The ⊰-order on submodels

Extract

The relation M ₁ ⊰ M ₂, where M ₁ and M ₂ are elementary submodels of a given model M, defines a partial ordering ⊰(M). A natural question is: What are the general properties of this ordering and what special properties can it have for given M? Here we give an example of a prime model M for which ⊰(M) is a dense linear order, in particular ⊰(M) is isomorphic to the natural order of (− ∞, ∞], the reals with a last element (Theorem 1). In fact, this is the only way for ⊰(M) to be a linear order when M is prime (Theorem 3).

We then remark that if N is a model of an ℵ₀-categorical theory then ⊰(N) cannot be a linear order, but we give an example where ⊰(N) is a dense partial order, N a model of an ℵ₀-categorical theory.

Independently, and about the same time, Benda [1] found nonprime models M where ⊰(M) is isomorphic to any one of a large class of orders, including (− ∞, ∞]. His methods are very similar to ours.

Notation. We assume the reader is familar with Vaught [2] where he will find the definitions and standard results pertaining to prime models, homogeneous models, and ℵ₀-categorical theories.