Generalized stability of the class of injective $S$-acts

The concept of $P$-stability is a particular case of generalized stability of complete theories. We study injective $S$-acts with a $P$-stable theory. It is proved that the class of injective $S$-acts is $(P,1)$-stable only if $S$ is a one-element monoid. Also we describe commutative and linearly ordered monoids $S$ the class of injective $S$-acts over which is $(P,s)$-, $(P,a)$-, and $(P,e)$-stable.