Abstract
The notion of sequential purity for acts over the monoid $\mathbb{N}^\infty$, called projection algebras, was introduced and studied by Mahmoudi and Ebrahimi. This paper is devoted to the study of this notion and its relation to injectivity of $S$-acts for a semigroup $S$. We prove that in general injectivity implies absolute sequential purity and they are equivalent for acts over some classes of semigroups.
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