Let S be a semigroup with a left multiplier  on S. A new product on S is defined by  related to S and  such that S and the new semigroup ST have the same underlying set as S. It is shown that if  is injective then where, is the extension of  on  Also, we show that if  is bijective then is amenable if and only if is so. Moreover, if  S completely regular, then is weakly amenable. 

The largest volume ratio of a given convex body $K \subset \mathbb R^n$ is defined as $$ \mathrm{lvr}(K):= \mathrm {sup}\_{L R^n} {vr}(K,L), where the sup runs over all bodies $L$. We prove following sharp lower bound: c \sqrt{n} \leq \mathrm{lvr}(K), for every $K$ (where $c > 0$ an absolute constant). This result improves former best known bound, order $\sqrt{{n}/{\log \log(n)}}$. also study e...

Let $mathcal{X}$ be a class of $R$-modules‎. ‎In this paper‎, ‎we investigate ;$mathcal{X}$-injective (projective) and DG-$mathcal{X}$-injective (projective) complexes which are generalizations of injective (projective) and DG-injecti‎‎ve (projective) complexes‎. ‎We prove that some known results can be extended to the class of ;$mathcal{X}$-injective (projective) and DG-$mathcal{X}$-injective ...

Some prosperities of matrix product are presented in the paper, Kronecker product, Khatri-Rao product, Hadamard product and outer product are involved. And we get some results that a multilinear tensor can be represented by the product of matrix product for a three order tensor. For higher tensor, we conjure that the same results also hold. By the representation of matrix, we give an iterative ...

let $mathcal{x}$ be a class of $r$-modules‎. ‎in this paper‎, ‎we investigate ;$mathcal{x}$-injective (projective) and dg-$mathcal{x}$-injective (projective) complexes which are generalizations of injective (projective) and dg-injecti‎‎ve (projective) complexes‎. ‎we prove that some known results can be extended to the class of ;$mathcal{x}$-injective (projective) and dg-$mathcal{x}$-injective ...

The theory of crystal bases introduced by Kashiwara in [6] to study the category of integrable representations of quantized Kac–Moody Lie algebras has been a major development in the combinatorial approach to representation theory. In particular Kashiwara defined the tensor product of crystal bases and showed that it corresponded to the tensor product of representations. Later, in [7] he define...

The theory of crystal bases introduced by Kashiwara in [4] to study the category of integrable representations of quantized Kac–Moody Lie algebras has been a major development in the combinatorial approach to representation theory. In particular Kashiwara defined the tensor product of crystal bases and showed that it corresponded to the tensor product of representations. Later, in [5] he define...

EXTENT M an EXpert system for TENsor product formula Translation. In this paper we present a programming environment for automattc generation of parallel/vector programs from tensor product formulas. A tensor (Kronecker) product based programming methodology is used for deszgning high performance programs on vartous architectures. In thts programming methodology, block recurstve algordhms such ...