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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]

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, 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]

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]

Let S and T be two numerical semigroups. We say that T is an I (S) -semigroup if T ∖ { 0 } is an ideal of S. Given k a positive integer, we denote by Δ (k) the symmetric numerical semigroup generated by { 2 , 2 k + 1 } . In this paper we present a formula which calculates the number of I (S) -semigroups with genus g (Δ (k)) + h for some nonnegative integer h and which we will denote by i (Δ (k) , h) . As a consequence, we obtain that the sequence { i (Δ (k) , h) } h ∈ N is never decreasing. Besides, it becomes stationary from a certain term. [ABSTRACT FROM AUTHOR]

In this paper, we give a mathematical model of the logic of determination of objects (LDO) based on preordered sets, and a mathematical model of the logic of typical and atypical instances (LTA). We prove that LTA is an extension of LDO. It can manipulate several types of "exceptions". Finally, we show that the structural part of LTA can be modeled by a quasi topology structure (QTS). [ABSTRACT FROM AUTHOR]

This article is devoted to normed simple annihilator algebras. Their structure is investigated in the paper. Maximal families of orthogonal irreducible idempotents of normed simple annihilator algebras are scrutinized. Division subalgebras of annihilator algebras are studied. Realizations of these algebras by operator algebras in Banach spaces are described. For this purpose, quasi finite dimensional operators are investigated. [ABSTRACT FROM AUTHOR]

This article is devoted to the investigation of dual and annihilator normed algebras. Their structure is studied in the paper. Extensions of algebras and fields are considered and by using them, core radicals and radicals are investigated. Moreover, for this purpose ∗-algebras and finely regular algebras are also studied. Relations with operator theory and realizations of these algebras by operator algebras are outlined. [ABSTRACT FROM AUTHOR]

Let K n be a complete graph on n vertices. Denote by S K n the set of all subgraphs of K n . For each G , H ∈ S K n , the ring sum of G and H is a graph whose vertex set is V ( G ) ∪ V ( H ) and whose edges are that of either G or H, but not of both. Then S K n is a semigroup under the ring sum. In this paper, we study Green’s relations on S K n and characterize ideals, minimal ideals, maximal ideals, and principal ideals of S K n . Moreover, maximal subsemigroups and a class of maximal congruences are investigated. Furthermore, we prescribe the natural order on S K n and consider minimal elements, maximal elements and covering elements of S K n under this order. [ABSTRACT FROM AUTHOR]