## Abstract

It was proved by Jang et al. that various chains of one-parameter distributions converge to Benford’s law. We study chains of truncated distributions and propose another approach, using a recent convergence result of the Lerch transcendent function, to proving that they converge to Benford’s law for initial Beta distributions with parameters *α* and 1.

## 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.

## 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].

## Abstract

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

## Abstract

Let *q* be an integer greater than or equal to 2, and let *S*
* _{q}*(

*n*)denote the sum of digits of

*n*in base

*q*.For

let *S*
_{α}(*n*) denote the sum of digits in the Ostrowski *α*-representation of *n*. Let *m*
_{1},*m*
_{2} ≥ 2 be integers with

We prove that there exists *δ>* 0 such that for all integers *r*
_{1},*r*
_{2},

The asymptotic relation implied by this equality was proved by Coquet, Rhin & Toffin and the equality was proved for the case

## 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.

## 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.

## 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*).

## Abstract

Let *X* and *Y* be nonempty finite subsets of and *X* +*Y* its sumset. The structures of *X* and *Y* when *r*(*X, Y* ):= |*X* +*Y* |−|*X*|−|*Y* | is small have been widely studied; in particular the Generalized Freiman’s 3*k* − 4 Theorem describes *X* and *Y* when *r*(*X, Y* ) ≤ min{|*X*|, |*Y* |} − 4. However, not too much is known about *X* and *Y* when *r*(*X, Y* ) *>* min{|*X*|, |*Y* |} − 4. In this paper we study the structure of *X* and *Y* for arbitrary *r*(*X, Y* ).

## 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.