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For M:Σ-model which satisfies T, is it proper to name it “Σ,T-model” or “Σ-model satisfying T”? In comparison, in the context of Lawvere theories, any Lawvere theory L incorporates a signature as well as axioms. If we construct L from Σ and T, any L-model obviously satisfies T. Therefore “Σ,T-model” seems as an appropriate name.

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up vote 6 down vote accepted

The usual terminology in model theory is that for a given language or signature $\Sigma$, one may speak of a structure in the signature $\Sigma$, or a $\Sigma$-structure, and this places no requirements on the theory of the model. When one has a theory $T$ in the language of $\Sigma$, one may have a model of $T$, which is a structure in that signature satisfying that theory.

One ambiguity, as you notice, is that if a theory is regarded as a set of sentences, then one cannot fully tell the language from the theory, since perhaps some parts of the language were not used in the theory. This ambiguity is not actually ambiguous in practice if the signature $\Sigma$ is clear from context, and when it is not, then you are right that one must say what the language is, in the cases that this is a difference that matters. So in practice, this amounts essentially to the same as the Lawvere practice.

An interesting example where the distinction is important is that it is possible for a theory $T$ to be decidable in a small language, but not in a larger language. For example, Presburger arithmetic is decidable in the language with just $+$, but not in the language having both plus and times, even when no additional axioms are added concerning times. Indeed, almost no theory will be decidable in a language that includes extra unmentioned binary or higher arity relation or function symbols, since the empty theory in the language with such relation or functions symbols is undecidable.

In practice, researchers often use the term model when no theory has been specified, and in such cases, they usually mean structure, unless a particular theory and signature is implicit.

Finally, regarding your proposal for the term "$\Sigma,T$-model", let me say that my own taste tends towards natural language formulations, where we could simply speak of a model of $T$ in the language $\Sigma$.

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Thank you, very interesting. – beroal Feb 28 '11 at 15:42
I found one source of confusion. In “Categorical Logic and Type Theory” by Bart Jacobs in Section 1.6 “Fibrations of signatures” a “model” means a structure as defined in your answer. I believe such confusion appears in other books. – beroal Mar 14 '11 at 20:06

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