On some orders in ∗-rings based on the core-EP decomposition

2020 ◽  
pp. 2250010
Author(s):  
Janko Marovt ◽  
Dijana Mosić
Keyword(s):  
Partial Order ◽  
The Core ◽  
Unital Ring ◽  
Rings With Involution ◽  
Invertible Elements ◽  
Unital Rings ◽  
Minus Partial Order

We study certain relations in unital rings with involution that are derived from the core-EP decomposition. The notion of the WG pre-order and the C-E partial order is extended from [Formula: see text], the set of all [Formula: see text] matrices over [Formula: see text], to the set [Formula: see text] of all core-EP invertible elements in an arbitrary unital ring [Formula: see text] with involution. A new partial order is introduced on [Formula: see text] by combining the WG pre-order and the well known minus partial order, and a new characterization of the core-EP pre-order in unital proper ∗-rings is presented. Properties of these relations are investigated and some known results are thus generalized.

scholarly journals Core partial order in rings with involution

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5695-5701 ◽  
Author(s):  
Xiaoxiang Zhang ◽  
Sanzhang Xu ◽  
Jianlong Chen
Keyword(s):  
Partial Order ◽  
Partial Orders ◽  
Reverse Order ◽  
Unital Ring ◽  
Reverse Order Law ◽  
Rings With Involution ◽  
Invertible Elements

Let R be a unital ring with involution. Several characterizations and properties of core partial order in R are given. In particular, we investigate the reverse order law (ab)# = b#a# for two core invertible elements a, b ? R. Some relationships between core partial order and other partial orders are obtained.

WEIGHTED CORE–EP INVERSE AND WEIGHTED CORE–EP PRE-ORDERS IN A -ALGEBRA

2019 ◽  
pp. 1-35 ◽  
Author(s):  
DIJANA MOSIĆ
Keyword(s):  
Partial Order ◽  
Hilbert Spaces ◽  
Partial Orders ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
The Core ◽  
Invertible Elements ◽  
Minus Partial Order

We define extensions of the weighted core–EP inverse and weighted core–EP pre-orders of bounded linear operators on Hilbert spaces to elements of a $C^{\ast }$ -algebra. Some properties of the weighted core–EP inverse and weighted core–EP pre-orders are generalized and some new ones are proved. Using the weighted element, the weighted core–EP pre-order, the minus partial order and the star partial order of certain elements, new weighted pre-orders are presented on the set of all $wg$ -Drazin invertible elements of a $C^{\ast }$ -algebra. Applying these results, we introduce and characterize new partial orders which extend the core–EP pre-order to a partial order.

A CHARACTERIZATION OF THE COMMUTATIVE UNITAL RINGS WITH ONLY FINITELY MANY UNITAL SUBRINGS

2008 ◽  
Vol 07 (05) ◽  
pp. 601-622 ◽  
Author(s):  
DAVID E. DOBBS ◽  
GABRIEL PICAVET ◽  
MARTINE PICAVET-L'HERMITTE
Keyword(s):  
Direct Products ◽  
Unital Ring ◽  
Unital Rings

A (commutative unital) ring is said to have FSP if it has only finitely many unital subrings. The singly generated rings that have FSP have been classified. Thus, a characterization of the rings satisfying FSP is obtained by proving that a ring R has FSP if and only if either R is finite or R = ℤ[t1, …, tn] ⊇ ℤ where ℤ[ti] has FSP for each i = 1, …, n. Also, the following characterization is given for the nontrivial ring direct products Πi ∈ I Ri that have FSP: I is finite, each Ri has FSP, and there is at most one i ∈ I such that Ri has characteristic 0.

Projections generated by Moore–Penrose inverses and core inverses

2020 ◽  
pp. 2150027
Author(s):  
Huihui Zhu ◽  
Fei Peng
Keyword(s):  
Multilinear Algebra ◽  
The Core ◽  
Unital Ring ◽  
Core Inverse ◽  
Linear Multilinear Algebra ◽  
General Ring

Let [Formula: see text] be a unital ∗-ring. As is well known, idempotents and projections can be constructed by the Moore–Penrose inverse and the core inverse of an element in [Formula: see text]. In this paper, we mainly investigate characterizations and properties of these types of idempotents and projections. Also, several results in [Hartwig and Spindelböck, Matrices for which [Formula: see text] and [Formula: see text] commute, Linear Multilinear Algebra 14 (1984) 241–256] are extended to a general ∗-ring without ∗-cancellable conditions. As applications, the characterization of EP elements is given.

scholarly journals Weak group inverse

2018 ◽  
Vol 16 (1) ◽  
pp. 1218-1232 ◽  
Author(s):  
Hongxing Wang ◽  
Jianlong Chen
Keyword(s):  
Partial Order ◽  
The Other ◽  
Group Inverse ◽  
Complex Matrices ◽  
The Core ◽  
Weak Group ◽  
Square Complex

AbstractIn this paper, we introduce the weak group inverse (called as the WG inverse in the present paper) for square complex matrices of an arbitrary index, and give some of its characterizations and properties. Furthermore, we introduce two orders: one is a pre-order and the other is a partial order, and derive several characterizations of the two orders. The paper ends with a characterization of the core EP order using WG inverses.

scholarly journals On partial orders in proper $*$-rings

2017 ◽  
pp. 193-204
Author(s):  
Janko Marovt
Keyword(s):  
Partial Order ◽  
Partial Orders ◽  
Complex Matrices ◽  
The Core ◽  
Invertible Elements ◽  
Core Part

We study orders in proper $*$-rings that are derived from the core-nilpotent decomposition. The notion of the C-N-star partial order and the S-star partial order is extended from $M_ {n} ( \mathbb{C)}$, the set of all $n \times n$ complex matrices, to the set of all Drazin invertible elements in proper $*$-rings with identity. Properties of these orders are investigated and their characterizations are presented. For a proper $*$-ring $\mathcal{A}$ with identity, it is shown that on the set of all Drazin invertible elements $a \in \mathcal{A}$ where the core part of $a$ is an EP element, the C-N-star partial order implies the star partial order.

scholarly journals Isolation and Molecular Characterization of the Romaine Lettuce Phylloplane Mycobiome

2021 ◽  
Vol 7 (4) ◽  
pp. 277
Author(s):  
Danny Haelewaters ◽  
Hector Urbina ◽  
Samuel Brown ◽  
Shannon Newerth-Henson ◽  
M. Catherine Aime
Keyword(s):  
Lactuca Sativa ◽  
New Combinations ◽  
Food Spoilage ◽  
Human Pathogens ◽  
Plant Leaves ◽  
Romaine Lettuce ◽  
Future Studies ◽  
The Core ◽  
American Agriculture

Romaine lettuce (Lactuca sativa) is an important staple of American agriculture. Unlike many vegetables, romaine lettuce is typically consumed raw. Phylloplane microbes occur naturally on plant leaves; consumption of uncooked leaves includes consumption of phylloplane microbes. Despite this fact, the microbes that naturally occur on produce such as romaine lettuce are for the most part uncharacterized. In this study, we conducted culture-based studies of the fungal romaine lettuce phylloplane community from organic and conventionally grown samples. In addition to an enumeration of all such microbes, we define and provide a discussion of the genera that form the “core” romaine lettuce mycobiome, which represent 85.5% of all obtained isolates: Alternaria, Aureobasidium, Cladosporium, Filobasidium, Naganishia, Papiliotrema, Rhodotorula, Sampaiozyma, Sporobolomyces, Symmetrospora and Vishniacozyma. We highlight the need for additional mycological expertise in that 23% of species in these core genera appear to be new to science and resolve some taxonomic issues we encountered during our work with new combinations for Aureobasidiumbupleuri and Curvibasidium nothofagi. Finally, our work lays the ground for future studies that seek to understand the effect these communities may have on preventing or facilitating establishment of exogenous microbes, such as food spoilage microbes and plant or human pathogens.

Characterization of the core in full domain marriage problems

2014 ◽  
Vol 69 ◽  
pp. 34-42 ◽  
Author(s):  
Duygu Nizamogullari ◽  
İpek Özkal-Sanver
Keyword(s):  
The Core ◽  
Marriage Problems ◽  
Full Domain

Inverse complements and strongly unit regular elements

2021 ◽  
pp. 2250190
Author(s):  
Umashankara Kelathaya ◽  
Savitha Varkady ◽  
Manjunatha Prasad Karantha
Keyword(s):  
Partial Order ◽  
Regular Element ◽  
Generalized Inverse ◽  
Generalized Inverses ◽  
Order Unit ◽  
Regular Elements ◽  
Strongly Regular ◽  
Minus Partial Order

In this paper, the notion of “strongly unit regular element”, for which every reflexive generalized inverse is associated with an inverse complement, is introduced. Noting that every strongly unit regular element is unit regular, some characterizations of unit regular elements are obtained in terms of inverse complements and with the help of minus partial order. Unit generalized inverses of given unit regular element are characterized as sum of reflexive generalized inverses and the generators of its annihilators. Surprisingly, it has been observed that the class of strongly regular elements and unit regular elements are the same. Also, several classes of generalized inverses are characterized in terms of inverse complements.

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