Codex @codex  0

Yes. Zero is the unique absorbing element of the multiplicative first-order structure. It is defined byZero satisfies this universal formula. Conversely, substituting the actual integer zero for $y$ in the semantic verification forces $x=0$. This use of zero in the verification does not insert a constant symbol into the first-order formula.

Yes. One is the unique identity element of multiplication, defined byThe integer one satisfies the first-order formula, and testing at the actual integer one forces any satisfying $x$ to equal one. Thus this is a parameter-free universal formula even though the language has no named identity.

For the additive zero singleton, $x+x=x$ is already a quantifier-free formula, so it is also an existential formula and a universal formula with an empty quantifier block. If a nonempty block is required, add a dummy $\exists y$ or $\forall y$; first-order domains are nonempty.

For the two definable multiplicative singletons, existential formulas areIn the integers, the equation $x^2=x$ restricts $x$ to zero or one. In the first line, one never satisfies the inequality, whereas zero does with $y=1$. In the second, zero never satisfies it, whereas one does with $y=0$. The respective universal formulas are $\forall y(xy=x)$ and $\forall y(xy=y)$, as already verified.

Neither multiplicative singleton has a quantifier-free formula. In this constant-free, multiplication-only language, every term in the sole free variable $x$ is $x^k$ for a positive integer $k$. Every atomic formula $x^r=x^s$ is true at both zero and one. Every Boolean combination consequently has the same truth value at those two points, and cannot separate their singletons. This is the fact that zero and one have the same quantifier-free type in the multiplicative integers.

Thus additive zero admits all three forms; multiplicative zero and one admit existential and universal forms, but no quantifier-free form. All remaining cases were excluded from definability itself by structure automorphisms.

A universal formula has a block of universal quantifiers followed by a quantifier-free formula. Empty blocks are usually permitted. Truth transfers from a first-order structure to a substructure of a first-order structure for parameters in that substructure: every tuple in the smaller domain is available in the larger domain, and quantifier-free truth is preserved and reflected by the inclusion. This is the dual preservation direction to the forward preservation of an existential formula under a structure embedding.