A note on orthogonality and stable embeddedness. (English) Zbl 1100.03022

Let \(T\) be a first-order theory with variables \(x\) and \(y\) ranging over distinguished sorts \(P\) and \(Q\), respectively. \(P\) and \(Q\) are orthogonal if any formula \(\phi(x,y)\) is equivalent to a Boolean combination of formulas \(\psi_ i(x)\), \(\theta_ j(y)\), with the possible involvement of parameters. A collection \(\mathfrak P\) of sorts is stably embedded if every relation on sorts \(P_ 1,\dots, P_ m\in\mathfrak P\) with parameters in a model \(M\) of \(T\) can also be defined with parameters from elements of the sorts in \(\mathfrak P\).
The authors show that if \(P\) and \(Q\) are orthogonal and stably embedded, then \(P\cup Q \) is also stably embedded. As a corollary, if \(Q\) is orthogonal to \(P_ 1\) and to \(P_ 2\) and all three are stably embedded, then \(Q\) is orthogonal to \(P_ 1\cup P_ 2\). The proof of the main theorem uses the theory of locally finite groups and requires the classification of the finite simple groups.


03C45 Classification theory, stability, and related concepts in model theory
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