We prove that every quasi-Hopfian finitely presented structure A has a d- $$\Sigma _2$$ Scott sentence, and if in addition is computable Aut(A) satisfies natural condition, then sentence. This unifies several known results on sentences of structures it used to other not previously considered algebraic interest have sentences. In particular, we show right-angled Coxeter group finite rank as well...

In this paper we introduce the notion of e-gH modules which is a proper generalization Hopfian and defined as, module called if, any surjective -endomorphism has an e-small kernel, ring e-gH. We give some characterizations properties modules.

In this paper and its sequel, we prove that injective homomorphisms between Teichmüller modular groups of compact orientable surfaces are necessarily isomorphisms, if an appropriately measured “size” of the surfaces in question differs by at most one. In particular, we establish the co-Hopfian property for modular groups of surfaces of positive genus.

We introduce and characterize extremal 2-cuts for good fractal necklaces. Using this characterization the related topological properties of 2-cuts, we prove that every necklace has a unique IFS in certain sense. Also, two necklaces admit only rigid homeomorphisms thus group self-homeomorphisms is countable. In addition, weaker co-Hopfian property also obtained.