Abelian Monoid Research Articles

Morita homotopy theory of [formula omitted]-categories

In this article we establish the foundations of the Morita homotopy theory of C⁎-categories. Concretely, we construct a cofibrantly generated simplicial symmetric monoidal Quillen model structure (denoted by MMor) on the category C1⁎cat of small unital C⁎-categories. The weak equivalences are the Morita equivalences and the cofibrations are the ⁎-functors which are injective on objects. As an application, we obtain an elegant description of Brown–Green–Rieffelʼs Picard group in the associated homotopy category Ho(MMor). We then prove that Ho(MMor) is semi-additive. By group completing the induced abelian monoid structure at each Hom-set we obtain an additive category Ho(MMor)−1 and a composite functor C1⁎cat→Ho(MMor)→Ho(MMor)−1 which is characterized by two simple properties: inversion of Morita equivalences and preservation of all finite products. Finally, we prove that the classical Grothendieck group functor becomes co-represented in Ho(MMor)−1 by the tensor unit object.

A new graph based on the semi-direct product of some monoids

In this paper, firstly, we define a new graph based on the semi-direct product of a free abelian monoid of rank n by a finite cyclic monoid, and then discuss some graph properties on this new graph, namely diameter, maximum and minimum degrees, girth, degree sequence and irregularity index, domination number, chromatic number, clique number of � (PM). Since graph theoretical studies (including such above graph parameters) consist of some fixed point techniques, they have been applied in fields such as chemistry (in the meaning of atoms, molecules, energy etc.) and engineering (in the meaning of signal processing etc.), game theory and physics. MSC: 05C10; 05C12; 05C25; 20E22; 20M05

Some array polynomials over special monoid presentations

In a recent joint paper (Cevik et al. in Hacet. J. Math. Stat., acceptted), the authors have investigated the p-Cockcroft property (or, equivalently, efficiency) for a presentation, say , of the semi-direct product of a free abelian monoid rank two by a finite cyclic monoid. Moreover, they have presented sufficient conditions on a special case for to be minimal whilst it is inefficient. In this paper, by considering these results, we first show that the presentations of the form can actually be represented by characteristic polynomials. After that, some connections between representative characteristic polynomials and generating functions in terms of array polynomials over the presentation will be pointed out. Through indicated connections, the existence of an equivalence among each generating function in itself is claimed studied in this paper. MSC:11B68, 11S40, 12D10, 20M05, 20M50, 26C05, 26C10.

Constrained intervals and interval spaces

Constrained intervals, intervals as a mapping from [0, 1] to polynomials of degree one (linear functions) with non-negative slopes, and arithmetic on constrained intervals generate a space that turns out to be a cancellative abelian monoid albeit with a richer set of properties than the usual (standard) space of interval arithmetic. This means that not only do we have the classical embedding as developed by H. Radstrom, S. Markov, and the extension of E. Kaucher but the properties of these polynomials. We study the geometry of the embedding of intervals into a quasilinear space and some of the properties of the mapping of constrained intervals into a space of polynomials. It is assumed that the reader is familiar with the basic notions of interval arithmetic and interval analysis.

Homotopy type of space of maps into a $K(G,n)$

Let X be a connected CW complex and let K(G;n) be an Eilenberg-Mac Lane CW complex where G is abelian. As K(G;n) may be taken to be an abelian monoid, the weak homotopy type of the space of continuous functions X → K(G;n) depends only upon the homology groups of X. The purpose of this note is to prove that this is true for the actual homotopy type. Precisely, the space map ( X;K(G;n) ) of pointed continuous maps X → K(G;n) is shown to be homotopy equivalent to the Cartesian product ∏

A note on the weakly mixing of semigroup actions

This paper deals with the weak mixing of semigroup actions. We show that a system of semigroup action (S,X) is weakly mixing iff it has a uniform positive sequence entropy and it is topologically transitive, where X is a compact Hausdorff space and S is an abelian monoid semigroup.

Enumeration of classes of linear systems via equations and via partitions in an ordered abelian monoid

We prove that the feedback classes of locally Brunovsky linear systems are given by the decomposition of state space into direct summands. Hence the problem of enumerating feedback classes of systems is equivalent to the problem of enumerating all solutions of a linear equation over a commutative zerosumfree monoid. If this monoid is also cancelative then the problem is solved by all the partitions.

Quotients of dimension effect algebras

We show that the quotient of a dimension effect algebra by its dimension equivalence relation is a unital bounded lattice-ordered positive partial abelian monoid that satisfies a version of the Riesz decomposition property. For a dimension effect algebra of finite type, the quotient is a centrally orthocomplete Stone–Heyting MV-effect algebra; moreover, an orthocomplete effect algebra in which equality is a dimension equivalence relation is the same thing as a complete Stone–Heyting MV-effect algebra.

Leavitt path algebras of separated graphs

The construction of the Leavitt path algebra associated to a directed graph $E$ is extended to incorporate a family $C$ consisting of partitions of the sets of edges emanating from the vertices of $E$. The new algebras, $L_K(E,C)$, are analyzed in terms of their homology, ideal theory, and K-theory. These algebras are proved to be hereditary, and it is shown that any conical abelian monoid occurs as the monoid $\mon{L_K(E,C)}$ of isomorphism classes of finitely generated projective modules over one of these algebras. The lattice of trace ideals of $L_K(E,C)$ is determined by graph-theoretic data, namely as a lattice of certain pairs consisting of a subset of $E^0$ and a subset of $C$. Necessary conditions for $\mon{L_K(E,C)}$ to be a refinement monoid are developed, together with a construction that embeds $(E,C)$ in a separated graph $(E_+,C^+)$ such that $\mon{L_K(E_+,C^+)}$ has refinement.

Sensitivity and chaos of semigroup actions

We prove that if an action of a C-semigroup S on a Polish space is syndetic transitive, then the system is either minimal and equicontinuous, or sensitive. Additionally, we show that if an action of an abelian monoid S on a Polish space has a transitive point x and a periodic orbit O such that \(\overline{Hx}\) is perfect where H={s∈S:s|O is an identity map}, then the system is chaotic.

GRADED PRIMAL SUBMODULES OF GRADED MODULES

Let G be an abelian monoid with identity e. Let R be a G-graded commutative ring, and M a graded R-module. In this paper we first introduce the concept of graded primal submodules of M an give some basic results concerning this class of submodules. Then we characterize the graded primal ideals of the idealization R(+)M.

Monoids in the fundamental groups of the complement of logarithmic free divisors in [formula omitted

We study monoids generated by certain Zariski–van Kampen generators in the 17 fundamental groups of the complement of logarithmic free divisors in C 3 listed by Sekiguchi. These monoids admit positive homogeneous presentations (Theorem 1). Five of them are Artin monoids and eight of them are free abelian monoids. The remaining four monoids are not Gaußian and, hence, are neither Garside nor Artin (Theorem 2). However, we introduce the concept of fundamental elements for positive homogeneously presented monoids, and we show that all 17 monoids possess fundamental elements (Theorem 3). As an application of the study of monoids, we solve some decision problems for the fundamental groups except in three cases.

A NEW EXAMPLE FOR MINIMALITY OF MONOIDS

By considering the split extension of a free abelian monoid having finite rank by a finite monogenic monoid, the main purposes of this paper are to present examples of efficient monoids and, also, minimal but inefficient monoids. Although results presented in this paper seem as in the branch of pure mathematics, they are actually related to applications of Combinatorial and Geometric Group-Semigroup Theory, especially computer science, network systems, cryptography and physics etc., which will not be handled here.

Special ideals in partial abelian monoids

Under some conditions, the special congruences of partial abelian monoid are those induced by the special ideals, and a class of special ideals of partial abelian monoid has some upper and lower bound properties.

The regular algebra of a poset

LetKKbe a fixed field. We attach to each finite posetP\mathbb Pa von Neumann regularKK-algebraQK(P)Q_K(\mathbb P)in a functorial way. We show that the monoid of isomorphism classes of finitely generated projectiveQK(P)Q_K(\mathbb P)-modules is the abelian monoid generated byP\mathbb Pwith the only relations given byp=p+qp=p+qwheneverq>pq>pinP\mathbb P. This extends the class of monoids for which there is a positive solution to the realization problem for von Neumann regular rings.

Atoms of the relative block monoid

Let G be a finite abelian group with subgroup H and let F(G) denote the free abelian monoid with basis G. The classical block monoid B(G) is the collection of sequences in F(G) whose elements sum to zero. The relative block monoid BH(G), defined by Halter-Koch, is the collection of all sequences in F(G) whose elements sum to an element in H. We use a natural transfer homomorphism θ : BH(G) → B(G/H) to enumerate the irreducible elements of BH(G) given an enumeration of the irreducible elements of B(G/H). MSC 2000: 11P70, 20M14

Adding inverses to diagrams encoding algebraic structures

We modify a previous result, which showed that certain diagrams of spaces are essentially simplicial monoids, to construct diagrams of spaces which model simplicial groups. Furthermore, we show that these diagrams can be generalized to models for Segal groupoids. We then modify Segal’s model for simplicial abelian monoids in such a way that it becomes a model for simplicial abelian groups. Much research has been done on various structures equivalent to topological or simplicial groups. Classical examples include Thomason’s work with delooping machines [27] and Stashefi’s group-like A1-spaces [26], while more modern examples make use of the structure of algebraic theories [1], [2]. In [7], we focus instead on simplicial monoids. In particular, we consider diagrams of spaces which essentially give one of the spaces the structure of a monoid. While such a structure is given when the diagram is the theory of monoids, we obtain a simpler diagram which, from the viewpoint of homotopy theory, encodes the same structure. Here we return to the original situation and ask if this construction can be modifled to obtain a diagram encoding the structure of a simplicial group rather than a simplicial monoid. After all, we need only to flnd a way to include inverses. In this paper, we show that indeed we can represent simplicial groups in this manner. Furthermore, as the construction for simplicial monoids generalizes to the manyobject case of simplicial categories, we can obtain models for simplicial groupoids as well. In doing so, we extend a result relating simplicial categories to Segal categories (essentially categories with composition only given up to higher homotopy) to one relating simplicial groupoids to Segal groupoids. One might ask, then, if there are other algebraic structures which can be modelled in a similar way, in particular by some kind of diagram which is simpler than the one given by the corresponding algebraic theory. An answer for the case of simplicial abelian monoids is given by Segal’s category i [24], although it does not provide the

Gödel incompleteness in AF C*-algebras

For any (possibly, non-unital) AF C*-algebra A with comparability of projections, let D(A) be the Elliott partial monoid of A, and G(A) the dimension group of A with scale D(A). For D ⊆ D(A) a generating set of G(A) let 𝒫 be the set of all formal inequalities a1 + ⋯ + ak ≤ b1 + ⋯ + bl satisfied by G(A), for any ai, bj ∈ D. By Elliott's classification, 𝒫 together with the list of all sums a1 + ⋯ + ak ∈ D(A) uniquely determines A. Can 𝒫 be Gödel incomplete, i.e., effectively enumerable but undecidable? We give a negative answer in case D is finite, and a positive answer in the infinite case. We also show that the range of the map A ↦ D(A) precisely consists of all countable partial abelian monoids satisfying the following three conditions: (i) a + b = a + c ⇒ b = c, (ii) a + b = 0 ⇒ a = b = 0 and (iii) ∀a, b ∈ E ∃c ∈ E such that either a + c = b or b + c = a.

Minimal But Inefficient Presentations for Self Semidirect Products of the Free Abelian Monoid on Two Generators

Let A and K both be copies of the free Abelian monoid on two generators. For any connecting monoid homomorphism θ: A → End(K), let M = K ⋊θ A be the corresponding monoid semidirect product. In Çevik (2005), it has been proved necessary and sufficient conditions for the efficiency of a standard presentation for M in terms of the matrix representation for θ. As a main result of this article, we give sufficient conditions for this presentation of M to be minimal but not efficient. To do that we will use the same method as given in Çevik (2003b).

The support of a uniform semigroup valued measure

In this paper we introduce the notion of orthogonality for arbitrary families of elements of a partial abelian monoid, we study uniform semigroup valued measures on a partial abelian monoid, and we establish interesting results about the existence of the support of such measures.