Let R be an (m,n)-hyperring. The Γ∗-relation on R in the sense of Mirvakili and Davvaz [?] is the... more Let R be an (m,n)-hyperring. The Γ∗-relation on R in the sense of Mirvakili and Davvaz [?] is the smallest strong compatible relation such that the quotient R∕Γ∗ is an (m,n)-ring. We use Γ∗-relation to define a fundamental functor, F from the category of (m,n)-hyperrings to the category of (m,n)-rings. Also, the concept of a fundamental (m,n)-ring is introduced and it is shown that every (m,n)-ring is isomorphic to R∕Γ∗ for a nontrivial (m,n)-hyperring R. Moreover, the notions of partitionable and quotientable are introduced and their mutual relationship is investigated. A functor G from the category of classical (m,n)-rings to the category of (m,n)-hyperrings is defined and a natural transformation between the functors F and G is given.
The purpose of this paper is the study of direct limits in category of Krasner (m, n)-hyperrings.... more The purpose of this paper is the study of direct limits in category of Krasner (m, n)-hyperrings. In this regards we introduce and study direct limit of a direct system in category (m, n)-hyperrings. Also, we consider fundamental relation , as the smallest equivalence relation on an (m, n)-hyperring R such that the quotient space is an (m, n)-ring, to introduce the fundamental functor from category of Krasner (m, n)-hyperrings to the category of (m, n)-rings. Finally, we study the relationship between fundamental functor and direct limit on Krasner (m, n)-hyperrings. In particular, we prove that the fundamental functor is exact and obtain some its basic properties.
In this paper, we generalize the concepts of ${\\mathcal{N}}$-subnexuses of types $(\\in, q)$, $(... more In this paper, we generalize the concepts of ${\\mathcal{N}}$-subnexuses of types $(\\in, q)$, $(\\in, \\in \\vee{q})$ and $(q, \\in \\vee {q})$, and introduce the notions of ${\\mathcal{N}}$-subnexuses of types $(\\in, q_{k})$, $(\\in, \\in \\vee{q_{k}})$ and $(q, \\in \\vee {q_{k}})$. We investigate their basicproperties, characterize subnexuses by ${\\mathcal{N}}$-subnexuses of type $(\\in, \\in \\vee {q_{k}})$, and give some characterizations for${\\mathcal{N}}$-subnexuses of types $(\\in, q_{k})$ and $(q, \\in \\vee{q_{k}})$. Moreover, we define ${\\mathcal{N}}$-subnexuses of type $(\\overline{\\in}, \\overline{\\in} \\vee \\overline{q_{k}})$ and discusson their different properties.
The notion of a subnexus based on ${\mathcal{N}}$-function (briefly, ${\mathcal{N}}$-subnexus) is... more The notion of a subnexus based on ${\mathcal{N}}$-function (briefly, ${\mathcal{N}}$-subnexus) is introduced, and related properties are investigated. Also, the notions of ${\mathcal{N}}$-subnexus of type $(\alpha, \beta)$, where $(\alpha, \beta)$ is $(\in, \in)$, $(\in, q)$, $(\in, \in\! \vee \, {q})$, $(q, \in)$, $(q,q)$, $(q, \in\! \vee \, {q})$, $(\overline{\in}, \overline{\in})$ and $(\overline{\in}, \overline{\in} \vee \overline{q})$, are introduced, and their basic properties are investigated. Conditions for an ${\mathcal{N}}$-structure to be an ${\mathcal{N}}$-subnexus of type $(q, \in\! \vee \, {q})$ are given, and characterizations of ${\mathcal{N}}$-subnexus of type $(\in, \in\! \vee \, {q})$ and $(\overline{\in}, \overline{\in} \vee \overline{q})$ are provided. Homomorphic image and preimage of ${\mathcal{N}}$-subnexus are discussed.
The notion of a subnexus based on ${\mathcal{N}}$-function (briefly, ${\mathcal{N}}$-subnexus) is... more The notion of a subnexus based on ${\mathcal{N}}$-function (briefly, ${\mathcal{N}}$-subnexus) is introduced, and related properties are investigated. Also, the notions of ${\mathcal{N}}$-subnexus of type $(\alpha, \beta)$, where $(\alpha, \beta)$ is $(\in, \in)$, $(\in, q)$, $(\in, \in\! \vee \, {q})$, $(q, \in)$, $(q,q)$, $(q, \in\! \vee \, {q})$, $(\overline{\in}, \overline{\in})$ and $(\overline{\in}, \overline{\in} \vee \overline{q})$, are introduced, and their basic properties are investigated. Conditions for an ${\mathcal{N}}$-structure to be an ${\mathcal{N}}$-subnexus of type $(q, \in\! \vee \, {q})$ are given, and characterizations of ${\mathcal{N}}$-subnexus of type $(\in, \in\! \vee \, {q})$ and $(\overline{\in}, \overline{\in} \vee \overline{q})$ are provided. Homomorphic image and preimage of ${\mathcal{N}}$-subnexus are discussed.
Background Salinity is one of the major limiting abiotic stresses that decrease crop production w... more Background Salinity is one of the major limiting abiotic stresses that decrease crop production worldwide. To recommend genotypes for cultivation under saline stress conditions, a comprehensive understanding of the genetic basis and plant responses to this stress is needed. In the present study, a total of 20 barley genotypes were investigated to identify potential salt-tolerant genotypes, both at the early growth stage using a hydroponic system, and in adult plants under field conditions. For these purposes, the multi-trait genotype-ideotype distance index (MGIDI) was used to identify salt-tolerant barley genotypes at the seedling stage, and the weighted average of absolute scores (WAASB) index was used to identify the high-yielding and stable genotypes in adult plant stage. At the early growth stage, barley seedlings were treated with two salinity levels: 0 mM NaCl (as control conditions) and 200 mM NaCl (as stress conditions) for 30 days, and during this period different growth a...
In this paper, we generalize the concepts of N -subnexuses of types (∈, q), (∈,∈ ∨q) and (q,∈ ∨q)... more In this paper, we generalize the concepts of N -subnexuses of types (∈, q), (∈,∈ ∨q) and (q,∈ ∨q), and introduce the notions of N -subnexuses of types (∈, qk), (∈,∈ ∨qk) and (q,∈ ∨qk). We investigate their basic properties, characterize subnexuses by N -subnexuses of type (∈,∈ ∨qk), and give some characterizations for N -subnexuses of types (∈, qk) and (q,∈ ∨qk). Moreover, we define N subnexuses of type (∈,∈ ∨ qk) and discuss on their different properties. MSC: 08A72; 03E20; 03G15
Let R be an (m,n)-hyperring. The Γ∗-relation on R in the sense of Mirvakili and Davvaz [?] is the... more Let R be an (m,n)-hyperring. The Γ∗-relation on R in the sense of Mirvakili and Davvaz [?] is the smallest strong compatible relation such that the quotient R∕Γ∗ is an (m,n)-ring. We use Γ∗-relation to define a fundamental functor, F from the category of (m,n)-hyperrings to the category of (m,n)-rings. Also, the concept of a fundamental (m,n)-ring is introduced and it is shown that every (m,n)-ring is isomorphic to R∕Γ∗ for a nontrivial (m,n)-hyperring R. Moreover, the notions of partitionable and quotientable are introduced and their mutual relationship is investigated. A functor G from the category of classical (m,n)-rings to the category of (m,n)-hyperrings is defined and a natural transformation between the functors F and G is given.
The purpose of this paper is the study of direct limits in category of Krasner (m, n)-hyperrings.... more The purpose of this paper is the study of direct limits in category of Krasner (m, n)-hyperrings. In this regards we introduce and study direct limit of a direct system in category (m, n)-hyperrings. Also, we consider fundamental relation , as the smallest equivalence relation on an (m, n)-hyperring R such that the quotient space is an (m, n)-ring, to introduce the fundamental functor from category of Krasner (m, n)-hyperrings to the category of (m, n)-rings. Finally, we study the relationship between fundamental functor and direct limit on Krasner (m, n)-hyperrings. In particular, we prove that the fundamental functor is exact and obtain some its basic properties.
In this paper, we generalize the concepts of ${\\mathcal{N}}$-subnexuses of types $(\\in, q)$, $(... more In this paper, we generalize the concepts of ${\\mathcal{N}}$-subnexuses of types $(\\in, q)$, $(\\in, \\in \\vee{q})$ and $(q, \\in \\vee {q})$, and introduce the notions of ${\\mathcal{N}}$-subnexuses of types $(\\in, q_{k})$, $(\\in, \\in \\vee{q_{k}})$ and $(q, \\in \\vee {q_{k}})$. We investigate their basicproperties, characterize subnexuses by ${\\mathcal{N}}$-subnexuses of type $(\\in, \\in \\vee {q_{k}})$, and give some characterizations for${\\mathcal{N}}$-subnexuses of types $(\\in, q_{k})$ and $(q, \\in \\vee{q_{k}})$. Moreover, we define ${\\mathcal{N}}$-subnexuses of type $(\\overline{\\in}, \\overline{\\in} \\vee \\overline{q_{k}})$ and discusson their different properties.
The notion of a subnexus based on ${\mathcal{N}}$-function (briefly, ${\mathcal{N}}$-subnexus) is... more The notion of a subnexus based on ${\mathcal{N}}$-function (briefly, ${\mathcal{N}}$-subnexus) is introduced, and related properties are investigated. Also, the notions of ${\mathcal{N}}$-subnexus of type $(\alpha, \beta)$, where $(\alpha, \beta)$ is $(\in, \in)$, $(\in, q)$, $(\in, \in\! \vee \, {q})$, $(q, \in)$, $(q,q)$, $(q, \in\! \vee \, {q})$, $(\overline{\in}, \overline{\in})$ and $(\overline{\in}, \overline{\in} \vee \overline{q})$, are introduced, and their basic properties are investigated. Conditions for an ${\mathcal{N}}$-structure to be an ${\mathcal{N}}$-subnexus of type $(q, \in\! \vee \, {q})$ are given, and characterizations of ${\mathcal{N}}$-subnexus of type $(\in, \in\! \vee \, {q})$ and $(\overline{\in}, \overline{\in} \vee \overline{q})$ are provided. Homomorphic image and preimage of ${\mathcal{N}}$-subnexus are discussed.
The notion of a subnexus based on ${\mathcal{N}}$-function (briefly, ${\mathcal{N}}$-subnexus) is... more The notion of a subnexus based on ${\mathcal{N}}$-function (briefly, ${\mathcal{N}}$-subnexus) is introduced, and related properties are investigated. Also, the notions of ${\mathcal{N}}$-subnexus of type $(\alpha, \beta)$, where $(\alpha, \beta)$ is $(\in, \in)$, $(\in, q)$, $(\in, \in\! \vee \, {q})$, $(q, \in)$, $(q,q)$, $(q, \in\! \vee \, {q})$, $(\overline{\in}, \overline{\in})$ and $(\overline{\in}, \overline{\in} \vee \overline{q})$, are introduced, and their basic properties are investigated. Conditions for an ${\mathcal{N}}$-structure to be an ${\mathcal{N}}$-subnexus of type $(q, \in\! \vee \, {q})$ are given, and characterizations of ${\mathcal{N}}$-subnexus of type $(\in, \in\! \vee \, {q})$ and $(\overline{\in}, \overline{\in} \vee \overline{q})$ are provided. Homomorphic image and preimage of ${\mathcal{N}}$-subnexus are discussed.
Background Salinity is one of the major limiting abiotic stresses that decrease crop production w... more Background Salinity is one of the major limiting abiotic stresses that decrease crop production worldwide. To recommend genotypes for cultivation under saline stress conditions, a comprehensive understanding of the genetic basis and plant responses to this stress is needed. In the present study, a total of 20 barley genotypes were investigated to identify potential salt-tolerant genotypes, both at the early growth stage using a hydroponic system, and in adult plants under field conditions. For these purposes, the multi-trait genotype-ideotype distance index (MGIDI) was used to identify salt-tolerant barley genotypes at the seedling stage, and the weighted average of absolute scores (WAASB) index was used to identify the high-yielding and stable genotypes in adult plant stage. At the early growth stage, barley seedlings were treated with two salinity levels: 0 mM NaCl (as control conditions) and 200 mM NaCl (as stress conditions) for 30 days, and during this period different growth a...
In this paper, we generalize the concepts of N -subnexuses of types (∈, q), (∈,∈ ∨q) and (q,∈ ∨q)... more In this paper, we generalize the concepts of N -subnexuses of types (∈, q), (∈,∈ ∨q) and (q,∈ ∨q), and introduce the notions of N -subnexuses of types (∈, qk), (∈,∈ ∨qk) and (q,∈ ∨qk). We investigate their basic properties, characterize subnexuses by N -subnexuses of type (∈,∈ ∨qk), and give some characterizations for N -subnexuses of types (∈, qk) and (q,∈ ∨qk). Moreover, we define N subnexuses of type (∈,∈ ∨ qk) and discuss on their different properties. MSC: 08A72; 03E20; 03G15
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