesian closed.     

The role of normally hyperbolic invariant manifolds (NHIMS) in the context of the phase space setting for chemical reaction dynamics

In this paper we give an introduction to the notion of a normally hyperbolic invariant manifold (NHIM) and its role in chemical reaction dynamics.We do this by considering simple examples for one-, two-, and three-degree-of-freedom systems where explicit calculations can be carried out for all of the relevant geometrical structures and their properties can be explicitly understood. We specifically emphasize the notion of a NHIM as a “phase space concept”. In particular, we make the observation that the (phase space) NHIM plays the role of “carrying” the (configuration space) properties of a saddle point of the potential energy surface into phase space.We also consider an explicit example of a 2-degree-of-freedom system where a “global” dividing surface can be constructed using two index one saddles and one index two saddle. Such a dividing surface has arisen in several recent applications and, therefore, such a construction may be of wider interest.     

On the Description of the Content-Equality by a Relation

In the geometry of polyhedra we understand by an elementary content-functional a real valued, non-negative, finite additive measure on the set of polyhedra which is invariant under isometries. There are close relations between the content-measurement and the relation of equidecomposability. Two polyhedra are called equidecomposable if they are decomposed into pairwise congruent pieces. For an example we consider the set of all polygons in the euclidean plane. It is well known that planar polygons have the same area if and only if they are equidecomposable. In the three-dimensional euclidean space one also can describe the content-equality of polyhedra by a relation. Two polyhedra have the same volume if they are equidecomposable with respect to equiaffine mappings (see [3]). In [4] the concept of an invariant content of polyhedra in a topological Klein space is introduced. Each regular closed quasicompact set ot the space is called polyhedron. Under this supposition two polyhedra have equal contents if they are equivalent by decomposition. The relation “equivalent by decomposition” is closely related to the relation “equidecomposable”.     

Diversity in monoids

Let M be a (commutative cancellative) monoid. A nonunit element q ∈ M is called almost primary if for all a, b ∈ M, q | ab implies that there exists k ∈ ? such that q | a ^k or q | b ^k . We introduce a new monoid invariant, diversity, which generalizes this almost primary property. This invariant is developed and contextualized with other monoid invariants. It naturally leads to two additional properties (homogeneity and strong homogeneity) that measure how far an almost primary element is from being primary. Finally, as an application the authors consider factorizations into almost primary elements, which generalizes the established notion of factorization into primary elements.     

Central factorial numbers; their main properties and some applications.

The purpose of this paper is to present a systematic treatment of central factorial numbers (cfn), including their main properties, as well as to employ them in a variety of applications. The cfn are related more closely to the Stirling numbers than to the other well-known numbers of Bernoulli, Euler, etc., and they are at least as important as Stirling's numbers, said to be “as important as Bernoulli's, or even more so”.     

图的强收缩核与图的强自同态幺半群的正则性

本文研究图及其强自同态幺半群．首先刻画了图的强自同态幺半群的正则元，然后给出了此幺半群正则的充要条件．这推广了［１］和［２］中关于有限图的强自同态幺半群正则的结果．     

On metrizability of the free monoids

A complete solution is given of the problem of S. Marcus concerning the construction of a “better” distance in the free monoids from the viewpoint of measuring the difference of contextual behaviour with respect to a given language.     

The cardinalities of the Green classes of the free objects in varieties of bands

We compute the cardinalities of the Green classes of the free objects in the varieties of bands. We also compare the cardinalities obtained and analyse the “collapsing” of the finitely generated free objects. We deduce corresponding results for varieties of band monoids.     

About non-coincidence of invariant manifolds and intrinsic low dimensional manifolds (ILDM)

The present paper contains an analysis of some aspects of a well known method of Intrinsic Low-Dimensional Manifolds (ILDM), which is regularly used for model reduction purposes in a number of combustion problems. One of these aspects relates to an existence of additional solutions (so-called “ghost”-manifolds), which represent intrinsic low-dimensional manifolds and do NOT represent any slow invariant manifold even for two-dimensional singularly perturbed systems (for a small but finite singular parameter). These “ghost”-manifolds are examples that contradict to the conjecture about the coincidence of ILDM and slow invariant manifolds published previously. Another aspect of the ILDM-method concerns the so-called transition zones (turning manifolds) between different invariant manifolds. It is shown that transition manifolds can not be correctly described by the ILDM-method. This statement is illustrated by an example taken from the mathematical theory of combustion.     

Endomorphisms of graphs II. Various unretractive graphs

In this part of the article we investigate graphs for which different endomorphism monoids coincide. We consider endomorphisms, strong endomorphisms and automorphisms. Coincidences are investigated for joins of graphs and some lexicographic products. In an additional section the graphs with the respective properties are listed up to 8 vertices in two cases and up to 5 vertices in the remaining case.     

On Flatness Properties of Torsion Free Right Rees Factor Acts

In this paper, we investigate the monoids over which all torsion free right Rees factor acts satisfy some properties that follow from projectivity(such as (weak) flatness, strong flatness, condition (P), etc.). These results answer the questions in [1].