= Zero and one have the same quantifier-free type in the multiplicative integers
In the constant-free language with multiplication as its sole nonlogical symbol, every term in one free variable is $x^k$ for some $k\geq1$. Every <atomic formula> $x^r=x^s$ holds at both zero and one. Hence every <quantifier-free formula>, being a finite Boolean combination of those atoms, has the same truth value at both points. They have the same <quantifier-free type>, and neither singleton is quantifier-free definable. Quantifiers distinguish them: $\forall y(xy=x)$ defines zero, while $\forall y(xy=y)$ defines one. These formulas do not rely on named zero or one constants.
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