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Giving the proper citations, it is shown that, except of Lemma 2.4 and Theorem 2.6, almost all the results of the paper in the title have been previously published for ordered hypersemigroups in Eur. J. Pure Appl. Math. and they are not new. There are also two results obtained from ordered semigroups just putting a "o" instead of "o" (that isn't a correct way to work), without reference to ordered semigroups on which the results on ordered hypersemigroups are based. One of them can be obtained as corollary to a theorem in Eur. J. Pure Appl. Math. as well, and it is not new. [ABSTRACT FROM AUTHOR]

In this paper, first the fuzzy set and i-(m,n)-near ring then fuzzy subgroups, fuzzy (m,n)-subnear ring and finally fuzzy ideals are defined. We state and prove theorems in these cases. Also in the next chapter, normal fuzzy subnear ring expressed and theorems about normal fuzzy subnear rings is expressed and proved. [ABSTRACT FROM AUTHOR]

AbstractLet V be a vector space and U a fixed subspace of V. We denote the semigroup of all linear transformations on V under composition of functions by L(V). In this paper, we study the semigroup of all linear transformations on V whose restrictions belong to the general linear group GL(U), denoted by LGL(U)(V). More precisely, we consider the subsemigroup LGL(U)(V)={α∈L(V):α|U∈GL(U)} of L(V). In this work, Green’s relations and ideals of this semigroup are described. Then we also determine the minimal ideal and the set of all minimal idempotents of it. Moreover, we establish an isomorphism theorem when V is a finite dimensional vector space over a finite field. Finally, we find its generating set. [ABSTRACT FROM AUTHOR]

In this paper, we investigate r-ideals in commutative ordered semigroups with zero and identity. Moreover, we study some properties of r-ideals. Furthermore, we give several characterizations of r-ideals. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to introduce the notion of stabilizers in L-algebras and develop stabilizer theory in L-algebras. In this paper, we introduced the notions of left and right stabilizers and investigated some related properties of them. Then, we discussed the relations among stabilizers, ideal and co-annihilators. Also, we obtained that the set of all ideals of a CKL-algebra forms a relative pseudo-complemented lattice. In addition, we proved that right stabilizers in CKL-algebra are ideals. Then by using the right stabilizers we produced a basis for a topology on L-algebra. We showed that the generated topology by this basis is Baire, connected, locally connected and separable and we investigated the other properties of this topology. [ABSTRACT FROM AUTHOR]

This paper examines how the Durkheimian approach to the 'ideal' delineates a possible way of straddling the dilemma between the normative orientation of 'powerful knowledge' accounts and the critical orientation of 'knowledge of the powerful' accounts. It argues that the normative aims are embedded in the fabrics of the sociological description with which the Durkheimian notion of 'elementary form' is concerned. To see where this enterprise can lead, this paper turns to the sociology of education of Bourdieu and Bernstein. Both draw on Durkheim's writings on primitive classifications in education and society, working towards uncovering the regularities of the world of knowledge classifications. Keeping in line with Bourdieu and Bernstein, this paper argues that one has to make the same refusal to the advocates of an abstract ideal of educational knowledge that is dissociated from its social conditions of historical realization in pedagogic contexts, and to the advocates of a cynical relativism of ideal that rejects any necessities socially established. [ABSTRACT FROM AUTHOR]

In this paper we investigate the concept of ℐ {\mathcal{I}} -monotonic convergence of sequences of bi-complex numbers. We also discuss some partial order relations ≼ i 2 {\preccurlyeq_{i_{2}}} and ≼ i id {\preccurlyeq_{i_{\rm id}}} between two bi-complex numbers and ℐ {\mathcal{I}} -convergence, ℐ {\mathcal{I}} -monotonic sequences with respect to different partial order relations, ℐ {\mathcal{I}} -Cauchy sequences, ℐ {\mathcal{I}} -bounded monotonic sequences of bi-complex numbers. The analysis indicates that ℐ {\mathcal{I}} -monotonicity with respect to ≼ i 2 {\preccurlyeq_{i_{2}}} is not comparable with ℐ {\mathcal{I}} -monotonicity with respect to ≼ i id {\preccurlyeq_{i_{\rm id}}} . [ABSTRACT FROM AUTHOR]

In this paper, we study the ideal strong lacunary ward compactness of a subset of a 2-normed space and the ideal strongly lacunary ward continuity of a function on . Here a subset of is said to be ideal strong lacunary ward compact if any sequence in has an ideal strong lacunary quasi-Cauchy subsequence. Additionally, a function on is said to be ideal strong lacunary ward continuous if it preserves ideal strong lacunary quasi-Cauchy sequences; an ideal is defined to be a hereditary and additive family of subsets of . We find that a subset of with a countable Hamel basis is totally bounded if and only if it is ideal strong lacunary ward compact. [ABSTRACT FROM AUTHOR]

Ideals and filters are important notions with different meanings in the study of algebraic structures related to logical systems. In this paper we establish new connections between these concepts in residuated lattices. [ABSTRACT FROM AUTHOR]

Recently, Milles and Hammami presented and studied the concept of a neutrosophic topology generated by a neutrosophic relation. As a continuation in the same direction, this paper studies the concepts of neutrosophic ideals and neutrosophic filters on that topology. More precisely, we offer the lattice structure of neutrosophic open sets of a neutrosophic topology generated via a neutrosophic relation and examine its different characteristics. Furthermore, we enlarge to this lattice structure the notions of ideals (respectively, filters) and characterize them with regard to the lattice operations. We end this work by studying the prime neutrosophic ideal and prime neutrosophic filter as interesting types of neutrosophic ideals and neutrosophic filters. [ABSTRACT FROM AUTHOR]

In this paper, we first construct a convex structure by ideals of effect algebras. Then we discuss convex properties of morphisms and monomorphisms between effect algebras. Further, we prove that partial binary operations ⊕ and ⊖ are separately convexity-preserving for the first position with respect to ideal convex structures when the effect algebra is a lattice effect algebra. A lattice effect algebra equipped with an ideal convex structure is called a semi-ideal convex effect algebra. Finally, we obtain that finite product and quotient of semi-ideal convex effect algebras are also semi-ideal convex effect algebras. [ABSTRACT FROM AUTHOR]

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]

In this paper, the concept of Paradistributive Latticoid (PDL) as a generalization of a distributive lattice is introduced and investigated its properties. A set of equivalent conditions for a PDL to become a distributive lattice are given. The notions of an ideal and a filter in a PDL are introduced and studied their properties. Subdirect representation of a PDL is obtained [ABSTRACT FROM AUTHOR]

In this paper, convergences of a filter and a net have been characterized through ideal on topological spaces. Furthermore, we characterized the local function in an ideal topological space in terms of convergence of filter. Using Zorn's Lemma, we have found a maximal element in the collection of all proper ideals on a nonempty set which is called maximal ideal. We provide a convenient characterization of maximal ideals. We also consider simple properties of the image of an ideal, a net and various local functions under a homeomorphism. [ABSTRACT FROM AUTHOR]

This paper aims to examine the notion of λ-ideal statistical convergence and λ-ideal statistically Cauchy sequence over non-Archimedean 2-normed spaces. Moreover, we have also studied and proven some significant properties for λ-ideal statistically Cauchy sequence in 2-normed spaces to be λ-ideal statistically Cauchy with respect to II.II∞. And also we show that, λ-ideal statistical convergence in 2-normed spaces is λ-ideal statistical pre-Cauchy in 2-normed spaces over non-Archinedean field. Through this paper, K symbolizes a complete, non-trivially valued, non-Archinedean field. [ABSTRACT FROM AUTHOR]

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]

Let A be an MV-algebra. An (⊙ , ∨) -derivation on A is a map d : A → A satisfying: d (x ⊙ y) = (d (x) ⊙ y) ∨ (x ⊙ d (y)) for all x , y ∈ A . This paper initiates the study of (⊙ , ∨) -derivations on MV-algebras. Several families of (⊙ , ∨) -derivations on an MV-algebra are explicitly constructed to give realizations of the underlying lattice of an MV-algebra as lattices of (⊙ , ∨) -derivations. Furthermore, (⊙ , ∨) -derivations on a finite MV-chain are enumerated and the underlying lattice is described. [ABSTRACT FROM AUTHOR]

In a large number of multivalued logic and fuzzy logic of algebraic systems, residuated lattices play a prominent role and have considerable applications. States operators have been introduced on residuated lattices, and their properties are useful for the development of an algebraic theory of probabilistic models of those algebras. In this paper, we introduce the notion of state ideal in the framework of state residuated lattices, investigate some related properties, and provide several examples. Also, we present two types of state residuated lattices: state i-simple residuated lattices and state i-local residuated lattices, and characterize them. Moreover, the relationship between state ideals and state filters is analyzed using the set of complement elements. Furthermore, we prove that the lattice of all state ideals of a given state residuated lattice is a complete lattice. The notion of obstinate state ideals in state residuated lattices is also introduced, and several characterizations are presented. [ABSTRACT FROM AUTHOR]

In this paper new notions of dimensions for ideal topological spaces are inserted, called *-quasi covering dimension and ideal quasi covering dimension. We study several of their properties and investigate their relations with types of covering dimensions like the-covering dimension and the ideal covering dimension. [ABSTRACT FROM AUTHOR]

Through this paper, via the operators (.)* and ψ, we presented notion of *-Locally set in an ideal topological space Zi as a new stronger form of locally closed set, and considered relations with various existing weak form of locally closed set. Preservations of direct images as well as inverse images of (.)*, ψ, *-perfect and various weak forms of locally closed set including *-Locally closed set are important investigating part. Besides, we pointed out that consideration of 'bijectivity' in Lemma 3.1 of [24] is sufficient, and the Lemma 3.3 of [24] is wrong. We demonstrated two modifications of the last one. [ABSTRACT FROM AUTHOR]

In this paper, a new diamond–silicon carbide ceramic composite material—Ideal—is studied and its mechanical characteristics are determined. For the first time, a comprehensive determination of Poisson's ratio, shear modulus, bulk modulus, and transverse sound velocity is carried out. Poisson's ratio is in the region of 0.008 to 0.01, which, in turn, indicates the absolutely brittle nature of the failure of the Ideal ceramic under loading. The criteria that allow evaluating different materials used for body armor are calculated. [ABSTRACT FROM AUTHOR]

We solve the last standing open problem from the seminal paper by J. Gerlits and Zs. Nagy [Topology Appl. 14 (1982), pp. 151–161], which was later reposed by A. Miller, T. Orenshtein, and B. Tsaban. Namely, we show that under \mathfrak {p}=\mathfrak {c} there is a \delta-set that is not a \gamma-set. Thus we constructed a subset A of reals such that the space \mathrm {C}_p(A) of all real-valued continuous functions on A is not Fréchet–Urysohn, but possesses the Pytkeev property. Moreover, under \mathbf {CH} we construct a \pi-set that is not a \delta-set solving a problem by M. Sakai. In fact, we construct various examples of \delta-sets that are not \gamma-sets, satisfying finer properties parametrized by ideals on natural numbers. Finally, we distinguish ideal variants of the Fréchet–Urysohn property for many different Borel ideals in the realm of functional spaces. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an equiprime fuzzy graph of a nearring with respect to the level ideal of a fuzzy ideal. We interrelate graph theoretical properties of the graph and ideal theoretical properties of nearring. We show that the properties like vertex cut, connectedness of the graph depend on the properties of the fuzzy ideal. We define ideal symmetry of the graph and find conditions for the graph to be ideal symmetric. If the fuzzy ideal is equiprime then we show that the level set induces a fuzzy clique. We find conditions required for the level set to be the vertex cover of the graph. We find interrelation between equiprime fuzzy graph and fuzzy graph of nearring with respect to level ideal. We study properties of the graph under neaaring homomorphism. We prove that the connectedness of the graph in homomorphic image depends on properties of ideal. We obtain conditions required for homomorphic image of an equiprime fuzzy ideal to be an equiprime fuzzy ideal. [ABSTRACT FROM AUTHOR]

Over the last few years, numerous researchers have contributed significantly to summability theory by connecting various notions of convergence concepts of sequences. In this paper, we introduce the concepts of ℐ -statistical supremum and ℐ -statistical infimum of a real-valued sequence and study some fundamental features of the newly introduced notions. We also introduce the concept of ℐ -statistical monotonicity and establish the condition under which an ℐ -statistical monotonic sequence is ℐ -statistical convergent. We end up giving a necessary and a sufficient condition for the ℐ -statistical convergence of a real-valued sequence. [ABSTRACT FROM AUTHOR]

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]

In this paper, we focus on the structure of the variety of Lie algebras with a finite number of ideals and their graph representations using Hasse diagrams. The large number of necessary conditions on the algebraic structure of this type of algebras leads to the explicit description of those algebras in the variety with trivial Frattini subalgebra. To illustrate our results, we have included and discussed lots of examples throughout the paper. [ABSTRACT FROM AUTHOR]

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]

As a continuation of the paper "Adjunction Identity to Hypersemigroup" in Turk J Math 2022; 46 (7): 2834-2853, it has been proved here that the adjunction of a greatest element to an ordered hypersemigroup is actually an embedding problem. The concept of pseudoideal has been introduced and has been proved that for each ordered hypersemigroup S an ordered hypersemigroup V having a greatest element (poe-hypersemigroup) can be constructed in such a way that there exists a pseudoideal T of S such that S is isomorphic to T. If S does not have a greatest element, then this can be regarded as the embedding of an ordered hypersemigroup in an ordered semigroup with greatest element. [ABSTRACT FROM AUTHOR]

In this paper, we characterize when, for any infinite cardinal α , the Fremlin tensor product of two Archimedean Riesz spaces (see Fremlin in Am J Math 94:777–798, 1972) is Dedekind α -complete. We also provide an example of an ideal I in an Archimedean Riesz space E such that the Fremlin tensor product of I with itself is not an ideal in the Fremlin tensor product of E with itself. [ABSTRACT FROM AUTHOR]

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, the inner (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]

This is from Birkhoff, the "father of lattice theory" in Trends in Lattice Theory. Van Nostrand 1970: "Lattices can do things for you, no matter what kind of mathematician you are!". The aim of this paper is to show that the le-semigroups (lattice ordered semigroups possessing a greatest element) play the main role in studying the ordered hypersemigroups. From many results on lattice ordered semigroups corresponding results on ordered semigroups can be obtained. The converse is also possible but the beauty and simplicity of "order" makes it easier to investigate the lattice ordered semigroup at first. After getting the results on ordered semigroups, by an easy modification corresponding results on ordered hypersemigroups can be obtained. [ABSTRACT FROM AUTHOR]

In this paper, we have introduced the notions of rough I-convergence of sequences and defined I-cluster point of a sequence and proved some results associated with this notion in 2-normed spaces. Also, we have defined rough I-limit set of a sequence and shown that this set is closed and convex. Further, we have defined the notion of rough I-Cauchy sequence in the same space. [ABSTRACT FROM AUTHOR]

Indonesian central and regional government relationship often changes along with domestic political development and people's needs. Law no. 1/1945 contained the concept of Indonesia's autonomy with people's sovereignty, and Law no. 22 of 1948 contained the concept of 'the broadest economy.' During the 1950 Provisional Constitution era, Law no. 1/1957 and Law no. 6/159 contained the concept of real and broadest autonomy. During the 1945 Constitution in New Order Era, presidential decree No. 6 of 1959 stipulated the real and broadest autonomy, in addition to Law no. 18/1965 with a similar concept and deconcentration principle, Law no. 5/1974 on real and responsible autonomy, stipulating that decentralization is equally important to deconcentration, Law no. No. 22 of 1999 stipulated the real and broadest autonomy, in addition to Law no. 32/2004 on the amendment of real autonomy. Law no. 23 of 2014 tends to adhere to the deconcentration principle. This normative legal study employed secondary data sources, including the 1945 Constitution, Law no. 22/1999 on local government, Law no. 32/2004 on local government, Law no. 23/2014 on local government, Law no. 9/2015 on the second amendment of Law no. 23/2014 on local government, and relevant books, articles, and papers. [ABSTRACT FROM AUTHOR]

Many uncertainties arise in real-world problems, making them impossible to solve using conventional approaches. Researchers all over the world have developed new mathematical theories like fuzzy set theory and rough set theory to better understand the uncertainties that occur in various fields. Soft set theory, which was recently introduced, offers a novel approach to real-world problem solving by removing the need to set the membership function. This is helpful in resolving a variety of issues, and much progress is being made these days. Recently, Jun introduced the concept of a hybrid structure, which blends the concepts of a fuzzy set as well as a soft set. In this paper, we define the hybrid k-sum and hybrid k-product of k-ideals of semiring and investigate their properties. We illustrate with an example that the hybrid sum and hybrid product of two k-ideals are not always hybrid ideals. We also describe semiring regularity constraints in terms of hybrid k-ideal structures. [ABSTRACT FROM AUTHOR]

A ring R is called filial (left weakly filial) if it satisfies the following condition: for all subrings J and I of R, if J is an ideal of I and I is an ideal of R, then J is an ideal (left ideal) of R. If every subring of R is filial (left weakly filial), then R is called fully filial (fully left weakly filial). In this paper, some new characterizations of left weakly filial rings in terms of the prime radical are given. Left weakly filial rings that are matrix rings, polynomial rings and the direct sum of copies of a ring are also investigated. Recently, in [3], [4] and [13], classification theorems for semiprime torsion fully filial rings and for torsion-free fully filial rings were presented. We study fully left weakly filial rings and show that semiprime torsion rings and semiprime torsion-free rings are fully left weakly filial if and only if they are fully filial. [ABSTRACT FROM AUTHOR]

BL-algebras are algebraic structures corresponding to Hajek's basic fuzzy logic. The aim of this paper is to analyze the structure of BL-algebras using commutative rings. Due to computational considerations, we are interested in the finite case. We present new ways to generate finite BL-algebras using commutative rings and provide summarizing statistics. Furthermore, we investigated BL-rings, i.e., commutative rings whose the lattice of ideals can be equipped with a structure of BL-algebra. A new characterization for these rings and their connections to other classes of rings is established. Furthermore, we give examples of finite BL-rings for which the lattice of ideals is not an MV-algebra and, using these rings, we construct BL-algebras with 2 r + 1 elements, r ≥ 2 , and BL-chains with k elements, k ≥ 4. In addition, we provide an explicit construction of isomorphism classes of BL-algebras of small n size ( 2 ≤ n ≤ 5 ). [ABSTRACT FROM AUTHOR]

In this paper using the connections between some subvarieties of residuated lattices, we investigated some properties of the lattice of ideals in commutative and unitary rings. We give new characterizations for commutative rings A in which Id(A) is an MV-algebra, a Heyting algebra or a Boolean algebra and we establish connections between these types of rings. We are very interested in the finite case and we present summarizing statistics. We show that the lattice of ideals in a finite commutative ring is a Boolean algebra or an MV-algebra (which is not Boolean). Using this result we generate the binary block codes associated to the lattice of ideals in finite commutative rings and we present a new way to generate all (up to an isomorphism) finite MV-algebras using rings. [ABSTRACT FROM AUTHOR]

In this paper we describe commutative monoids S containing a zero element in which every ideal is the annihilator of an element of S. [ABSTRACT FROM AUTHOR]

It is shown that some embedding problems on hypersemigroups are actually problems of adjunction. According to the theorem of this paper, for every hypersemigroup S which does not have identity element, an hypersemigroup T having identity element can be constructed in such a way that S is an ideal of T. Moreover, if S is regular, intra-regular, right (left) regular, right (left) quasi-regular or semisimple, then so is T. If A is an ideal, subidempotent bi-ideal or quasi-ideal of S, then it is an ideal, bi-ideal, quasi-ideal of T as well. Illustrative examples are given. [ABSTRACT FROM AUTHOR]

In this paper we have introduced and studied the notion of rough I-convergence in intuitionistic fuzzy normed spaces. Also we have defined rough I-cluster point of a sequence and proved some related results in the same space. [ABSTRACT FROM AUTHOR]

In this paper we introduce the notion of I*-α-uniform equal convergence and I*-α-strong uniform equal convergence of sequences of functions and then investigate some lattice properties of ΦI*-α-u:e: and I*-α-s:u:e:, the classes of all functions which are I*-α-uniform equal limits and I*-α-strong uniform equal limits of sequences of functions respectively obtained from a class of functions Φ. We have also shown that I-exhaustiveness, I-uniform and I--convergence of sequences of functions are preserved under uniform conjugacy. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the connection between ideals of semisimple MV-algebras and set-theoretic filters. In particular, we obtain that there is a bijection between closed ideals (with respect to a specified closure operation) and all filters on some associated set X. Finally, we establish that the space of closed prime ideals is homeomorphic to a well-described space having the Stone-Čech compactification βX as a subspace. [ABSTRACT FROM AUTHOR]

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]

Many researchers have introduced several generalizations of somewhat continuous functions. In our paper, we want to define a new type of somewhat continuous function by using the definition of a feebly ideal open set. The relationship of the somewhat continuous function has been established and justified by giving relevant illustrative examples. Specific characterizations and explanations of the somewhat continuous function have also been explored. [ABSTRACT FROM AUTHOR]

In this paper, we characterize residuated lattices for which the topological space of prime ideals is a Noetherian space. The notion of i-Noetherian residuated lattice is introduced and related properties are investigated. We proved that a residuated lattice is i-Noetherian iff every ideal is principal. Moreover, we show that a residuated lattice has the spectrum of a Noetherian space iff it is i-Noetherian. [ABSTRACT FROM AUTHOR]

In this paper the notions of radical, prime radical and quasi prime radical are extended to seminearrings and the results analogue to those on prime radical and quasi prime radical are presented. We also obtained the class of right duo seminearrings in various characteristic in terms of ideals. [ABSTRACT FROM AUTHOR]

In order to enhance the calculation efficiency of steady-state incompressible fluid flow, based on the existing IDEAL (i.e. the inner doubly iterative efficient algorithm for linked equations) algorithm, a novel segregated pressure-based algorithm (named as IDEALC) has been developed in this paper. The novel algorithm inherits the superiorities of the original IDEAL algorithm and at the same time improves its potential shortcomings. The specific promotions of the IDEALC algorithm consist of three aspects, including the first approximation of the SIMPLE algorithm, the approach to solving the momentum equation and the second approximation of the SIMPLE algorithm. Detailed comparisons between the IDEAL and the IDEALC algorithms are carried out by three numerical experiments on structured collocated grids. The numerical results demonstrate that the IDEALC algorithm can accurately predict the information of the flow field and has substantial advantages over the IDEAL algorithm in terms of the efficiency because of the consistent property of the former. The superiority of the IDEALC algorithm over the IDEAL algorithm is especially remarkable when the time step multiplier gets a moderate value. [ABSTRACT FROM AUTHOR]

Let C n denote the order-preserving and decreasing full transformation semigroup on X n = { 1 , 2 , ... , n } . In this paper, first we describe each principal ideal of C n . Second, we discuss left and right abundance of the principal ideals of C n . Further, necessary and sufficient conditions for the principal ideals in C n to be abundant and regular are given. Finally, we study abundance of the ideals in C n . [ABSTRACT FROM AUTHOR]

The study of symmetry is one of the most important and beautiful themes uniting various areas of contemporary arithmetic. Algebraic structures are useful structures in pure mathematics for learning a geometrical object's symmetries. In this paper, we introduce new concepts in an algebraic structure called BCI-algebra, where we present the concepts of bipolar fuzzy (closed) BCI-positive implicative ideals and bipolar fuzzy (closed) BCI-commutative ideals of BCI-algebras. The relationship between bipolar fuzzy (closed) BCI-positive implicative ideals and bipolar fuzzy ideals is investigated, and various conditions are provided for a bipolar fuzzy ideal to be a bipolar fuzzy BCI-positive implicative ideal. Furthermore, conditions are presented for a bipolar fuzzy (closed) ideal to be a bipolar fuzzy BCI-commutative ideal. [ABSTRACT FROM AUTHOR]

We give a counterexample in this amendment to show that there is an error in consideration of the statement "if f: X → Y and J is an ideal on Y, then f-1(J) = ff-1(J): J ℇ J- is an ideal on X" by Hamlett in his paper "Lindelöf with respect to an ideal" [New Zealand J. Math. 42, 115-120, 2012]. We also modify it here in a new way and henceforth put forward correctly all the results that were based on the said statement derived therein. [ABSTRACT FROM AUTHOR]