## 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.

### MSC:

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

### Keywords:

stably embedded; orthogonal sorts
Full Text:

### References:

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