Аннотация:
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
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T. G. Mustafin, “On stability theory of polygons”, Trudy Inst. Mat., 8 (1988), 92–108
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Е. А. Палютин, “$E^*$-стабильные теории”, Алгебра и логика, 42:2 (2003), 194–210
; Algebra and Logic, 42:2 (2003), 112–120 -
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 -
А. А. Степанова, А. И. Красицкая, “$P$-стабильность некоторых классов $S$-полигонов”, Сиб. матем. журн., 62:2 (2021), 441–449
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