Regular $S$-acts with primitive normal and antiadditive theories

In this work, we investigate the commutative monoids over which the axiomatizable class of regular $S$-acts is primitive normal and antiadditive. We prove that the primitive normality of an axiomatizable class of regular $S$-acts over the commutative monoid $S$ is equivalent to the antiadditivity of this class and it is equivalent to the linearity of the order on a semigroup $R$ such that an $S$-act $_SR$ is a maximal (under the inclusion) regular subact of the $S$-act $_SS$.