For W a finite set of words, we consider the Rees quotient of a free monoid with respect to the ideal consisting of all words that are not subwords of W . This monoid is denoted by S(W ). It is shown that for every finite set of words W , there are sets of words U ⊃ W and V ⊃ W such that the identities satisfied by S(V ) are finitely based and those of S(U) are not finitely based (regardless of...

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. An $R$-module $M$ is called a multiplication module if for every submodule $N$ of $M$ there exists an ideal $I$ of $R$ such that $N = IM$. It is shown that over a Noetherian domain $R$ with dim$(R)leq 1$, multiplication modules are precisely cyclic or isomorphic to an invertible ideal of $R$. Moreover, we give a charac...

In this paper first of all we introduce Property $U$-($G$-$PWP$) of acts, which is an extension of Condition $(G$-$PWP)$ and give some general properties. Then we give a characterization of monoids when this property of acts implies some others. Also we show that the strong (faithfulness, $P$-cyclicity) and ($P$-)regularity of acts imply the property $U$-($G$-$PWP$). Finally, we give a necessar...

HNN extensions of inverse semigroups, where the associated inverse subsemigroups are order ideals of the base, are defined by means of a construction based upon the isomorphism between the categories of inverse semigroups and inductive groupoids. The resulting HNN extension may conveniently be described by an inverse semigroup presentation, and we determine when an HNN extension with finitely g...

The optical light generated simultaneously with x-rays and gamma rays during a gamma-ray burst (GRB) provides clues about the nature of the explosions that occur as massive stars collapse. We report on the bright optical flash and fading afterglow from powerful burst GRB 130427A. The optical and >100-megaelectron volt (MeV) gamma-ray flux show a close correlation during the first 7000 seconds, ...

‎A rank equality is established for the sum of finitely many tripotent matrices via elementary block matrix operations‎. ‎Moreover‎, ‎by using this equality and Theorems 8 and 10 in [Chen M‎. ‎and et al‎. ‎On the open problem related to rank equalities for the sum of finitely many idempotent matrices and its applications‎, ‎The Scientific World Journal 2014 (2014)‎, ‎Article ID 702413‎, ‎7 page...

We prove here that a finitely presented group with a free quotient of rank n is an HNN-extension with n stable letters of a finitely generated group where the associated subgroups are finitely generated. This theorem has a number of consequences. In particular, in the event that the free quotient is cyclic it reduces to an elementary and quick proof of a theorem of Bieri and Strebel. 1. Finitel...

A group is called morphic if for each normal endomorphism α in end(G),there exists β such that ker(α)= Gβ and Gα= ker(β). In this paper, we consider the case that there exist normal endomorphisms β and γ such that ker(α)= Gβ and Gα = ker(γ). We call G quasi-morphic, if this happens for any normal endomorphism α in end(G). We get the following results: G is quasi-morphic if and only if, for any ...

We report on magnetorheological finishing (MRF) spotting experiments performed on glasses and ceramics using a zirconia-coated carbonyl-iron (CI)-particle-based magnetorheological (MR) fluid. The zirconia-coated magnetic CI particles were prepared via sol-gel synthesis in kilogram quantities. The coating layer was approximately 50-100 nm thick, faceted in surface structure, and well adhered. Co...

For any module M over an associative ring R, let σ[M ] denote the smallest Grothendieck subcatgory of Mod-R containingM . If σ[M ] is locally finitely presented the notions of purity and pure injectivity are defined in σ[M ]. In this paper the relationship between these notions and the corresponding notions defined in Mod-R are investigated, and the connection between the resulting Ziegler spec...