If $p\in N$, then a $p$-semigroup, introduced in [3], is a generalization of the notion of an anti-inverse semigroup [2]. A similar notion is a $p$-semiring. The aim of the paper was to investigate the closeness of classes of these algebras under the operators $H$ (homomorphisms), $S$ (subalgebras) and $P$ (direct products). It is proved that for every $p\in N$ each of these classes is closed under $H$ and $P$. Conditions under which closeness under $S$ also hold are presented. It turns out that for $p$ even or $p=4k+3$ both the class of $p$-semigroups and the one of $p$-semirings are varieties. The corresponding identities are presented.