Questions: Amalgamation Property and Joint Embedding Property

Question 1

Classes K₁ and K₂ each satisfy a condition: K₁ has the property that any two members embed into a common larger member (with no required shared substructure), while K₂ has the property that whenever two members share a common substructure, they can always be embedded into a common extension. Which properties do K₁ and K₂ have?

• A. K₁ has AP but not JEP; K₂ has JEP but not AP
• B. K₁ has JEP but not necessarily AP; K₂ has AP (and therefore JEP)
• C. Both have AP, since both can embed members into a common structure
• D. K₁ has JEP but not AP; K₂ has AP but not JEP

Answer: (B)

JEP requires only that any two members embed into a common structure — no shared substructure is needed. K₁ meets exactly this condition. AP is stronger: it requires the 'diamond' to be completable whenever two members share a common substructure (a span A ← B, A ← C must complete to a commuting square). K₂ meets exactly this condition. Since AP implies JEP (take A to be the empty or initial structure), K₂ has both. Option D reverses the relation between AP and JEP.

Question 2

In Fraïssé limit construction, why does the amalgamation property ensure the back-and-forth argument never reaches an impasse?

• A. AP guarantees that every structure in K embeds into every other structure in K
• B. AP guarantees that any two partial extensions sharing a common sub-part can always be merged into a single larger extension
• C. AP guarantees that the Fraïssé limit is the unique countable model of the theory
• D. AP guarantees that the limit structure contains only finitely many isomorphism types

Answer: (B)

At each step of the back-and-forth construction, you have two partial extensions of a common 'current structure.' The amalgamation property says exactly that this diamond can always be completed — the two extensions can always be reconciled into a single larger structure. Without AP, you could reach a state where two required extensions are incompatible, and the construction halts. Options A and D describe different properties (universality and ω-categoricity, roughly), and C is a consequence of additional axioms, not of AP alone.

Question 3

The joint embedding property for a class K implies that any two structures in K have a common substructure.

• T. True
• F. False

Answer: False

JEP says the opposite: any two structures in K embed into a common superstructure — they can both be found inside some larger member of K. JEP says nothing about shared substructures. AP, the stronger property, is the one that involves shared substructures: it requires that two structures with a common substructure can be embedded into a common extension.

Question 4

The amalgamation property is strictly stronger than the joint embedding property — any class with AP automatically has JEP, but not vice versa.

• T. True
• F. False

Answer: True

AP implies JEP: given any two structures A, B ∈ K, take the empty (or initial) structure as their 'common substructure.' AP then provides a structure C into which both embed, which is exactly JEP. The converse fails: a class can have JEP (any two structures embed somewhere together) but fail AP (two structures with a specific common substructure cannot be jointly extended in a compatible way).

Question 5

Explain in your own words why the amalgamation property — rather than just the joint embedding property — is required for Fraïssé's theorem to produce a homogeneous universal limit.

Model answer: The back-and-forth construction at each step extends a partial isomorphism, creating a situation where two extensions share a specific common 'base' structure already built. JEP only guarantees that two structures can be embedded somewhere together when there is no pre-existing common part — it says nothing about reconciling two extensions that have diverged from a fixed starting point. AP provides exactly the needed guarantee: any two extensions of the same structure can be merged into a single larger extension. This prevents the construction from getting 'stuck' when the two sides of the back-and-forth have made incompatible choices above a shared foundation.

The key distinction is that the back-and-forth method is not combining arbitrary structures but always combining extensions of something already built. JEP handles the global question of whether two things can coexist; AP handles whether two things can coexist given that they already agree below a shared foundation — which is what the construction actually requires at every step.