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

One of the main characteristics of the theory is stability. T. G. Mustafin [1] proposed a generalization of this concept. The concept of $ P $-stability [2] is a special case of generalized stability of complete theories. In [3], $S$-acts with a $ (P, 1) $, $ (P, s) $-, $ (P, a) $-, and $ (P, e) $-stable theory are considered. In [4], the description of the monoids $S$ over which the classes of free, projective, strongly flat, divisible, regular $S$-acts are $P$-stable is given.
In this paper, we prove that if the class of all injective $S$-acts is $(P,1)$-stable then $|S|=1.$ Besides that, we prove that the property of being $ (P, s) $-, $ (P, a) $-, and $ (P, e) $-stable for the class of all injective $S$-acts is equivalent to $ S $ being a group.

Список литературы

1. T. G. Mustafin, “On stability theory of polygons”, Trudy Inst. Mat., 8 (1988), 92–108
2. Е. А. Палютин, “$E^*$-стабильные теории”, Алгебра и логика, 42:2 (2003), 194–210      ; Algebra and Logic, 42:2 (2003), 112–120    
3. Д. О. Птахов, “Полигоны с $(P,1)$-стабильной теорией”, Алгебра и логика, 56:6 (2017), 712–720    ; Algebra and Logic, 56:6 (2018), 473–478      
4. А. А. Степанова, А. И. Красицкая, “$P$-стабильность некоторых классов $S$-полигонов”, Сиб. матем. журн., 62:2 (2021), 441–449