Intermediate Model Theory

In general, a product has a different theory to that of the structures in the set Mi. For example a product of fields is just a ring. Note also that if φ is an atomic formula then ∏ iMi |= φ((ai )i, · · · , (ai )i) iff {i ∈ I : Mi |= φ(ai , · · · , ai )} = I. In particular this shows that if J ⊆ I then the restiction function (ai)i∈I → (ai)i∈J is a surjective homomorphism from ∏ i∈IMi to ∏ i∈JMi. We will define next the notion of reduced product which generalizes the notion of product in a sense that if φ is atomic then the reduced product satisfies φ((ai )i, · · · , (ai )i) iff {i ∈ I :Mi |= φ(ai , · · · , ai )} is “large” inside I. Let us first define a notion of “large” inside I.