# Has this generalization of quasi-equations been studied in the mathematical literature?

In universal algebra, there is a lot of talk of quasi-equations, which are conditionals, where the antecedent is a conjunction of finitely many (possibly even $$0$$) equations, and the consequent is a single equation. However, I have come up with a generalization of quasi-equations, which I call semi-quasi-equations. These are conditionals, where the antecedent is a conjunction of finitely many (possibly even $$0$$) equations, and the consequent is a disjunction of finitely many (possibly even $$0$$) equations. To give an example of their use, every field $$F$$ satisfies the semi-quasi-equation, $$x*y=0 \rightarrow (x=0 \vee y=0)$$. So, has this notion been studied in the mathematical literature, especially the universal algebra literature?

Let $$\varphi$$ be a universal sentence. Then $$\varphi$$ is equivalent to $$\forall \overline{x}\, \psi$$, where $$\psi$$ is quantifier-free (i.e., a boolean combination of atomic formulas). Writing $$\psi$$ in conjunctive normal form, and separating the atomic formulas from the negated atomic formulas in each disjunctive clause, $$\psi$$ is equivalent to $$\bigwedge_{i=1}^n \left(\bigvee_{j=1}^k \lnot \theta_{i,j}\lor \bigvee_{j={k+1}}^m\theta_{i,j}\right),$$ where each $$\theta_{i,j}$$ is atomic (i.e., an equation). Finally, note that $$\bigvee_{j=1}^k \lnot \theta_{i,j}\lor \bigvee_{j={k+1}}^m\theta_{i,j}$$ is equivalent to $$\left(\bigwedge_{j=1}^k\theta_{i,j}\right)\to \left(\bigvee_{j=k+1}^m\theta_{i,j}\right).$$
Altogether, $$\varphi$$ is equivalent to the finite set of semi-quasi-equations $$\left\{\forall \overline{x}\left(\bigwedge_{j=1}^k\theta_{i,j}\right)\to \left(\bigvee_{j=k+1}^m\theta_{i,j}\right)\mid 1\leq i\leq n\right\}.$$
It's worth noting that implications from conjunctions to disjunctions are very natural from the point of view of mathematical logic, so much so that a standard notational convention is that $$\varphi_1,\dots,\varphi_n\vdash \psi_1,\dots,\psi_m$$ means that the conjunction of the $$\varphi_i$$ entails the disjunction of the $$\psi_j$$.
• Oh, so this also answers the bonus question in the linked post. But there is still the main question, is there a $\forall$-sentence which is not equivalent to just a single semi-quasi-equation? Apr 5 at 0:48