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Consider the structure $(\mathbb{R}, +, r)$, where r is a nonzero real number. Are the commutative and associative identities already sufficient to derive all universally valid equations in that structure? Basically, is 0 the only number that behaves in a special manner under addition?

J.-E. Pin
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user107952
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    What do you mean by 'the structure $(\mathbb{R}, +, r)$'? – Michael Albanese Feb 10 '16 at 10:52
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    @MichaelAlbanese: The structure (in the sense of first-order logic or universal algebra) over the signature consisting of a binary operation and a constant symbol, where the underlying set is $\mathbb{R}$, the binary operation is $+$, and the constant is $r$. – Eric Wofsey Feb 10 '16 at 11:04
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    This doesn't directly address your question, but it's a way of answering the motivating question "is $0$ the only number that behaves in a special manner under addition?", so it might be worth pointing out. For any nonzero $a$ and $b$ in $\mathbb{R}$, multiplication by $b/a$ is an automorphism of $(\mathbb{R},+)$ sending $a$ to $b$. This shows that all nonzero elements are indistinguishable via formulas in the language of addition. – Alex Kruckman Feb 17 '16 at 20:26

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Yes, this is true. Suppose $s(x_0,x_1,\dots,x_n)$ and $t(x_0,x_1,\dots,x_n)$ are terms in the language of addition such that $s(r,x_1,\dots,x_n)=t(r,x_1,\dots,x_n)$ for all $x_1,\dots,x_n\in\mathbb{R}$. We can choose $a_1,\dots,a_n\in\mathbb{R}$ such that $r,a_1,\dots,a_n$ are linearly independent over $\mathbb{Q}$. Let $F\subset\mathbb{R}$ be the subsemigroup generated by $r,a_1,\dots,a_n$. The linear independence of $r,a_1,\dots,a_n$ implies that $F$ is freely generated by $r,a_1,\dots,a_n$ as a commutative semigroup (here we use the fact that the free commutative semigroup on a set $\{x_0,x_1,\dots,x_n\}$ is the set of formal expressions $\sum m_i x_i$ where $m_i\in\mathbb{N}$ and at least one $m_i$ is nonzero). Thus the identity $s(r,a_1,\dots,a_n)=t(r,a_1,\dots,a_n)$ implies that actually $s(x_0,x_1,\dots,x_n)=t(x_0,x_1,\dots,x_n)$ whenever $x_0,\dots,x_n$ are elements of any commutative semigroup.

Eric Wofsey
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