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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]

In this paper, the notion of minimal prime ideal is introduced in residuated lattices and related properties are investigated. Also, new equivalent characterizations and properties for prime and maximal ideals are obtained and the relation between these ideals and minimal prime ideals is discussed for De Morgan residuated lattices. Moreover, we prove that it is possible to introduce and study, by a standard way, Zariski topology on the lattice P (L) of prime ideals of any residuated lattice L. Also, since m P (L) , the set of minimal prime ideals of L , and M (L) , the set of maximal ideals of L , are subsets of P (L) , we endow m P (L) and M (L) with the topology induced by the Zariski topology on P (L) and we characterize these topological spaces for residuated lattices. [ABSTRACT FROM AUTHOR]

This paper is devoted to studies of I wQN-spaces and some of their cardinal characteristics. Recently, Šupina in [32] proved that I is not a weak P-ideal if and only if any topological space is an I QN-space. Moreover, under p = c he constructed a maximal ideal I (which is not a weak P-ideal) for which the notions of I QN-space and QN-space do not coincide. In this paper we show that, consistently, there is an ideal I (which is not a weak P-ideal) for which the notions of I wQN-space and wQN-space do not coincide. This is a partial solution to [6, Problem 3.7] . We also prove that for this ideal the ideal version of Scheepers Conjecture does not hold (this is the first known example of such weak P-ideal). We obtain a strictly combinatorial characterization of non ( I wQN-space ) similar to the one given in [32] by Šupina in the case of non ( I QN-space ) . We calculate non ( I QN-space ) and non ( I wQN-space ) for some weak P-ideals. Namely, we show that b ≤ non ( I QN-space ) ≤ non ( I wQN-space ) ≤ d for every weak P-ideal I and that non ( I QN-space ) = non ( I wQN-space ) = b for every F σ ideal I as well as for every analytic P-ideal I generated by an unbounded submeasure (this establishes some new bounds for b ( I , I , Fin ) introduced in [31] ). As a consequence, we obtain some bounds for add ( I QN-space ) . In particular, we get add ( I QN-space ) = b for analytic P-ideals I generated by unbounded submeasures. By a result of Bukovský, Das and Šupina from [6] it is known that in the case of tall ideals I the notions of I QN-space ( I wQN-space) and QN-space (wQN-space) cannot be distinguished. Answering [6, Problem 3.2] , we prove that if I is a tall ideal and X is a topological space of cardinality less than co v ⁎ ( I ) , then X is an I wQN-space if and only if it is a wQN-space. [ABSTRACT FROM AUTHOR]

The paper deals with an algebraic extension of MV-algebras based on the definition of generalized Boolean algebras. We introduce a new class of structures, not necessarily with a top element, which are called EMV-algebras, in a way that every EMV-algebra contains an MV-algebra. First, we present basic properties of EMV-algebras. We give some examples, introduce and investigate congruence relations, ideals and filters on these algebras. We establish a basic representation result saying that each EMV-algebra can be embedded into an EMV-algebra with top element and we characterize EMV-algebras either as structures which are termwise equivalent to MV-algebras or as maximal ideals of EMV-algebras with top element. We study the lattice of ideals of an EMV-algebra and prove that every EMV-algebra has at least one maximal ideal. We define an EMV-clan of fuzzy sets as a special EMV-algebra where all operations are defined by points. We show that any semisimple EMV-algebra is isomorphic to an EMV-clan of fuzzy functions on a set. The set of EMV-algebras is neither a variety nor a quasivariety, but rather a special class of EMV-algebras which we call a q-variety of EMV-algebras. We present an equational base for each proper q-subvariety of the q-variety of EMV-algebras. We establish categorical equivalencies among the category of proper EMV-algebras, the category of MV-algebras with a fixed special maximal ideal, and a special category of Abelian unital ℓ -groups. [ABSTRACT FROM AUTHOR]

Abstract In this paper we primarily introduce certain new versions of groupability, namely star- I -groupability which is introduced using both star operation and the notion of ideals and then a more general notion of R -groupability. We establish several results regarding these groupability notions in the process of extending certain existing results on groupability. [ABSTRACT FROM AUTHOR]

We consider four classes of topological spaces which are defined with the aid of convergence with respect to ideals on N . All these classes are subclasses of countably compact spaces, and two of them are also subclasses of sequentially compact spaces. In the first part of the paper (Sections 1 and 2 ) we prove some properties of these classes. In the second part of the paper (Sections 3 and 4 ) we focus on spaces defined by two particular ideals connected with well known theorems in combinatorics, namely van der Waerden's theorem and Hindman's theorem. The main aim of this part of the paper is to show that two classes of the considered spaces coincide for the van der Waerden ideal and Hindman ideal respectively. [ABSTRACT FROM AUTHOR]

In this paper we primarily consider the ideal analogue of Hurewicz property introduced recently by Das in [6] and subsequently introduce the ideal analogues of several versions of this very important property studied in the last decade. We establish certain inter relationships between the new notions presenting suitable examples as far as possible as well as prove many related results and in particular preservation properties. [ABSTRACT FROM AUTHOR]

In order to accurately, robustly and efficiently simulate the three-dimensional (3D) interfacial flow problems with large density and viscosity ratio, we should, on the one hand, adopt the accurate and robust interface capturing method, on the other hand, use the efficient solution algorithm for coupling the velocity and pressure. In this paper, a coupled volume-of-fluid and level set (VOSET) method is applied to capture the two-phase interface, while an inner doubly iterative efficient algorithm for linked equations (IDEAL) is used to couple the velocity and pressure. Both of the VOSET method and the IDEAL algorithm were first proposed by the present author and further extended by other authors. Finally, the superiority of the VOSET method combined with the IDEAL algorithm (VOSET + IDEAL) is verified by three interfacial flow problems with large density and viscosity ratio. [ABSTRACT FROM AUTHOR]

• Numerical performances of IDEAL and SIMPLERR are analyzed and compared. • SIMPLERR can be regarded as a special form of IDEAL. • The robustness of IDEAL is much better than SIMPLERR. • In general, the performance of IDEAL is superior to SIMPLERR. In this paper, the numerical performances of the algorithm IDEAL proposed by the present authors in 2008, and the recently proposed (2017) SIMPLERR are systematically analyzed and compared via six incompressible fluid flow and heat transfer problems. It is found that the two algorithms are almost identical when the second inner iteration number of IDEAL is fixed at 2. In general, the robustness and convergence of IDEAL are much better than SIMPLERR. The reason why the performance of SIMPLERR deteriorates with the increase of the first inner iteration number is analyzed and detailed comparisons in the robustness and computing time are presented. [ABSTRACT FROM AUTHOR]

In this paper, the notions of (para, quasi, semi) topological MV-algebras are defined and their related properties are studied. Also, topologies with which an MV-algebra can be a (para, semi) topological MV-algebra are obtained. Clearly, a topological MV-algebra is a (para, quasi, semi) topological MV-algebra, but the converse is not true, as shown by an example. In addition, we study ideals and filters in (para, quasi) topological MV-algebras, and we show that a quasitopological MV-algebra is a topological MV-algebra if the ideal {0}, or equivalently, the filter {1} is open. [ABSTRACT FROM AUTHOR]

(Bounded integral) residuated lattices (which need not be commutative) form a large class of algebras containing some classes of algebras behind many-valued and fuzzy logics. Congruences of such algebras are usually defined and investigated by means of their normal filters. In the paper we introduce and investigate ideals of residuated lattices. We show that one can define, in some cases, congruences also using ideals and that the corresponding quotient residuated lattices are involutive. [ABSTRACT FROM AUTHOR]

In this paper we discuss the following Tkachuk's question in the sense of ideal convergence [16,18] : Is any Tychonoff connected sequential space a quotient image of a connected metric space? It is proved that let I be an ideal on the set N then a topological space X is an I -connected space with an I - csf -network if and only if X is a continuous I -covering image of a connected metric space. It follows that a topological space X is a connected I -sequential space with an I - csf -network if and only if X is a quotient I -covering image of a connected metric space. [ABSTRACT FROM AUTHOR]

Following the concept of topological theory of S.E. Rodabaugh, this paper introduces a new approach to (lattice-valued) bornology, which is based in bornological theories, and which is called variety-based bornology. In particular, motivated by the notion of topological system of S. Vickers, we introduce the concept of variety-based bornological system, and show that the category of variety-based bornological spaces is isomorphic to a full reflective subcategory of the category of variety-based bornological systems. [ABSTRACT FROM AUTHOR]

We study the eigenscheme of a matrix which encodes information about the eigenvectors and generalized eigenvectors of a square matrix. The two main results in this paper are a decomposition of the eigenscheme of a matrix into primary components and the fact that this decomposition encodes the numeric data of the Jordan canonical form of the matrix. We also describe how the eigenscheme can be interpreted as the zero locus of a global section of the tangent bundle on projective space. This interpretation allows one to see eigenvectors and generalized eigenvectors of matrices from an alternative viewpoint. [ABSTRACT FROM AUTHOR]

In this paper we follow the line of recent works of Das [4,5,7] (see also [3] for similar investigation), where a more general approach was made to study certain results on open covers and selection principles by using the notion of ideals and ideal convergence, which automatically extends similar classical results (where finite sets are used), and also recent statistical variants studied by Di Maio and Kočinac [10] . Here we further introduce the notions of cω -covers and I -large covers which extend the notions of ω -covers and large covers in a topological space. We then study the I -groupability of different open covers and some of its consequences. [ABSTRACT FROM AUTHOR]

The Tamari lattice of order n can be defined on the set T n of binary trees endowed with the partial order relation induced by the well-known rotation transformation. In this paper, we restrict our attention to the subset M n of Motzkin trees. This set appears as a filter of the Tamari lattice. We prove that its diameter is 2 n − 5 and that its radius is n − 2 . Enumeration results are given for join and meet irreducible elements, minimal elements and coverings. The set M n endowed with an order relation based on a restricted rotation is then isomorphic to a ranked join-semilattice recently defined in Baril and Pallo (2014). As a consequence, we deduce an upper bound for the rotation distance between two Motzkin trees in T n which gives the exact value for some specific pairs of Motzkin trees. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we primarily consider the notion of -covers recently introduced in [11] as well as related notion of -covers [9] and make some further investigations. In particular we consider the notion of κ-covers studied by Di Maio, Kočinac, Meccariello and Caserta in [8,18], and establish some results involving -covers ( -covers) and κ-covers. Our results extend the earlier results proved in [6,8,18,29] and present a more general version using the notion of ideals. [Copyright &y& Elsevier]

Let L be a completely regular frame and, as customary, let R L denote the ring of continuous real-valued functions on L. In the first part of the paper we characterize the maximal regular ideal of R L. We show that it consists precisely of the functions α such that the open sublocale of L associated with coz α is clopen and is a P -frame. We also give a characterization of this ideal in terms of the notion of the localic remainder. In contrast, the maximal regular ideal of the ring Z L of continuous integer-valued functions on L is shown to be always the zero ideal. Purity and regularity of ideals are germane in the description and characterization of the maximal regular ideal of R L. It is for this reason that in the last part of the paper we consider ideals of ideals of R L because, as we prove, it is precisely the pure ideals of R L all of whose ideals are ideals in the whole ring. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce the generalized ideal based zero divisor graph of a poset , denoted by . A representation theorem is obtained for generalized zero divisor graphs. It is proved that a graph is complete -partite with if and only if it is a generalized zero divisor graph of a poset. As a consequence of this result, we prove a form of a Beck’s Conjecture for generalized zero divisor graphs of a poset. Further, it is proved that a generalized zero divisor graph of a section semi-complemented poset with respect to the ideal is a complete graph. [Copyright &y& Elsevier]

Abstract: In this paper we make some further investigations on ideal convergence and in particular we concentrate on -limit points and -cluster points. We try to establish the characterization of the set of -limit points (which has not been done in any structure so far) and show that this set can be characterized as an -set for a large class of ideals, namely analytic P-ideals and then make certain interesting observations on -cluster points. [Copyright &y& Elsevier]

Abstract: Multipliers and factorizations for bounded statistically convergent sequences were studied by Connor et al. [J. Connor, K. Demirci, C. Orhan, Multipliers and factorizations for bounded statistically convergent sequences, Analysis 22 (2002) 321–333] and for bounded -convergent sequences by Yardımcı [Ş. Yardımcı, Multipliers and factorizations for bounded -convergent sequences, Math. Commun., 11 (2006) 181–185]. In this paper, we get analogous results of multipliers for bounded -convergent double sequences. [Copyright &y& Elsevier]

Abstract: The paper deals with states on commutative basic algebras that are a non-associative generalization of MV-algebras or, in other words, the algebraic semantics for a fuzzy logic which generalizes the Łukasiewicz logic in that the conjunction is not associative. States are defined in the same way as Mundici''s states on MV-algebras as normalized finitely additive [0,1]-valued functions, and some results analogous to the results that are known from MV-algebras are proved. [Copyright &y& Elsevier]

Abstract: In this paper we introduce certain new sequence spaces of fuzzy numbers defined by -convergence using sequences of Orlicz functions and a difference operator of order . We study some basic topological and algebraic properties of these spaces. Also we investigate the relations related to these spaces. [Copyright &y& Elsevier]

Abstract: An efficient segregated algorithm for non-linear incompressible fluid flow and heat transfer problems, called IDEAL (Inner Doubly-Iterative Efficient Algorithm for Linked-Equations) for short, was proposed in reference [9]. Subsequently, it was extended to the 3D staggered/collocated grid systems. IDEAL includes inner doubly-iterative processes for solving pressure equations at each iteration level, and it could adjust the inner iteration times to control the convergence rate and the stability of iteration process, which is greatly different from other segregated algorithms. The objective of this paper is to analyze the effects of inner iteration times on the performance of IDEAL by four incompressible fluid flow problems, two of which belong to open systems, and the others refer to closed systems. It is found that: (1) the robustness of IDEAL is enhanced greatly with the increase of inner iteration times; (2) at the same time step multiple, the outer iteration number decreases with the increase of inner iteration times and the computation time is not related to the inner iteration times; (3) at the optimal time step multiple, the large inner iteration times of 4&4 and 7&7 could reduce the outer iteration number by about 70% and the computation time by about 40% over the small inner iteration times of 1&1. [Copyright &y& Elsevier]

Abstract: In this paper, following the line of Savas and Das (2011) , we provide a new approach to two well-known summability methods by using ideals, introduce new notions, namely, -statistical convergence and -lacunary statistical convergence, investigate their relationship, and make some observations about these classes. [Copyright &y& Elsevier]

Abstract: In this paper we continue our investigation of nets using ideals in line of our earlier work where we had studied -Cauchy nets and asked when -Cauchy nets in complete uniform spaces are -convergent in line of a problem by Di Maio and Kočinac who asked in 2008 when statistically Cauchy sequences are statistically convergent in uniform spaces. To answer this, here we consider another type of Cauchy condition of nets, namely -Cauchy condition and examine its basic properties and in particular its relation with the concept of -Cauchy nets. This helps us to give an answer to the above mentioned open question. [Copyright &y& Elsevier]

Abstract: In this paper we make a new approach to the notions of -summability and -statistical convergence by using ideals and introduce new notions, namely, -summability and -statistical convergence. We mainly examine the relation between these two new methods as also the relation between -statistical convergence and -statistical convergence introduced by the authors recently. We carry out the whole investigation in normed linear spaces. [Copyright &y& Elsevier]

Abstract: With L a complete lattice and M a continuous lattice, this paper demonstrates an adjunction between M -valued L-topological spaces (i.e. (L,M )-topological spaces) and Idl(M )-valued L-topological spaces where Idl(M ) is the complete lattice of all ideals of M . It is shown that the right adjoint functor provides a procedure of generating (L,M )-topologies from antitone families of (L,M )-topologies. This procedure is then applied to give an internal characterization of joins in the complete lattice of all (L,M )-topologies on a given set. [Copyright &y& Elsevier]

Abstract: In this paper we extend the idea of usual Cauchy condition of nets to I-Cauchy condition by using the concept of ideals. This Cauchy condition arises naturally from the notion of I-convergence of nets introduced by Lahiri and Das (2008). As the underlying structure for the whole study we take a uniform space so that our notion and results extend the idea of statistical Cauchy sequences very recently introduced in uniform spaces by Di Maio and Kočinac (2008). In particular we try to give partial answers to an open problem posed by Di Maio and Kočinac and examine the relationship between this new Cauchy condition and usual completeness of a uniform space. [Copyright &y& Elsevier]

Abstract: Given an infinite group G and an infinite cardinal , we say that a subset A of G is κ-large (κ-small) if there exists such that ( is κ-large for each ). The subject of the paper is the family of all κ-small subsets. We describe the left ideal of the right topological semigroup βG determined by . We study interrelations between κ-small and other (-small and κ-thin) subsets of groups, and prove that G can be generated by some 2-thin subsets. We partition G in countable many subsets which are κ-small for each . We show that is dual to provided that either κ is regular and , or G is Abelian and κ is a limit cardinal, or G is a divisible Abelian group. [Copyright &y& Elsevier]

We denote by C s (X) the set C (X) of real-valued continuous functions defined on X endowed with the topology of the uniform convergence on the closed separable subspaces of X. In this paper we continue the study of C s (X) initiated in Pseudouniform topologies on C (X) given by ideals (Pichardo-Mendoza et al. (2013) [6]). We prove that C s (X) is a k -space if and only if C s (X) is metrizable, and that compactness, sequential compactness and countable compactness coincide in subspaces of C s (X). In addition, we study the cellularity, density, weight and character of C s (X). We prove that (1) d ((R 2 λ ) s) = λ if λ = λ ω , and (2) the Continuum Hypothesis is equivalent to the statement: Every non-separable space X satisfies χ (C s (X)) = w (C s (X)). [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose the application of formal methods to Software Engineering. The most used data model is the relational model and we present, within the general framework of lattice theory, this analysis of functional dependencies. For this reason, we characterize the concept of -family by means of a new concept which we call non-deterministic ideal operator (nd.ideal-o). The study of nd.ideal-o.s allows us to obtain results about functional dependencies as trivial particularizations, to clarify the semantics of the functional dependencies and to progress in their efficient use, and to extend the concept of schema. Moreover, the algebraic characterization of the concept of Key of a schema allows us to propose new formal definitions in the lattice framework for classical normal forms in relation schemata. We give a formal definition of the normal forms for functional dependencies more frequently used in the bibliography: the second normal form (2FN), the third normal form(3FN) and Boyce–Codd''s normal form (FNBC). [Copyright &y& Elsevier]

Let X be a Tychonoff space. Associated with every subset A ⊆ β X is the ideal O A of the ring C (X) consisting of all functions that vanish in a neighborhood of X ∩ A. Now, viewing Tychonoff spaces as objects in the category CRLoc of completely regular locales, we have ideals of the form O A , where A is a sublocale of βX. In this paper we study properties of such ideals not only for Tychonoff spaces, but for any object in CRLoc. Carrying out the discussion in this category, we have more function rings (they are denoted R L , for any L ∈ CRLoc) than the class of the rings C (X). Pure ideals of C (X) are known to be exactly the ideals O A , for A a closed subset of βX. We characterize the spaces for which the ideals O A , for A a closed subset of X (note that it need not be closed in βX) are pure ideals. They properly contain the normal ones. We describe the socle of any ring R L as the ideal O A , with A equal to the join of all nowhere dense sublocales of βL. We show that the socle is zero precisely when βL is dense in itself, and essential if and only if the smallest dense sublocale of βL has a complement in the lattice of sublocales of βL. If βL is scattered, then this is so if and only if L has a smallest nowhere dense sublocale. [ABSTRACT FROM AUTHOR]