Model Companion and Model Completion
A companion of a theory T is a theory T* such that every model of T can be embedded in a model of T* and vice versa.
A model companion of a theory T is a companion of T that is model complete. Robinson proved that a theory has at most one model companion.
A model completion for a theory T is a model companion T* such that for any model M of T, the theory of T* together with the diagram of M is complete. Roughly speaking, this means every model of T is embeddable in a model of T* in a unique way.
If T* is a model companion of T then the following conditions are equivalent:
- T* is a model completion of T
- T has the amalgamation property.
If T also has universal axiomatization, both of the above are also equivalent to:
- T* has elimination of quantifiers
Read more about this topic: Model Complete Theory
Famous quotes containing the words model and/or companion:
“The best way to teach a child restraint and generosity is to be a model of those qualities yourself. If your child sees that you want a particular item but refrain from buying it, either because it isn’t practical or because you can’t afford it, he will begin to understand restraint. Likewise, if you donate books or clothing to charity, take him with you to distribute the items to teach him about generosity.”
—Lawrence Balter (20th century)
“I have found it a singular luxury to talk across the pond to a companion on the opposite side.”
—Henry David Thoreau (1817–1862)