######
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].

###### 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 Equality of Certain Derivations of Lie Algebras

## Abstract

Let *L* be a Lie algebra. A derivation α of *L* is a commuting derivation (central derivation), if α (*x*) ∈ *CL* (*x*) (α (*x*) ∈ *Z* (*L*)) for each *x* ∈ *L*. We denote the set of all commuting derivations (central derivations) by 𝒟 (*L*) (*Derz* (*L*)). In this paper, first we show 𝒟 (*L*) is subalgebra from derivation algebra *L*, also we investigate the conditions on the Lie algebra *L* where commuting derivation is trivial and finally we introduce the family of nilpotent Lie algebras in which *Der _{z}* (

*L*) = 𝒟 (

*L*).

######
On Properties of *t*-Fuzzy Modules and Uniform *t*-Fuzzy Modules

## Abstract

In this paper, we extend the idea of fuzzy modules and uniform fuzzy modules to the concepts of *t*-fuzzy modules and uniform *t*-fuzzy modules, respectively. We give some characterizations and properties of *t*-fuzzy modules and uniform *t*-fuzzy modules.

###### On the Genus of the Co-Annihilating Graph of Commutative Rings

## Abstract

Let *R* be a commutative ring with identity and 𝒰* _{R}* be the set of all nonzero non-units of

*R*. The

*co-annihilating graph*of

*R*, denoted by 𝒞𝒜

*, is a graph with vertex set 𝒰*

_{R}*and two vertices*

_{R}*x*and

*y*are adjacent whenever

*ann*(

*x*) ∩

*ann*(

*y*) = (0). In this paper, we characterize all commutative Artinian non-local rings

*R*for which the 𝒞𝒜

*has genus one and two. Also we characterize all commutative Artinian non-local rings*

_{R}*R*for which 𝒞𝒜

*has crosscap one. Finally, we characterize all finite commutative non-local rings for which*

_{R}*g*(Г

_{2}(

*R*)) =

*g*(𝒞𝒜

*) = 0 or 1.*

_{R}###### Topological Up-Algebras

## Abstract

In this paper, we introduce the notion of topological UP-algebras and several types of subsets of topological UP-algebras, and prove the generalization of these subsets. We also introduce the notions of quotient topological spaces of topological UP-algebras and topological UP-homomorphisms. Furthermore, we study the relation between topological UP-algebras, Hausdor spaces, discrete spaces, and quotient topological spaces, and prove some properties of topological UP-algebras.

###### Uniqueness Theorem in Complete Residuated Almost Distributive Lattices

## Abstract

Important properties of primary elements in a complete residuated ADL *L* and the uniqueness theorem in a complete complemented residuated ADL *L* are proved.