scholarly journals The Decomposition Theorems of AG-Neutrosophic Extended Triplet Loops and Strong AG-(l, l)-Loops

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 268 ◽  
Author(s):  
Xiaoying Wu ◽  
Xiaohong Zhang
Keyword(s):  
Disjoint Union ◽  
Decomposition Theorem ◽  
Left Identity ◽  
Maximal Subgroups ◽  
Left Inverse ◽  
Decomposition Theorems

In this paper, some new properties of Abel Grassmann‘s Neutrosophic Extended Triplet Loop (AG-NET-Loop) were further studied. The following important results were proved: (1) an AG-NET-Loop is weakly commutative if, and only if, it is a commutative neutrosophic extended triplet (NETG); (2) every AG-NET-Loop is the disjoint union of its maximal subgroups. At the same time, the new notion of Abel Grassmann’s (l, l)-Loop (AG-(l, l)-Loop), which is the Abel-Grassmann’s groupoid with the local left identity and local left inverse, were introduced. The strong AG-(l, l)-Loops were systematically analyzed, and the following decomposition theorem was proved: every strong AG-(l, l)-Loop is the disjoint union of its maximal sub-AG-groups.

The Arithmetic of the Quasi-Uniserial Semigroups without Zero

1971 ◽  
Vol 23 (3) ◽  
pp. 507-516 ◽  
Author(s):  
Ernst August Behrens
Keyword(s):  
Simple Semigroup ◽  
Disjoint Union ◽  
Maximal Subgroups ◽  
Ordered Semigroup ◽  
Partially Ordered ◽  
Completely Simple Semigroup ◽  
Image Position ◽  
Previous Paragraph ◽  
Completely Simple ◽  
Identity Elements

An element a in a partially ordered semigroup T is called integral ifis valid. The integral elements form a subsemigroup S of T if they exist. Two different integral idempotents e and f in T generate different one-sided ideals, because eT = fT, say, implies e = fe ⊆ f and f = ef ⊆ e.Let M be a completely simple semigroup. M is the disjoint union of its maximal subgroups [4]. Their identity elements generate the minimal one-sided ideals in M. The previous paragraph suggests the introduction of the following hypothesis on M.Hypothesis 1. Every minimal one-sided ideal in M is generated by an integral idempotent.

A Lebesgue Decomposition Theorem for C* Algebras

1972 ◽  
Vol 15 (1) ◽  
pp. 87-91 ◽  
Author(s):  
Michael Henle
Keyword(s):  
Absolute Continuity ◽  
Linear Functional ◽  
Decomposition Theorem ◽  
Lebesgue Decomposition ◽  
C Algebras ◽  
Positive Linear Functional ◽  
Decomposition Theorems

This paper, by generalizing von Neumann's proof of the Radon-Nikodym and Lebesgue decomposition theorems [3], obtains analogous results for positive linear functional on a C* algebra. The concept of "absolute continuity" used and the Radon-Nikodym portion of the resulting theorem are due to Dye [2].

scholarly journals Decompositions of vector measures in Riesz spaces and Banach lattices

1986 ◽  
Vol 29 (1) ◽  
pp. 23-39 ◽  
Author(s):  
Klaus D. Schmidt
Keyword(s):  
Bounded Variation ◽  
Banach Lattice ◽  
Decomposition Theorem ◽  
Riesz Space ◽  
Jordan Decomposition ◽  
Countable Additivity ◽  
Vector Measures ◽  
Central Result ◽  
Decomposition Theorems ◽  
Natural Way

The present paper is mainly concerned with decomposition theorems of the Jordan, Yosida-Hewitt, and Lebesgue type for vector measures of bounded variation in a Banach lattice having property (P). The central result is the Jordan decomposition theorem due to which these vector measures may alternately be regarded as order bounded vector measures in an order complete Riesz space or as vector measures of bounded variation in a Banach space. For both classes of vector measures, properties like countable additivity, purely finite additivity, absolute continuity, and singularity can be defined in a natural way and lead to decomposition theorems of the Yosida-Hewitt and Lebesgue type. In the Banach lattice case, these lattice theoretical and topological decomposition theorems can be compared and combined.

Decomposition of totally transcendental modules

1980 ◽  
Vol 45 (1) ◽  
pp. 155-164 ◽  
Author(s):  
Steven Garavaglia
Keyword(s):  
Quantifier Elimination ◽  
Decomposition Theorem ◽  
Noetherian Rings ◽  
Projective Modules ◽  
Artinian Modules ◽  
Injective Modules ◽  
Unique Decomposition ◽  
Special Cases ◽  
Decomposition Theorems ◽  
Coherent Rings

The main theorem of this paper states that if R is a ring and is a totally transcendental R-module, then has a unique decomposition as a direct sum of indecomposable R-modules. Natural examples of totally transcendental modules are injective modules over noetherian rings, artinian modules over commutative rings, projective modules over left-perfect, right-coherent rings, and arbitrary modules over Σ – α-gens rings. Therefore, our decomposition theorem yields as special cases the purely algebraic unique decomposition theorems for these four classes of modules due to Matlis; Warfield; Mueller, Eklof, and Sabbagh; and Shelah and Fisher. These results and a number of other corollaries about totally transcendental modules are covered in §1. In §2, I show how the results of § 1 can be used to give an improvement of Baur's classification of ω-categorical modules over countable rings. In §3, the decomposition theorem is used to study modules with quantifier elimination over noetherian rings.The goals of this section are to prove the decomposition theorem and to derive some of its immediate corollaries. I will begin with some notational conventions. R will denote a ring with an identity element. LR is the language of left R-modules described in [4, p. 251] and TR is the theory of left R-modules. “R-module” will mean “unital left R-module”. A formula will mean an LR-formula.

scholarly journals Purity and decomposition theorems for staggered sheaves

2012 ◽  
Vol 11 (4) ◽  
pp. 695-745
Author(s):  
Pramod N. Achar ◽  
David Treumann
Keyword(s):  
Decomposition Theorem ◽  
Derived Category ◽  
Perverse Sheaves ◽  
Perverse Sheaf ◽  
Coherent Sheaves ◽  
Direct Sum ◽  
Constructible Sheaves ◽  
Decomposition Theorems ◽  
Pure Complex

AbstractTwo major results in the theory of ℓ-adic mixed constructible sheaves are the purity theorem (every simple perverse sheaf is pure) and the decomposition theorem (every pure object in the derived category is a direct sum of shifts of simple perverse sheaves). In this paper, we prove analogues of these results for coherent sheaves. Specifically, we work with staggered sheaves, which form the heart of a certain t-structure on the derived category of equivariant coherent sheaves. We prove, under some reasonable hypotheses, that every simple staggered sheaf is pure, and that every pure complex of coherent sheaves is a direct sum of shifts of simple staggered sheaves.

scholarly journals A Unified Approach to the Decomposition Theorems in Baer $$*$$-Rings

2021 ◽  
Vol 76 (3) ◽  
Author(s):  
Zbigniew Burdak ◽  
Marek Kosiek ◽  
Patryk Pagacz ◽  
Marek Słociński
Keyword(s):  
Hilbert Space ◽  
Linear Algebra ◽  
Decomposition Theorem ◽  
Unified Approach ◽  
Hereditary Property ◽  
Hilbert Space Operators ◽  
Baer Rings ◽  
Decomposition Theorems

AbstractThe aim of the paper is to generalize decomposition theorems showed in Bagheri-Bardi et al. (Linear Algebra Appl 583:102–118, 2019; Linear Algebra Appl 539:117–133, 2018) by a unified approach. We show a general decomposition theorem with respect to a hereditary property. Then the vast majority of decompositions known in the algebra of Hilbert space operators is generalized to elements of Baer $$*$$ ∗ -rings by this theorem. The theorem yields also results which are new in the algebra of bounded Hilbert space operators. Additionally, the model of summands in Wold–Słociński decomposition is given in Baer $$*$$ ∗ -rings.

scholarly journals Quasi-dual-continuous modules

1985 ◽  
Vol 39 (3) ◽  
pp. 287-299 ◽  
Author(s):  
Saad Mohamed ◽  
Bruno J. Müller ◽  
Surjeet Singh
Keyword(s):  
Short Proof ◽  
Decomposition Theorem ◽  
Structure Theorems ◽  
Decomposition Theorems

AbstractQuasi-dual-continuous modules, which generalize the concept of dual-continuous modules, are studied Mohamed, Müller and Singh had obtained some decomposition theorems and their partial converses for dual-continuous modules. It is shown that these results can be extended to quasi-dual-continuous modules. Further, a short proof of a decomposition theorem for quasi-dual-continuous modules established recently by Oshiro is given. Some more structure theorems for such modules are established. Finally, quasi-dual-continuous covers are studied, and duals for results of Müller and Rizvi are derived.

scholarly journals A decomposition theorem for submeasures

1985 ◽  
Vol 26 (1) ◽  
pp. 69-74
Author(s):  
A. R. Khan ◽  
K. Rowlands
Keyword(s):  
Decomposition Theorem ◽  
Lebesgue Decomposition ◽  
Decomposition Theorems

In recent years versions of the Lebesgue and the Hewitt-Yosida decomposition theorems have been proved for group-valued measures. For example, Traynor [4], [6] has established Lebesgue decomposition theorems for exhaustive groupvalued measures on a ring using (1) algebraic and (2) topological notions of continuity and singularity, and generalizations of the Hewitt-Yosida theorem have been given by Drewnowski [2], Traynor [5] and Khurana [3]. In this paper we consider group-valued submeasures and in particular we have established a decomposition theorem from which analogues of the Lebesgue and Hewitt-Yosida decomposition theorems for submeasures may be derived. Our methods are based on those used by Drewnowski in [2] and the main theorem established generalizes Theorem 4.1 of [2].

Decomposition Theorem of Generalized Interval-Valued Intuitionistic Fuzzy Sets

2014 ◽  
pp. 174-180
Author(s):  
Amal Kumar Adak ◽  
Monoranjan Bhowmik ◽  
Madhumangal Pal
Keyword(s):  
Fuzzy Sets ◽  
Decomposition Theorem ◽  
Intuitionistic Fuzzy Sets ◽  
Fundamental Theory ◽  
Intuitionistic Fuzzy ◽  
Decomposition Theorems ◽  
Interval Valued

In this chapter, the authors establish decomposition theorems of Generalized Interval-Valued Intuitionistic Fuzzy Sets (GIVIFS) by use of cut sets of generalized interval-valued intuitionistic fuzzy sets. First, new definitions of eight kinds of cut sets generalized interval-valued intuitionistic fuzzy sets are introduced. Second, based on these new cut sets, the decomposition generalized interval-valued intuitionistic fuzzy sets are established. The authors show that each kind of cut sets corresponds to two kinds of decomposition theorems. These results provide a fundamental theory for the research of generalized interval-valued intuitionistic fuzzy sets.

scholarly journals Resolvent Decomposition Theorems and Their Application in Denumerable Markov Processes with Instantaneous States

2019 ◽  
Vol 33 (4) ◽  
pp. 2089-2118
Author(s):  
Anyue Chen
Keyword(s):  
Markov Processes ◽  
Existence And Uniqueness ◽  
Decomposition Theorem ◽  
Subsequent Paper ◽  
Probabilistic Interpretation ◽  
Analytic Proof ◽  
Present Part ◽  
Decomposition Theorems ◽  
Uniqueness Criteria

Abstract The basic aim of this paper is to provide a fundamental tool, the resolvent decomposition theorem, in the construction theory of denumerable Markov processes. We present a detailed analytic proof of this extremely useful tool and explain its clear probabilistic interpretation. We then apply this tool to investigate the basic problems of existence and uniqueness criteria for denumerable Markov processes with instantaneous states to which few results have been obtained even until now. Although the complete answers regarding these existence and uniqueness criteria will be given in a subsequent paper, we shall, in this paper, present part solutions of these very important problems that are closely linked with the subtle Williams S and N conditions.

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