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Abstract

Let $\tau$ be a type of algebras without nullary fundamental operation symbols. We call an identity $\varphi\approx\psi$ of type $\tau$ clone compatible if $\varphi$ and $\psi$ are the same variable or the sets of fundamental operation symbols in $\varphi$ and $\psi$ are nonempty and identical. For a variety $\mathcal V$ of type $\tau$ we denote by ${\mathcal V}^{c}$ the variety of type $\tau$ defined by all clone compatible identities from $\mathop{\rm Id}\nolimits(\mathcal V)$. We call ${\mathcal V}^{c}$ the clone extension of $\mathcal V$. In this paper we describe algebras and minimal generics of all subvarieties of ${\mathcal{B}}^{c}$, where $\mathcal B$ is the variety of Boolean algebras.