Associative $H$-pseudoalgebras with a semigroup

Family algebraic structures indexed by a semigroup arise naturally in renormalizations of quantum field theory. In this paper, we first define the notion of $\Omega$-associative $H$-pseudoalgebra, where the operations are indexed by pairs of elements from a semigroup $\Omega$. Then we construct $\Omega$-associative $H$-pseudoalgebras from associative $H$-pseudoalgebras, $\Omega$-associative algebras, Rota-Baxter family algebras, $\Omega$-type $H$-pseudoalgebras and family-type $H$-pseudoalgebras. Moreover, we investigate the cohomology of $\Omega$-associative $H$-pseudoalgebras and establish that it both induces the cohomology of pseudo-$\mathcal{O}$-operator families and governs the associated formal deformations. As an application, we show that the first-order deformation of a commutative $\Omega$-associative $H$-pseudoalgebra yields an $\Omega$-Poisson $H$-pseudoalgebra.