# A generalization of quasi-identities

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!

• Aren't these just the universal theories in algebraic languages (meaning languages with no relation symbols except equality)? The point is that, given a universal prenex sentence in such a language, we can put its matrix into conjunctive normal form and distribute the universal quantifiers over the conjunctions. What remains, universally quanntified disjunctions of equations and negated equations, are just the formulas you asked about. – Andreas Blass Aug 14 '13 at 15:54
• Yes, at least in the classical setting. Is there a special name for these? – François G. Dorais Aug 14 '13 at 16:02
• I don't know any special name unless "universal" counts as special. – Andreas Blass Aug 14 '13 at 16:03
• Nothing else came to mind either but "universal algebraic theory" sounds weird (to me). – François G. Dorais Aug 14 '13 at 16:08
• @The Masked Avenger: A class of structures is a variety iff it is closed under $H$, $S$, and $P$. It is a quasivariety iff it is closed under $I$, $S$, $P$, and $P_U$. Both happen to imply the 1-element algebra (being the empty product). A class is axiomatizable by a universal theory iff it is closed under $I$, $S$, and $P_U$. This happens not to yield the 1-element algebra. I don’t see anything unnerving about that, on the contrary it would seem quite untidy to arbitrarily throw the 1-element algebra in the mix. – Emil Jeřábek Aug 14 '13 at 18:35