In universal algebra, a variety is axiomatized by identities $t \approx s$ between terms $t$ and $s$. More general are quasi-varieties that are axiomatized by quasi-identities of the form $$u_1 \approx v_1,\dots,u_n \approx v_n \Rightarrow t \approx s,$$ which is intended to mean $$(\forall \bar{x})[u_1 = v_1 \land \cdots \land u_n = v_n \to t = s].$$ These are universal Horn theories which are therefore exceptionally well behaved and worthy of a special name.

A natural next step in the hierarchy are theories axiomatized by gadgets of the form $$u_1 \approx v_1,\dots,u_n \approx v_n \Rightarrow t_1 \approx s_1,\dots,t_m \approx s_m,$$ which are intended to mean $$(\forall \bar{x})[u_1 = v_1 \land \cdots \land u_n = v_n \to t_1 = s_1 \lor \cdots \lor t_m = s_m].$$ (When $m = 0$ the right hand side is understood to be $\bot$.) Unfortunately, such theories are not as well behaved as the above so it is not clear they are worthy of a special name. However, if there is one name that has been in use for these theories, I would like to know!