Are there any new identities when we go from subtraction to subtraction with a nonzero constant?

Question

This is the subtraction counterpart to my previous universal algebra question on addition with a nonzero constant, here: No simplifying identities for any single nonzero number under addition.. I know that the structure $(\mathbb{R};-)$ of the binary operation of subtraction on reals has a finite basis $E$ of identities. I also know that when we expand that structure by adding $0$, we need new identities. My question is, suppose $r$ is a nonzero real number. Is the equational identities of the structure $(\mathbb{R};-,r)$ generated by $E$ alone?

Answer 1

As often, think about automorphisms! For any $r,s\in\mathbb{R}_{\not=0}$, there is an automorphism of $(\mathbb{R};-)$ sending $r$ to $s$ (this is a good exercise) and so the structures $(\mathbb{R};-,r)$ and $(\mathbb{R};-,s)$ are isomorphic. In particular, if $$\forall\overline{x}(t(\overline{x},c)=s(\overline{x},c))$$ is an equation true in $(\mathbb{R};-,1)$ (say) where $c$ is a fresh constant symbol and $t,s$ are $\{-\}$-terms, it is in fact true for every nonzero value of $c$. But since the functions corresponding to $t$ and $s$ are continuous, this means that we will also have $\forall\overline{x}(t(\overline{x},0)=s(\overline{x},0)$, which is to say that in fact $$\forall y\forall\overline{x}(t(\overline{x},y)=s(\overline{x}, y))$$ is true in $(\mathbb{R};-)$. So the equational theory of $(\mathbb{R};-,a)$ for any nonzero $a$ is the equational-logic deductive closure of the equational theory of $(\mathbb{R};-)$.

Of course this breaks down once there are two or more named reals in play, since the analogue of the isomorphism fact above is not true.