Embeddings and variables. Let $\sigma$ be an embedding, $t$ be a variable. Prove $\sigma(t^\mathcal{M}(a))=\sigma(a)=t^\mathcal{N}(\sigma(a)).$

Question

I was reading Notes on Model Theory by Gabriel Conant https://www3.nd.edu/~cmnd/programs/cmnd2016/undergrad/Conant_MTnotes.pdf, and on page $6$ it had a proof of the following proposition:

Suppose $\mathcal{M}$ and $\mathcal{N}$ are $\mathcal{L}$-structures, and $\sigma:\mathcal{M}\to\mathcal{N}$ is an $\mathcal{L}$-embedding. For any quantifier-free formula $\varphi(v_1,\dots,v_n)$ and $\bar{a}\in\mathcal{M}^n,$ $$ \mathcal{M}\vDash\varphi(a_1,\dots,a_n)\iff\mathcal{N}\vDash\varphi(\sigma(a_1),\dots,\sigma(a_n)). $$ During the proof, it was asserted that if $t$ is a variable and $a\in M$, then $\sigma(t^\mathcal{M}(a))=\sigma(a)=t^\mathcal{N}(\sigma(a)).$

The author seemed to think this was a very obvious line of working, but I don't see how it immediately follows from the definition the author gave of an $\mathcal{L}$-embedding:

Let $\mathcal{L}$ be a language and let $\mathcal{M}\to\mathcal{N}$ be $\mathcal{L}$-structures. A function $\sigma:M\to N$ is an $\mathcal{L}$-embedding if $\sigma$ is injective and:

• for any function symbol $f$ in $\mathcal{L}$ of arity $n$, and $a_1,\dots,a_n\in M,$ $$ \sigma(f^\mathcal{M}(a_1,\dots,a_n))=f^\mathcal{N}(\sigma(a_1),\dots,\sigma(a_n)); $$
• for any relation symbol $R$ in $\mathcal{L}$ of arity $n$, and $a_1,\dots a_n\in M,$ $$ (a_1,\dots a_n)\in R^\mathcal{M}\iff(\sigma(a_1),\dots,\sigma(a_n))\in R^\mathcal{N}; $$
• for any constant symbol $c$ in $\mathcal{L}$, $$ \sigma(c^\mathcal{M})=c^\mathcal{N}. $$

This definition only seems to speak of the effect of embeddings on constants, functions and relations given in the language. So exactly does this definition have any implication on variables? I was tempted to just accept this as fact and move on, but it seems like such a basic point, that I don't want to start my study of logic on uncertain grounds. Any help in understanding would be greatly appeciated!

Let $\sigma$ be an $\mathcal{L}$-embedding between $\mathcal{M}$ and $\mathcal{N}$, and let $t$ be a variable and $a\in M$. Prove $\sigma(t^\mathcal{M}(a))=\sigma(a)=t^\mathcal{N}(\sigma(a)).$

Answer 1

This has nothing to do with the definition of embedding. Instead it's about the definition of evaluation of terms.

If $t$ is the variable $x$ and $a$ is an element of a structure $M$, then $t^M(a)$ is defined to be $a$ (see Definition 2.4 (ii) on p.4). Do you agree now that the chain of equalities is obvious?