18 results
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2. Ideal lattices in semigroup of doubly stochastic matrices.
- Author
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Jose, Riya and Romeo, P. G.
- Abstract
In this paper, we illustrate the rule for finding number of idempotents in the doubly stochastic matrix semigroup Dn and also locate the idempotents in the case of D3 and D4. Further, we describe idempotent generated ideals of these semigroups and it is shown that the idempotent generated ideals form a lattice. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
3. Abundance of the ideals of the monoid of all orientation-preserving extensive full transformations.
- Author
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Zhao, Ping
- Abstract
Let 풪풫ℰn be the monoid of all orientation-preserving and extensive full transformations on {1,…,n}. In this paper, first we discuss left and right abundance of the principal ideals of 풪풫ℰn. Second, we give necessary and sufficient conditions for the principal ideals of 풪풫ℰn to be abundant and regular. Finally, we study abundance of the ideals in 풪풫ℰn. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
4. α-Ideals in Bounded Commutative Residuated Lattices.
- Author
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Kakeu, Ariane G. Tallee, Strüngmann, Lutz, Njionou, Blaise B. Koguep, and Lele, Celestin
- Subjects
- *
RESIDUATED lattices , *HEYTING algebras , *PRIME ideals - Abstract
This study aims to introduce the concept of α -ideal in bounded commutative residuated lattices and establish some related properties. In this paper, we show that the set of α -ideals of a bounded commutative residuated lattice is a Heyting algebra, and an algebraic lattice. Moreover, we state the prime α -ideal theorem, and describe relations between α -ideals and some types of ideals of a bounded commutative residuated lattice. Finally, we discuss correspondences between α -ideals and α -filters of a bounded commutative residuated lattice. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
5. On the Inclusion Ideal Graph of Semigroups.
- Author
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Baloda, Barkha and Kumar, Jitender
- Subjects
- *
AUTOMORPHISM groups , *UNDIRECTED graphs , *CAYLEY graphs , *PLANAR graphs - Abstract
The inclusion ideal graph I n (S) of a semigroup S is an undirected simple graph whose vertices are all the nontrivial left ideals of S and two distinct left ideals I , J are adjacent if and only if either I ⊂ J or J ⊂ I. The purpose of this paper is to study algebraic properties of the semigroup S as well as graph theoretic properties of I n (S). We investigate the connectedness of I n (S) and show that the diameter of I n (S) is at most 3 if it is connected. We also obtain a necessary and sufficient condition of S such that the clique number of I n (S) is the number of minimal left ideals of S. Further, various graph invariants of I n (S) , viz. perfectness, planarity, girth, etc., are discussed. For a completely simple semigroup S , we investigate properties of I n (S) including its independence number and matching number. Finally, we obtain the automorphism group of I n (S). [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
6. Inner generalized Weyl algebras and their simplicity criteria.
- Author
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Bavula, V. V.
- Abstract
The aim of the paper is to introduce a new class of rings — the
inner generalized Weyl algebras (IGWA) — and to give simplicity criteria for them. For each IGWA A a derivative series of IGWAs, A → A′→ A″ →⋯ → A(α) →⋯, is attached where α is an arbitrary ordinal. In general, all rings A(α) are distinct. A new construction of rings, theinner (σ,τ,a)-extension of a ring, is introduced (where σ and τ are endomorphisms of a ring D and a ∈ D). [ABSTRACT FROM AUTHOR]- Published
- 2023
- Full Text
- View/download PDF
7. Hamiltonian trace graph of matrices.
- Author
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Sivagami, M. and Tamizh Chelvam, T.
- Subjects
- *
HAMILTONIAN graph theory , *MATRIX rings , *UNDIRECTED graphs , *MATRICES (Mathematics) , *CHARTS, diagrams, etc. , *COMMUTATIVE rings - Abstract
Let R be a commutative ring with identity, n ≥ 2 be a positive integer and M n (R) be the set of all n × n matrices over R. For a matrix A ∈ M n (R) , Tr (A) is the trace of A. The trace graph of the matrix ring M n (R) , denoted by Γ t (M n (R)) , is the simple undirected graph with vertex set { A ∈ M n (R) ∗ : there exists B ∈ M n (R) ∗ such that Tr (A B) = 0 } and two distinct vertices A and B are adjacent if and only if Tr (A B) = 0. The ideal-based trace graph of the matrix ring M n (R) with respect to an ideal I of R , denoted by Γ I t (M n (R)) , is the simple undirected graph with vertex set M n (R) ∖ M n (I) and two distinct vertices A and B are adjacent if and only if Tr (A B) ∈ I. In this paper, we investigate some properties and structure of Γ I t (M n (R)). Further, it is proved that both Γ t (M n (R)) and Γ I t (M n (R)) are Hamiltonian. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
8. Annihilating properties of ideals generated by coefficients of polynomials and power series.
- Author
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Kim, Nam Kyun, Lee, Yang, and Ziembowski, Michał
- Subjects
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POWER series , *POLYNOMIALS , *IDEALS (Algebra) , *POLYNOMIAL rings - Abstract
In this paper, we study the annihilating properties of ideals generated by coefficients of polynomials and power series which satisfy a structural equation. We first show that if f (x) R g (x) = 0 for polynomials f (x) = ∑ i = 0 m a i x i , g (x) = ∑ j = 0 m b j x j over any ring R , then for any i , j , there exist positive integers s (i , j) and t (i , j) such that a i 1 R a i 2 R ⋯ a i s (i , j) R b j = 0 and a i R b j 1 R b j 2 R ⋯ R b j t (i , j) = 0 , whenever i 1 , i 2 , ... , i s (i , j) ≤ i and j 1 , j 2 , ... , j t (i , j) ≤ j. Next we prove that if f (x) R g (x) = 0 for power series f (x) = ∑ i = 0 ∞ a i x i , g (x) = ∑ j = 0 ∞ b j x j over any ring R , then for any i , j , there exist positive integers s (i , j) and t (i , j) such that a i s (i , j) R ⋯ R a i 2 R a i 1 R b j 1 R b j 2 R ⋯ R b j t (i , j) = 0 when ∑ p = 1 s (i , j) i p + ∑ q = 1 t (i , j) j q < (s (i , j) + 1) (t (i , j) + 1) (i + 1) (j + 1) and i p ≤ i , j q ≤ j for each p , q. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
9. Ideals and Bosbach States on Residuated Lattices.
- Author
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Woumfo, Francis, Koguep Njionou, Blaise B., Temgoua Alomo, Etienne R., and Lele, Celestin
- Subjects
- *
NONCOMMUTATIVE algebras , *UTOPIAS , *MATHEMATICAL logic , *BOOLEAN algebra , *COMMUTATIVE algebra , *RESIDUATED lattices - Abstract
In random experiments, the fact that the sets of events has a structure of a Boolean algebra, i.e. it follows the rules of classical logic, is the main hypothesis of classical probability theory. Bosbach states have been introduced on commutative and non-commutative algebras of fuzzy logics as a way of probabilistically evaluating the formulas. In this paper, we focus on the relationship between some properties of ideals and Bosbach states in the framework of commutative residuated lattices. In particular, we introduce the concept of co-kernel of a Bosbach state which is an ideal and we establish the relationship between the notion of co-kernel and the kernel. Moreover, we define and characterize maximal ideals and maximal MV-ideals in residuated lattices. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
10. Ideals of Finite-Dimensional Pointed Hopf Algebras of Rank One.
- Author
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Wang, Yu, Wang, Zhihua, and Li, Libin
- Subjects
- *
HOPF algebras , *INDECOMPOSABLE modules , *IDEALS (Algebra) , *PRIME ideals - Abstract
Let H be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed field of characteristic zero. In this paper we show that any finite-dimensional indecomposable H -module is generated by one element. In particular, any indecomposable submodule of H under the adjoint action is generated by a special element of H. Using this result, we show that the Hopf algebra H is a principal ideal ring, i.e., any two-sided ideal of H is generated by one element. As an application, we give explicitly the generators of ideals, primitive ideals, maximal ideals and completely prime ideals of the Taft algebras. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
11. Ideals and Bosbach States on Residuated Lattices.
- Author
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Woumfo, Francis, Njionou, Blaise B. Koguep, Alomo, Etienne R. Temgoua, and Lele, Celestin
- Subjects
- *
NONCOMMUTATIVE algebras , *UTOPIAS , *MATHEMATICAL logic , *BOOLEAN algebra , *PROBABILITY theory , *RESIDUATED lattices , *COMMUTATIVE algebra - Abstract
In random experiments, the fact that the sets of events has a structure of a Boolean algebra, i.e. it follows the rules of classical logic, is the main hypothesis of classical probability theory. Bosbach states have been introduced on commutative and non-commutative algebras of fuzzy logics as a way of probabilistically evaluating the formulas. In this paper, we focus on the relationship between some properties of ideals and Bosbach states in the framework of commutative residuated lattices. In particular, we introduce the concept of co-kernel of a Bosbach state which is an ideal and we establish the relationship between the notion of co-kernel and the kernel. Moreover, we define and characterize maximal ideals and maximal MV-ideals in residuated lattices. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
12. The Tutte polynomial of ideal arrangements.
- Author
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Randriamaro, Hery
- Subjects
- *
POLYNOMIALS , *FINITE fields , *GRAPH coloring , *HYPERPLANES - Abstract
The Tutte polynomial was originally a bivariate polynomial enumerating the colorings of a graph and of its dual graph. But it reveals more of the internal structure of the graph like its number of forests, of spanning subgraphs, and of acyclic orientations. In 2007, Ardila extended the notion of Tutte polynomial to hyperplane arrangements, and computed the Tutte polynomials of the classical root systems for certain prime powers of the first variable at the same time. In this paper, we compute the Tutte polynomial of ideal arrangements. These arrangements were introduced in 2006 by Sommers and Tymoczko, and are defined for ideals of root systems. For the ideals of classical root systems, we bring a slight improvement of the finite field method by showing that it can applied on any finite field whose cardinality is not a minor of the matrix associated to the studied hyperplane arrangement. Computing the minor set associated to an ideal of classical root systems particularly permits us to deduce the Tutte polynomials of the classical root systems. For the ideals of the exceptional root systems of type G 2 , F 4 , and E 6 , we use the formula of Crapo. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
13. n-Fold Boolean, Implicative and Integral Ideals on Bounded Commutative Residuated Lattices.
- Author
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Tchoua Yinga, Fabrice, Koguep Njionou, Blaise B., Temgoua Alomo, Etienne R., and Lele, Celestin
- Subjects
- *
IDEALS (Algebra) , *RESIDUATED lattices , *FILTERS & filtration - Abstract
In this paper, we introduce the concepts of n -fold boolean ideals, n -fold implicative ideals and n -fold integral ideals in residuated lattices and we state and prove their properties. Several characterizations of these notions are derived and the relations between those notions are investigated. Also, we construct the correspondence between the notions of n -fold ideal and n -fold filter in residuated lattices. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
14. Bounds for the Genus of Generalized Total Graph of a Commutative Ring.
- Author
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Asir, T. and Mano, K.
- Subjects
- *
COMMUTATIVE rings , *UNDIRECTED graphs , *DIVISOR theory - Abstract
Let R be a commutative ring with non-zero identity and I its proper ideal. The total graph of R with respect to I, denoted by T (ΓI (R)), is the undirected graph with all elements of R as vertices, and where distinct vertices x and y are adjacent if and only if x + y ∈ S (I) = { a ∈ R : r a ∈ I for some r ∈ R \ I }. In this paper, some bounds for the genus of T(ΓI(R)) are obtained. We improve and generalize some results for the total graphs of commutative rings. In addition, we obtain an isomorphism relation between two ideal based total graphs. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
15. Leibniz algebras whose subideals are ideals.
- Author
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Kurdachenko, Leonid A., Yashchuk, Viktoriia S., and Subbotin, Igor Ya.
- Subjects
- *
ALGEBRA education , *ALGEBRA exercises , *ALGEBRAIC equations , *ALGEBRAIC functions , *ALGEBRAIC logic - Abstract
In this paper, we obtain the description of hyperabelian Leibniz algebras, whose subideals are ideals. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
16. -Consistency in signed total graphs of commutative rings.
- Author
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Pranjali, Gaur, Atul, and Acharya, Mukti
- Subjects
- *
GRAPH theory , *COMMUTATIVE rings , *GEOMETRIC vertices , *SUBGRAPHS , *ALGEBRA - Abstract
Motivated by the earlier study on the notion of signed total graph of a commutative ring, in this paper, we characterize all the commutative rings with unity for which signed total graph is -consistent and sign-compatible. To do this, first, we derive a formula to determine the degree of each vertex in (induced subgraph of the total graph), when and , where each 's is a field of characteristic , is an odd prime, and . [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
17. Upper and Lower Periodic Subsets of Semigroups.
- Author
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Hooshmand, M. H. and Fang, Xingui
- Subjects
- *
PERIODIC functions , *SET theory , *SEMIGROUPS (Algebra) , *BINARY number system , *IDEALS (Algebra) , *FUNDAMENTAL theorem of algebra , *REPRESENTATIONS of algebras , *MATHEMATICAL analysis - Abstract
In this paper, a new topic about a vast class of subsets of semigroups and binary systems, which contains all ideals, periodic subsets and sub-semigroups, is introduced and studied. In fact, the 'upper periodic subsets' can be considered as a generalization of the conception 'ideals'. We prove a fundamental theorem which states that if A is a (left) upper B-periodic subset of a semigroup S, then under some conditions, it has a unique direct representation $A=\mathfrak{B}\cdot D\,\dot{\cup}\,B^1\cdot E$, where B1=B ∪ {1} and B ⊆ 픅 ≤ S. Especially, we prove a unique direct representation for upper and lower T-periodic subsets, and classify all sub-semigroups of S containing a fixed element T to three classes. This classification gives us more interesting properties for the real semigroups. At last, we characterize upper and lower T-periodic subsets of semigroups and groups. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
18. ON m-FOLD STABLE IDEALS.
- Author
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CHEN, HUANYIN, CHEN, MIAOSEN, and Lam, T. Y.
- Subjects
- *
IDEALS (Algebra) , *RING theory , *MATRICES (Mathematics) , *ALGEBRA , *MATHEMATICS - Abstract
In this paper, we introduce m-fold stable ideals and show that m-fold stable range for ideals is invariant under matrix extension. Also we prove that every m-fold stable ideal is right and left symmetric. [ABSTRACT FROM AUTHOR]
- Published
- 2004
- Full Text
- View/download PDF
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