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Characterizations of Weakly Ordered k-Regular Hemirings by k-Ideals

Abstract

We study the concepts of left weakly ordered k-regular and right weakly ordered k-regular hemirings and give some of their characterizations using many types of their k-ideals.

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The Clone of K*(n, r)-Full Terms

Abstract

Let τn be a type of algebras in which all operation symbols have arity n, for a fixed n ≥ 1. For 0 < rn, this paper introduces a special kind of n-ary terms of type τn called K*(n, r)-full terms. The set of all K*(n, r)-full terms of type τn is closed under the superposition operation Sn; hence forms a clone denoted by clone K*(n,r)(τn). We prove that clone K* (n,r)(τn) is a Menger algebra of rank n. We study K*(n, r)-full hypersubstitutions and the related K*(n, r)-full closed identities and K*(n, r)-full closed varieties. A connection between identities in cloneK * (n,r)(τn) and K* (n, r)-full closed identities is established. The results obtained generalize the results of Denecke and Jampachon [K. Denecke and P. Jampachon, Clones of full terms, Algebra and Discrete Math. 4 (2004) 1–11].

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Generalized Derivative and Generalized Continuity

Abstract

The main objective of this article is to show that generalized differentiation can be understood as a process of comparing functions and their generalized continuity properties. We show it by working with generalized notions of derivative and continuity. The article covers wide range of types of generalized continuity.

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How to Obtain Maximal and Minimal Subranges of Two-Dimensional Vector Measures

Abstract

Let (X, ℱ) be a measurable space with a nonatomic vector measure µ =(µ 1, µ 2). Denote by R(Y) the subrange R(Y)= (Z): Z ∈ ℱ, ZY }. For a given pµ(ℱ) consider a family of measurable subsets ℱp = {Z ∈ ℱ : µ(Z)= p}. Dai and Feinberg proved the existence of a maximal subset Z*Fp having the maximal subrange R(Z*) and also a minimal subset M* ∈ ℱp with the minimal subrange R(M*). We present a method of obtaining the maximal and the minimal subsets. Hence, we get simple proofs of the results of Dai and Feinberg.

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I-Completeness in Function Spaces

Abstract

In this paper, we have studied the idea of ideal completeness of function spaces YX with respect to pointwise uniformity and uniformity of uniform convergence. Further, involving topological structure on X,wehaveobtained relationships between the uniformity of uniform convergence on compacta on YX and uniformity of uniform convergence on Y X in terms of I-Cauchy condition and I-convergence of a net. Also, using the notion of a k-space, we have given a sufficient condition for C(X, Y) to be ideal complete with respect to the uniformity of uniform convergence on compacta.

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An Injective Pseudo-BCI Algebra is Trivial

Abstract

Injective pseudo-BCI algebras are studied. There is shown that the only injective pseudo-BCI algebra is the trivial one.

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Ján Borsík (1955 – 2019)
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ℒ-Fuzzy Ideals of Residuated Lattices

Abstract

This paper mainly focuses on building the ℒ-fuzzy ideals theory of residuated lattices. Firstly, we introduce the notion of ℒ-fuzzy ideals of a residuated lattice and obtain their properties and equivalent characterizations. Also, we introduce the notion of prime fuzzy ideal, fuzzy prime ideal and fuzzy prime ideal of the second kind of a residuated lattice and establish existing relationships between these types of fuzzy ideals. Finally, we investigate the notions of fuzzy maximal ideal and maximal fuzzy ideal of a residuated lattice and present some characterizations.

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Monoids of ND-Full Hypersubstitutions

Abstract

An nd-full hypersubstitution maps any operation symbols to the set of full terms of type τn. Nd-full hypersubstitutions can be extended to mappings which map sets of full terms to sets of full terms. The aims of this paper are to show that the extension of an nd-full hypersubstitution is an endomorphism of some clone and that the set of all nd-full hypersubstitutions forms a monoid.

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