A note on equational theories (Q2710602)

The paper describes and compares some notions of equational theories occurring in past works of G. Srour and others. Here an equation in a complete first-order theory \(T\) is a formula \(\varphi (\vec{v}, \vec{w})\) of the language of \(T\) such that there is no infinite strictly descending chain of intersections of definable subsets \(\varphi (U, \vec{a})\) (where \(U\) denotes is the monster model of \(T\) and \(\vec{a}\) ranges over \(U\)); more properly, such a formula is an equation in \(\vec{v}\). A definable set is called Srour closed if its conjugates in \(U\) satisfy the same chain condition. \(T\) is called equational if every formula is a Boolean combination of equations, and weakly equational when every definable set is a Boolean combination of Srour closed sets. Of course, equationality implies weak equationality. But the paper shows that the converse is also true, so the two notions are equivalent. It is also proved that equationality is preserved under passing from \(T\) to \(T^{\text{eq}}\), and hence is invariant under biinterpretability. Furthermore it is shown that equationality is preserved under definitional equivalence.