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. 

In this paper, we characterize w -Noetherian rings and SM rings. More precisely, in terms of the id="M3"> u -operation on a commutative ring id="M4"> R , prove that id="M5"> is id="M6"> if only direct limit id="M7"> r GV -torsion-free injective id="M8"> -modules id="M9"> SM, whic...

A one-way permutation (OWP) is one of the most fundamental cryptographic primitives, and can be used as a building block for most of basic symmetric-key cryptographic primitives. However, despite its importance and usefulness, previous black-box separation results have shown that constructing a OWP from another primitive seems hopeless, unless building blocks already achieve “one-way” property ...

The (in)equational properties of the least fixed point operation on ( -)continuous functions on ( -)complete partially ordered sets are captured by the axioms of (ordered) iteration algebras, or iteration theories. We show that the inequational laws of the sum operation in conjunction with the least fixed point operation in continuous additive algebras have a finite axiomatization over the ineq...

A generalization of injective modules (noted GI-modules), distinct from p-injective modules, is introduced. Rings whose p-injective modules are GI are characterized. If M is a left GI-module, E = End(AM), then E/J(E) is von Neumann regular, where J(E) is the Jacobson radical of the ring E. A is semisimple Artinian if, and only if, every left A-module is GI. If A is a left p. p., left GI-ring su...

In this paper we study the notions of cogenerator and subdirectlyirreducible in the category of S-poset. First we give somenecessary and sufficient conditions for a cogenerator $S$-posets.Then we see that under some conditions, regular injectivityimplies generator and cogenerator. Recalling Birkhoff'sRepresentation Theorem for algebra, we study subdirectlyirreducible S-posets and give this theo...