Ideal Extension Research Articles

Semilattices of stratified extensions

Grillet (Semigroup Forum 50:25–36, 1995) introduced the concept of stratified semigroups as a kind of generalisation of finite nilsemigroups. We extend Grillet’s ideas by introducing the notion of the base of a semigroup and show that a semigroup is stratified if and only if its base is either empty or consists of only the zero element. The general structure of semigroups with non-trivial bases is studied and we show that these can be described in terms of ideal extensions of semigroups by stratified semigroups. We consider certain types of group-bound semigroups and also ideal extensions of Clifford semigroups, and show how to describe them as semilattices of ideal extensions by stratified semigroups and provide a number of interesting examples.

Idealization of a Semigroup by Using Bruck-Reilly Monoids

The aim of this paper is to create a new semigroup by defining a special idealization operation on semigroups. Additionally, by considering amalgamation, we will present novel distinguishing results on idealization over semigroups. Initially, we will combine the definitions of idealization and the Bruck-Reilly extension. Consequently, by integrating these two structures, we will provide a significant result on idealization in semigroups that will be crucial for future studies. Finally, we will conclude by discussing the ideal extension for the new semigroup structure, namely idealization.

Classification of Ideals in Banach Spaces

Let an operator T belong to an operator ideal J, then for any operators A and B which can be composed with T asBTA then BTA \(\in\) J. Indeed, J contains the class of finite rank Banach Space operators. Now given L(X; Y ). Then J(X; Y ) \(\subseteq\) L(X; Y ) such that J(X; Y ) = {T : X \(\gets\) Y : T \(\subseteq\) }. Thus an operator ideal is a subclass J of L containing every identity operator acting on a one-dimensional Banach space such that: S + T \(\in\) J(X; Y ) where S; T \(\in\) J(X; Y ). If W;Z;X; Y \(\in\) K;A \(\in\) L(W;X);B \(\in\) L(Y;Z) then BTA \(\in\) J(W;Z) whenever T \(\in\) J(X; These properties compare very well with the algebraic notion of ideals in Banach Algebras within whose classes lie compact operators, weakly compact operators, finitely strictly regular operators, completely continuous operators, strictly singular operators among others. Thus, the aim of this paper is to characterize the various classes of ideals in Banach spaces. Special attention is given to the characteristics involving the ideal properties, the metric approximation properties, the hereditary properties in relation to the ideal extensions in the Hahn-Banach space, projection and embedments in the biduals of the Banach Space.

The semitrivial ideal extension

In this paper, we aim to generalize the ideal duplication defined for numerical semigroups to commutative rings with unity. We introduce the semitrivial ideal extension, a construction that, starting with an ideal of a commutative ring [Formula: see text] with unity and a submodule of a module [Formula: see text] over [Formula: see text], under specific assumptions, produces an ideal of the semitrivial extension [Formula: see text]. Using this tool we characterize the homogeneous prime ideals of a semitrivial extension and we completely describe the family of the maximal ideals. Furthermore, as it was done for the numerical duplication, using the semitrivial ideal extension, we characterize the modules [Formula: see text] such that [Formula: see text] is nearly Gorenstein.

Effects of marketing information on coffee farmers and trader’s performance in Kibinge sub county, Bukomasimbi district of Central Uganda

Marketing channel choice is among the most complex and challenging decisions facing farmers. This study therefore evaluate the impact of marketing information available to coffee farmers and traders in Bukomansimbi district of Uganda. The specific objective was to identify the methods through which coffee farmer’s access market information and to determine the challenges they face in an effort to access market information. A cross-sectional research design was employed to collect data from 48 smallholder coffee farmers using semi-structured questionnaire, participatory group discussions and key informants interviews. Using a purposive and stratified simple random sampling procedure, 48 respondents comprising of coffee farmers and traders were selected and interviewed. The study results revealed that Coffee farmers don't access information on some of the opportunities and programmes that could enable them to improve the production process and hence access better markets. When farmers market their coffee through cooperatives, they gain more because there will be no middlemen involved in the value chain. Through Co-operatives, it is also easy to ensure product quality besides accessing good markets that pay premium prices for the coffee. It was discovered that 35% of the cooperatives in the study district produce speciality Arabica coffee and fine Robusta coffee, which fetch premium prices on the global market. The study also established that farmers use word of mouth from other farmers and traders as source of their information. The policy recommendation is that Coffee farmers should conduct market assessment and research to get update information about the market dynamics and current price changes. The ideal extension model should be farmer centered with necessary linkages to markets, researchers, subject matter specialists, inputs providers, credit facilities, and other support services. An appropriate extension system should focus on farmer groups rather than individuals with an emphasis on group extension methods. Thus a sustainable extension model which is farmer-owned with a strong market focus should be promoted. The Uganda Coffee Development Authority should conduct training programmes to equip farmers with the skills and knowledge on how to ensure quality along the coffee value chain as well as on coffee marketing.

Ascending chain conditions on right ideals of semigroups

ABSTRACT We call a semigroup S right noetherian if it satisfies the ascending chain condition on right ideals, and we say that S satisfies ACCPR if it satisfies the ascending chain condition on principal right ideals. We investigate the behavior of these two conditions with respect to ideals and ideal extensions, with a particular focus on minimal and 0-minimal one-sided ideals. In particular, we show that the property of satisfying ACCPR is inherited by right and left ideals. On the other hand, we exhibit an example of a right noetherian semigroup with a minimal ideal that is not right noetherian.

On the Numerical Solution of 1D and 2D KdV Equations Using Variational Homotopy Perturbation and Finite Difference Methods

The KdV equation has special significance as it describes various physical phenomena. In this paper, we use two methods, namely, a variational homotopy perturbation method and a classical finite-difference method, to solve 1D and 2D KdV equations with homogeneous and non-homogeneous source terms by considering five numerical experiments with initial and boundary conditions. The variational homotopy perturbation method is a semi-analytic technique for handling linear as well as non-linear problems. We derive classical finite difference methods to solve the five numerical experiments. We compare the performance of the two classes of methods for these numerical experiments by computing absolute and relative errors at some spatial nodes for short, medium and long time propagation. The logarithm of maximum error vs. time from the numerical methods is also obtained for the experiments undertaken. The stability and consistency of the finite difference scheme is obtained. To the best of our knowledge, a comparison between the variational homotopy perturbation method and the classical finite difference method to solve these five numerical experiments has not been undertaken before. The ideal extension of this work would be an application of the employed methods for fractional and stochastic KdV type equations and their variants.

Corrigendum to “Annihilators and extensions of idempotent-generated ideals”

In this note, we give a counterexample and modification of Lemma 4.4 (ii) in [1]. Moreover, the validity of Theorem 4.6 in [1] is confirmed without using Lemma 4.4 in its proof.

The complexity of a numerical semigroup

Let S and Δ be numerical semigroups. A numerical semigroup S is an I(Δ)-semigroup if S \\ {0} is an ideal of Δ. We denote by S is an I(Δ)-semigroup}, and we say that Δ is an ideal extension of S if S . In this work, we present an algorithm to build all the ideal extensions of a numerical semigroup. We recursively denote by and for all k ∈ N. The complexity of a numerical semigroup S is the minimum of the set . In addition, we introduce an algorithm to compute all the numerical semigroups with fixed multiplicity and complexity.

Ideal extensions and directly infinite algebras

Directly infinite algebras , those algebras, E which have a pair of elements x and y where 1 = x y ≠ y x , are well known to have a sub-algebra isomorphic to M ∞ ( K ) , the set of infinite Z + × Z + -indexed matrices which have only finitely many nonzero entries . When this sub-algebra is actually an ideal, we may analyze the algebra in terms of an extension of some algebra A by M ∞ ( K ) , that is, a short exact sequence of K -algebras 0 → M ∞ ( K ) → E → A → 0 . The present article characterizes all trivial (split) extensions of K [ x , x − 1 ] by M ∞ ( K ) by examining the extensions as sub-algebras of infinite matrix algebras . Furthermore, we construct an infinite family of pairwise non-isomorphic extensions { T i : i ≥ 0 } , all of which can be written as an extension 0 → M ∞ ( K ) → T i → K [ x , x − 1 ] → 0 .

Ideal ring extensions and trusses

It is shown that there is a close relationship between ideal extensions of rings and trusses, that is, sets with a semigroup operation distributing over a ternary abelian heap operation. Specifically, a truss can be associated to every element of an extension ring that projects down to an idempotent in the extending ring; every weak equivalence of extensions yields an isomorphism of corresponding trusses. Furthermore, equivalence classes of ideal extensions of rings by integers are in one-to-one correspondence with associated trusses up to isomorphism given by a translation. Conversely, to any truss T and an element of this truss one can associate a ring and its extension by integers in which T is embedded as a truss. Consequently any truss can be understood as arising from an ideal extension by integers. The key role is played by interpretation of ideal extensions by integers as extensions defined by double homothetisms of Redei (L. Redei (1952) [24]) or by self-permutable bimultiplications of Mac Lane (S. Mac Lane (1958) [18]), that is, as integral homothetic extensions. The correspondence between homothetic ring extensions and trusses is used to classify fully up to isomorphism trusses arising from rings with zero multiplication and rings with trivial annihilators.

Hyperextension following two different designs of fixed-bearing posterior-stabilized total knee arthroplasty.

Hyperextension after total knee arthroplasty (TKA) is related to worse functional outcomes. Among the reported predictive factors, it is unclear whether prosthesis design affects postoperative hyperextension. Hence, our objectives were to (1) compare time-dependent changes of the extension angle following multi-radius (MR) and single-radius (SR) TKA and (2) assess predictive factors for the hyperextension. In this study, consecutive 136 MR TKAs and consecutive 71 SR TKAs performed by a single surgeon were reviewed. The extension angle was evaluated by digitized full-extension lateral radiographs at preoperative, 3months and at 1 and 2years after TKA. A positive value was defined as hyperextension. A multiple regression analysis was used to determine predictive factors for the extension angle and the risk of hyperextension > 5°. The postoperative extension angle of both groups gradually increased during the first 2years and the SR group finally progressed with 4.04° greater extension than the MR group (p < 0.01). The incidence of hyperextension > 5° in the SR group was 18.3% and for the MR group, 7.4% (p = 0.02). However, the Oxford knee scores was nonsignificant different between the groups (p = 0.15). Preoperative extension angle, sagittal femoral component angle (SFA) and SR prosthesis were predictive factors for the extension angle at 2years after TKA and were also associated with hyperextension > 5°. The extension angle of the SR group became significantly more extended than that of the MR group during the first two years after TKA. Hence, ideal extension position immediately after SR TKA may be different from that of the MR TKA prosthesis. The preoperative extension angle, SFA and SR prosthesis were factors that were associated with postoperative hyperextension.

Ideal extensions of Olivier's theorem

Let $p, q$ be given positive numbers and $a, b$ non-negative ones. In this paper we study and characterize the class $\mathcal S(a,b,p,q)$ of all admissible ideals $\mathcal I \subset 2^\mathbb N$ with the following property $$\sum_{n \in \mathbb N} n^a a^p_n \lt \infty\quad \Rightarrow \quad \mathcal I − \lim n^ba^q_n = 0,$$ for all sequences $(a_n)$ of positive real numbers. In a series of corollaries we discuss special cases including, also, several previously published theorems on this topic.

Characterizations of maximal z°-ideals of C(X) and real maximal ideals of q(X)

In this paper we first give elementwise characterizations of real maximal ideals and maximal z°-ideals of C(X). Next, using this we characterize the real maximal ideals of the classical ring of quotients q(X) of C(X) and also maximal z°-ideals of factor rings of C(X). We show that every real maximal ideal of q(X) is precisely the extension of a real maximal ideal of C(X) which is also a z° -ideal, i.e., the extension of Mp for some almost P -point p of υX. Using this fact it turns out that in contrast to C(X), q(X) may not contain any real maximal ideal. We observe that every maximal ideal of q(X) is real if and only if X is a pseudocompact almost P -space, i.e., q(X) = C*(X). We also observe that the real maximal ideals of q(X) are precisely the extensions of the real maximal ideals of C(X) when and only when X is an almost P -space, i.e., q(X) = C(X).

Extensions of fuzzy soft sets and rings

In this note, first we introduce the concept of extension of fuzzy soft subset (set) by utilizing its extension of mapping and re-define some operations of fuzzy soft sets based on the extension. Furthermore, we also discuss the extension of fuzzy soft set with respect to t-level soft sets of fuzzy soft sets. We explore the connection between both types of the extensions in example4.1. As an evidence we present the extension of fuzzy soft subrings and fuzzy soft ideals while considering the extension through t-level soft sets.

An approach to the study of fuzzy rough prime ideal extensions of semigroups

Fuzzy rough set is a pair of lower and upper approximations of fuzzy set in a fuzzy approximation space. This paper studies the extension of fuzzy rough prime and semi-prime ideals in semigroups. Then we introduce the composition theory on fuzzy rough set. Finally some new characterization of semigroups by the properties of composition on fuzzy rough ideal extension are given.

Gamma ideal extension in gamma systems

Let γ be a nonempty set, be a commutative γ-semigroup contains γ-identity and M a unitary right -system (act). The notion of a γ-ideal extension on gamma systems are given and studied. An -system M is said to be a γ-ideal extension, if for each -homomorphism f : L → M such that , for each γ-ideal L of . A lot of characterizations and properties of gamma ideal extension of gamma systems have been given. In truth, we offer that γ-ideal extension of -system M is equal to the following similar terms 1. . 2. For all and implies that . 3. For all and . 4. For all , then , for some γ-compatible ρ on .

Ternary Γ-semihyperrings: Ideals and ideals extensions

The aim of this research work is to define a new class of multi-algebras that is called ternary [Formula: see text]-semihyperrings as a generalization of [Formula: see text]-semihyperrings and semihyperrings. Then, we define and study the concepts of ideals, mutual ideals, spectrum ideals, strong prime ideals and ultra-prime ideals of them. Also, we define a topology on the set of strong prime ideals. Finally, we introduce the concept of ideal extensions and obtain some related results.

ON $(m,n)$-CLOSED IDEALS IN AMALGAMATED ALGEBRA

Let $R$ be a commutative ring with $1 \ne 0$ and let $m$ and&#x0D; $n$ be integers with $1\leq n &lt; m$. A proper ideal $I$ of $R$ is&#x0D; called an $(m, n)$-closed ideal of $R$ if whenever $a^m \in I$ for&#x0D; some $a\in R$ implies $a^n \in I$. Let $ f:A\rightarrow B$ be a&#x0D; ring homomorphism and let $J$ be an ideal of $B.$ This paper&#x0D; investigates the concept of $(m,n)$-closed ideals in the&#x0D; amalgamation of $A$ with $B$ along $J$ with respect $f$ denoted by&#x0D; $A\bowtie^{f}J$. Namely, Section 2 investigates this notion to&#x0D; some extensions of ideals of $A$ to $A\bowtie^fJ$. Section 3&#x0D; features the main result, which examines when each proper ideal of&#x0D; $A\bowtie^fJ$ is an $(m,n)$-closed ideal. This allows us to give&#x0D; necessary and sufficient conditions for the amalgamation to&#x0D; inherit the radical ideal property with applications on the&#x0D; transfer of von Neumann regular, $\pi$-regular and semisimple&#x0D; properties.

Cartesian product of the extensions of fuzzy soft ideals over near-rings

Cartesian product of the extensions of fuzzy soft ideals over near-rings