In this paper we consider the normal modal logics of elementary classes defined by first-order formulas of the form $forall x_0 exists x_1 dots exists x_n bigwedge x_i R_lambda x_j$. We prove that many properties of these logics, such as finite axiomatisability, elementarity, axiomatisability by a set of canonical formulas or by a single generalised Sahlqvist formula, together with modal definability of the initial formula, either simultaneously hold or simultaneously do not hold.
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