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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]

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]

Recently Bukovský, Das and Šupina [Ideal quasi-normal convergence and related notions, Colloq. Math. 146 (2017), 265–281] started the study of sequence selection properties (𝓘, 𝓙-α1) and (𝓘, 𝓙-α4) of Cp(X) using the double ideals, where 𝓘 and 𝓙 are the proper admissible ideals of ω, which are motivated by Arkhangeľskii local αi-properties [The frequency spectrum of a topological space and the classification of spaces, Dokl. Akad. Nauk SSSR 13 (1972), 1185–1189]. In this paper, we obtain some characterizations of (𝓘, 𝓙-α1) and (𝓘, 𝓙-α4) properties of Cp(X) in the terms of covering properties and selection principles. Under certain conditions on ideals 𝓘 and 𝓙, we identify the minimal cardinalities of a space X for which Cp(X) does not have (𝓘, 𝓙-α1) and (𝓘, 𝓙-α4) properties. [ABSTRACT FROM AUTHOR]

Let R be a prime ring with involution ⋆, and let σ, τ be endomorphisms on R. For any x , y ∈ R, let (x, y)σ, τ = xσ (y) + τ(y)x and Cσ, τ(R) = {x ∈ R∣ xσ(y) = τ(y)x. An additive subgroup U of R is said to be a (σ, τ)-right Jordan ideal (resp. (σ, τ-left Jordan ideal) of R if (U, R)σ, τ ⊆ U (resp. (R, U)σ, τ ⊆ U, and U is called a (σ, τ)-Jordan ideal if U is both a (σ, τ)-right Jordan ideal and a (σ, τ)-left Jordan ideal of R. A (σ, τ)-Jordan ideal U of R is said to be a (σ, τ)- ⋆-Jordan ideal if U ⋆ = U. In the present paper, it is shown that if U is commutative, then R is commutative. The commutativity of R is also obtained if (U , U)σ, τ ⊆ Cσ, τ(R). Some more results are obtained on the ⋆-prime ring with a characteristic different from 2. [ABSTRACT FROM AUTHOR]

Basic algebras were introduced by Chajda, Halaš and Kühr as a common generalization of MV-algebras and orthomodular lattices, i.e. algebras used for formalization of non-classical logics, in particular the logic of quantum mechanics. These algebras were represented by means of lattices with section involutions. On the other hand, classical logic was formalized by means of Boolean algebras which can be converted into Boolean rings. A natural question arises if a similar representation exists also for basic algebras. Several attempts were already realized by the authors, see the references. Now we show that if a basic algebra is commutative then there exists a representation via certain semirings with involution similarly as it was done for MV-algebras by Belluce, Di Nola and Ferraioli. These so-called basic semirings, their ideals and congruences are studied in the paper. [ABSTRACT FROM AUTHOR]

Weighted uniform densities are a generalization of the uniform density, which is also known as the Banach density. In this paper, we introduce the concept of weighted uniform density ideals and consider the topological complexity of these ideals as well as when they have certain analytical properties related to the ideal convergence of sequences and series. Furthermore, we prove some inequalities between different upper and lower weighted uniform densities and give the answer to the problem concerning the Darboux property of these densities. [ABSTRACT FROM AUTHOR]

In this paper, the notion of derivation on partially ordered sets is introduced and studied. Several characterization theorems on derivations are presented. The properties of the fixed points based on the derivations are investigated. Finally, the properties of ideals and operations related with derivations are studied. [ABSTRACT FROM AUTHOR]

Let R be a ring and I is a nonzero ideal of R. A mapping F : R → R is called a multiplicative generalized derivation if there exists a mapping g : R → R such that F(xy) = F(x)y + xg(y), for all x, y ∈ R. In the present paper, we shall prove that R contains a nonzero central ideal if any one of the following holds: i) F([x, y]) = 0, ii) F(xoy) = 0, iii) F([x, y]) = ±[x, y], iv) F(xoy) = ±(xoy), v) F([x, y]) = ±(xoy), v) F([x, y]) = ±(xoy), for all x, y ∈ I. vii) F([x, y]) = ±[F(x), y], viii) F(xoy) = ±(F(x)oy), ix) F(xy) ± xy ∈ Z, x) F(xy) ± yx ∈ Z, xi) F(xy) ± [x, y] ∈ Z, xii) F(xy) ± (xoy) ∈ Z,. [ABSTRACT FROM AUTHOR]

In the present paper we introduce some applications of I-statistical convergence to form some new spaces of sequences using a sequence of Orlicz functions and matrix transformation over n-normed spaces. We also make an effort to study several properties relevant to the topological and algebraic structure of these spaces. We investigate some inclusion relations involving these spaces. [ABSTRACT FROM AUTHOR]