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:
“When you model yourself on people, you should try to resemble their good sides.”
—Molière [Jean Baptiste Poquelin] (1622–1673)
“One who was my companion in my two previous excursions to these woods, tells me that ... he found himself dining one day on moose-meat, mud turtle, trout, and beaver, and he thought that there were few places in the world where these dishes could easily be brought together on one table.”
—Henry David Thoreau (1817–1862)