Automorphism of a first-order structure

An automorphism is a bijection from a structure to itself that preserves every operation and relation in its language.

For a fixed prime $p$, put $s_p(0)=0$ and $s_p(n)=(-1{)}^{v_p(n)}n$ for $n\ne 0$, where $v_p(n)$ is its P-adic valuation. Additivity of the valuation on products proves $s_p(mn)=s_p(m)s_p(n)$. Changing sign does not change the valuation, so $s_p^2$ is the identity. Thus this is a structure automorphism of $(\mathbb{Z},\cdot )$. At $p=2$ it fixes one and sends two to minus two, so it does not preserve the addition graph or the nonnegative integers. Automorphism invariance of parameter-free definability therefore excludes both from definability in this multiplicative language.

The automorphism group $\mathrm{Aut}(M)$ consists of all automorphisms of $M$, with composition as its group operation.

After adjoining a unary function symbol $f$, injectivity, surjectivity, preservation of each operation, and preservation in both directions of each relation form an axiom scheme whose models interpret $f$ as an automorphism.

A family $(f_g{)}_{g\in G}$ of automorphisms satisfying $f_g{f}_h=f_{gh}$ defines a group action. It is faithful exactly when distinct indices define distinct automorphisms, equivalently when $G$ embeds in the automorphism group.

A first-order structure is rigid when its identity map is its only automorphism. Every finite linear order is rigid, whereas $(\mathbb{Z},<)$ is nonrigid because every translation is an order automorphism.

Rigidity here means absence of nonidentity automorphisms of a first-order structure; it does not assert the mechanical rigidity of a rod-and-hinge framework.

If a theory has arbitrarily large finite nonrigid models, expand its language by a unary function $f$, axiomatize that $f$ is an automorphism, require $f$ to move some element, and add sentences demanding at least $n$ elements for every $n$. Every finite fragment has a sufficiently large finite nonrigid model, so compactness supplies an infinite nonrigid model.