######
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.

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

## Abstract

Let *τ _{n}* be a type of algebras in which all operation symbols have arity

*n*, for a fixed

*n*≥ 1. For 0

*< r*≤

*n*, this paper introduces a special kind of

*n*-ary terms of type

*τ*called

_{n}*K**(

*n, r*)

*-full terms*. The set of all

*K**(

*n, r*)-full terms of type

*τ*is closed under the superposition operation

_{n}*S*; hence forms a clone denoted by

^{n}*clone*

_{K*(n,r)}(

*τ*). We prove that

_{n}*clone*

_{K* (n,r)}(

*τ*) is a Menger algebra of rank

_{n}*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

*clone*

_{K}_{* (n,r)}(

*τ*) and

_{n}*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].

###### 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.

###### 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*)= *{µ*(*Z*): *Z* ∈ ℱ, *Z* ⊆ *Y }*. For a given *p* ∈ *µ*(ℱ) consider a family of measurable subsets ℱ* _{p}* =

*{Z*∈ ℱ :

*µ*(

*Z*)=

*p}.*Dai and Feinberg proved the existence of a maximal subset

*Z**∈

*F*having the maximal subrange

_{p}*R*(

*Z**) and also a minimal subset

*M**∈ ℱ

*with the minimal subrange*

_{p}*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.

######
*I*-Completeness in Function Spaces

## Abstract

In this paper, we have studied the idea of ideal completeness of function spaces *Y ^{X}* 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

*Y*and uniformity of uniform convergence on

^{X}*Y*in terms of

^{X}*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.

###### 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.

###### 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.

###### Monoids of ND-Full Hypersubstitutions

## Abstract

An nd-full hypersubstitution maps any operation symbols to the set of full terms of type *τ _{n}*. 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.

###### On a Lindenbaum Composition Theorem

## Abstract

We discuss several questions about Borel measurable functions on a topological space. We show that two Lindenbaum composition theorems [Lindenbaum, A. *Sur les superpositions des fonctions représentables analytiquement*, Fund. Math. **23** (1934), 15–37] proved for the real line hold in perfectly normal topological space as well. As an application, we extend a characterization of a certain class of topological spaces with hereditary Jayne-Rogers property for perfectly normal topological space. Finally, we pose an interesting question about lower and upper Δ^{0}
_{2}-measurable functions.