In this paper, a commutative semigroup will be written as a disjoint union of its cancellative subsemigroups. Based on this fact we will define the left regular representation of a commutative separative semigroup and show that this representation is faithful. Finally concrete examples of commutative separative semigroups, their decompositions and their left regular representations are given.

We prove higher integrability of the spatial gradient weak solutions to parabolic systems with $\phi$-growth, where $\varphi=\varphi(t)$ is a general Orlicz function. The need be neither degenerate nor singular. Our result generalized version one J. Kinnunen and Lewis [Duke Math. 102 (2000), no. 2, 253--271] for $p$-Laplace systems.

‎‎let r be a non-commutative ring with unity‎. ‎the commuting graph‎ ‎of $r$ denoted by $gamma(r)$‎, ‎is a graph with a vertex set‎ ‎$rsetminus z(r)$ and two vertices $a$ and $b$ are adjacent if and only if‎ ‎$ab=ba$‎. ‎in this paper‎, ‎we investigate non-commutative rings with unity of order $p^n$ where $p$ is prime and $n in lbrace 4,5 rbrace$‎. it is shown that‎, ‎$gamma(r)$ is the disjoint ...

The aim of this paper is to present some results about the space $L^{\varPhi }(\nu ),$ where $\nu$ a vector measure on compact (not necessarily abelian) group and $\varPhi$ Young function. We show that under natural conditions, )$ becomes an $L^{1}(G)$ -module with respect usual convolution functions. also define one more structure ).$

In this short paper, we study sums of the shape $$\sum _{n\leqslant x}{f([x/n])}/{[x/n]},$$ where f is Euler totient function $$\varphi $$ , Dedekind $$\Psi sum-of-divisors $$\sigma or alternating $$\beta .$$ We improve previous results when $$f=\varphi and derive new estimates $$f=\Psi f=\sigma $$f=\beta

let $r$ be a commutative ring. in this paper, by using algebraic properties of $r$, we study the hase digraph of prime ideals of $r$.

Abstract This article is dedicated to the study of singular codimension-one foliations ℱ {\mathcal{F}} on a simplicial complete toric variety X and their pullbacks by dominant rational maps φ : ℙ n ⇢ ...