Cancellative semigroup

About: Cancellative semigroup is a research topic. Over the lifetime, 1320 publications have been published within this topic receiving 13319 citations.

Semigroup compactifications in terms of filters

Abstract: We present a study of semigroup compactifications of a semitopological semigroup $S$ using certain filters on $S$. We characterize closed subsemigroups and closed left, right, and two-sided ideals in any semigroup compactification of any semitopological semigroup $S$ in terms of these filters and in terms of ideals of the corresponding $m$-admissible subalgebra of $C(S)$. Furthermore, we characterize those points in any semigroup compactification of $S$ which belong either to the smallest ideal of the semigroup compactification or to the closure of this smallest ideal.

Applications of semigroup algebras to ideal extensions of semigroups

Abstract: Let D be a semigroup and Z the integers modulo p, P where p is a prime. In this article we show that information concerning the semigroup algebras Z D of D P over Z provide information concerning a class of P ideal extensions of D.

On Generalized Semigroup Near-Rings

Abstract: Using Meldrum and van der Walt's scheme we successfully define (generalized) semigroup near-rings which are the extensions of their ring counterpart. Some standard semigroup ring results are extended. We define contracted objects for (generalized) semigroup near-rings and show (generalized) matrix near-rings are just a special case as in rings. This theory encompasses (generalized) matrix, group, and (generalized) polynomial near-rings.

Some questions in combinatorial and elementary number theory

Abstract: This thesis is divided into two parts. Part I is about additive combinatorics. Part II deals with questions in elementary number theory. In Chapter 1, we generalize the Davenport transform to prove that if si S\mathbb A=(A, +)S is acancellative semigroup (either abelian or not) and SX, YS are non-empty subsets of SAS such that the subsemigroup generated by SYS is abelian, then SS|X+Y|\gc\min(\gamma(Y, |X|+|Y|-I)SS, where for SZ\subsetcq AS we let S\gamma(Z):=\sup_{z_0\in Z^\times}\in f_(z_0 c z\inZ) (vm ord)(z-z_0)S. This implies an extension of Chowla’s and Pillai’s theorems for cyclic groups and a stronger version of an addition theorem by Hamidoune and Karolyi for arbitrary groups. In Chapter 2, we show that if S(A, +) is a cancellative semigroup and SX, Y\subsetcq AS then SS|X+Y|\gc\min(\gammaX+Y), |X|+|Y|-I)SS. This gives a generalization of Kemperman’s inequality for torsion free groups and a stronger version of the Hamidoune-Karolyi theorem. In Chapter 3, we generalize results by Freiman et al. by proving that if S(A,\ctlot)S is a linearly orderable semigroup and SSS is a finite subset of SAS generating a non-abelian subsemigroup, then S|S^2-\gc3|S|-2S. In Chapter 4, we prove results related to conjecture by Gyory and Smyth on the sets SR_k^\pm(a,b)S of all positive integers SnS such that Sn^kS divides Sa^a \pmb^nS for fixed integers SaS, SbS and SkS with Sk\gc3S, S|ab|\gc2Set S\gcd(a,b) = 1S. In particular, we show that SR_k^pm(a,b)S is finite if Sk\gc\max(|a|.|b|)S. In Chapter 5, we consider a question on primes and divisibility somchow related to Znam’s problem and the Agoh-Giuga conjecture