‎‎‎in this paper‎, ‎the‎ notion of injectivity with respect to order dense embeddings in ‎‎the category of $s$-posets‎, ‎posets with a monotone action of a‎ pomonoid $s$ on them‎, ‎is studied‎. ‎we give a criterion‎, ‎like the baer condition for injectivity of modules‎, ‎or skornjakov criterion for injectivity of $s$-sets‎, ‎for the order dense injectivity‎. ‎also‎, ‎we consider such injectivit...

‎for the subclasses ${mathcal m}_1$ and ${mathcal m}_2$ of‎‎monomorphisms in a concrete category $mathcal c$‎, ‎if ${mathcal‎‎m}_2subseteq {mathcal m}_1$‎, ‎then ${mathcal m}_1$-injectivity‎‎implies ${mathcal m}_2$-injectivity‎. ‎the baer type criterion is about‎ ‎the converse of this fact‎. ‎in this paper‎, ‎we apply injectivity to the classes of {it $c$-dense‎, ‎$c$-closed}‎ ‎monomorphisms‎. ...

It is known that injective Cellular Automata (CA) are surjective. In general, the converse is not true, and there are many algebraic CA counterexamples. However, there are interesting subclasses where this might be true. For example, if a surjective CA has entropy 0 then it is almost injective (Moothathu (2011)) and it is not known if it is actually injective. The author believes there are some...

‎‎‎In this paper‎, ‎the‎ notion of injectivity with respect to order dense embeddings in ‎‎the category of $S$-posets‎, ‎posets with a monotone action of a‎ pomonoid $S$ on them‎, ‎is studied‎. ‎We give a criterion‎, ‎like the Baer condition for injectivity of modules‎, ‎or Skornjakov criterion for injectivity of $S$-sets‎, ‎for the order dense injectivity‎. ‎Also‎, ‎we consider such injectivit...

‎in this paper some properties of weak regular injectivity for $s$-posets‎, ‎where $s$ is a pomonoid‎, ‎are studied‎. ‎the behaviour‎ ‎of different kinds of weak regular injectivity with products‎, ‎coproducts and direct sums is considered‎. ‎also‎, ‎some‎ ‎characterizations of pomonoids over which all $s$-posets are of‎ ‎some kind of weakly regular injective are obtained‎. ‎further‎, ‎we‎ ‎giv...

Let G be a residually finite group and let A be a finite set. We prove that if X ⊂ A is a strongly irreducible subshift of finite type containing a periodic configuration then periodic configurations are dense in X. The density of periodic configurations implies in particular that every injective endomorphism of X is surjective and that the group of automorphisms of X is residually finite. We a...

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 ...

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 ...