Signatures and L-Structures

Question

Consider the field of real numbers $\mathbb{R}$. This is an $L$-structure. Is there such a thing as an $S$-structure (i.e. a signature structure)? Or because we can recover a first order language from its signature then an $L$-structure and an "$S$-structure" can be regarded as equivalent? The signature is $S = \{0,-,+, \cdot \}$ with arities (I think) $0$, $1$, $2$, and $2$. Is the reason why $-$ has arity $1$ because we only need one input (i.e. $-: \mathbb{R} \to \mathbb{R}$ defined as $-(x) = -x$ (I know this is circular)? Couldn't we also have things like $4-5$, $7-6$ etc.? Also why even include $-$ in the signature set? Wouldn't $+$ be sufficient?

Answer 1

Actually, $+$ and $\cdot$ are enough to define all other familiar field constants and operations ($0,1,-$) as well as express all the axioms of the real field. However, we usually include all the familiar symbols in the language so that we don't have to think about whether or not we can use a particular symbol in given context.

Furthermore, sometimes adding or removing definable constants, operations and relations can affect some syntactic properties, without affecting semantics. For example, the field of real numbers in the signature you proposed doesn't have quantifier elimination, however it does if you add $1$ and $\leq$ to the language (and they're both definable in reals with addition and multiplication). From that we can extract some information of more semantic nature: for example, any subfield of the field of real numbers which is elementarily equivalent (like ${\bf Q}^{alg}\cap {\bf R}$) is automatically an elementary substructure.

As for your question about signatures, we usually equate signatures and languages. The actual symbols we use for constants, functions and relations are of no formal bearing (except to the extent that we can tell them apart), though of course, for example, using $\leq$ to denote a strict ordering would be a very bad idea in most, if not all cases.