Sequentially injective hull of acts over idempotent semigroups. (Q2373409)

Let \(S\) be a semigroup. A `Cauchy' sequence over an \(S\)-act is a family \((a_s)_{s\in S}\) of elements of \(A\) with \(a_st=a_{st}\) for all \(s,t\in S\). An element \(b\) of some extension \(B\) of \(A\) is called a `limit' of a Cauchy sequence \((a_s)_{s\in S}\) if \(bs=a_s\) for all \(s\in S\). An \(S\)-act \(A\) is said `s-complete' if every Cauchy sequence over \(A\) has a limit in \(A\). An `s-closure' \(C_S(A)\) of a subact \(A\leq B\) in \(B\) is defined as \(C_S(A)=\{b\in B:bS\subseteq A\}\). \(A\) is called `s-dense in \(B\)' if \(C_S(A)=B\); an \(S\)-map \(f\colon A\to B\) is called `s-dense' if \(f(A)\) is an s-dense subact of \(B\). An \(S\)-act is called `sequentially injective' or `s-injective' if it is injective with respect to s-dense monomorphisms, and `s-absolute retract' if it is a retract of each of its s-dense extensions. Equivalence of the following properties is proved for an \(S\)-act \(A\): 1) \(A\) is s-complete; 2) \(A\) is s-injective; 3) \(A\) is an s-absolute retract. Over an idempotent semigroup, s-injective hulls are described as well.