1
$\begingroup$

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?

$\endgroup$

1 Answer 1

2
$\begingroup$

Every universal sentence is equivalent to a finite set of "semi-quasi-equations". This is why "semi-quasi-equations" do not frequently appear in universal algebra or model theory books: the notion of semi-quasi-equational theory is the same as the notion of universal theory.


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 $\psi_j$.

$\endgroup$
1
  • $\begingroup$ 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? $\endgroup$
    – user107952
    Apr 5 at 0:48

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .