arity n and each ⃗a ∈ An, we have RA( ->a) ⇔ R B(h(->a)). Conclude the analogous statement for σ-embeddings. b. Deduce that if σ has no relation symbols, then σ-isomorphism
Let σ be a signature and Let A := (A, σA), B := (B, σB) be σ-structures. Let h : A → B be a map.
a. Prove: h is a σ-isomorphism if and only if h is a bijective σ-homomorphism with ⇔ in condition (iii) of the definition of homomorphism, i.e. for each relation symbol R of arity n and each ⃗a ∈ An, we have RA( ->a) ⇔ R B(h(->a)). Conclude the analogous statement for σ-embeddings.
b. Deduce that if σ has no relation symbols, then σ-isomorphism is the same as a bijective σ-homomorphism, and σ-embedding is the same as an injective σ-homomorphism.
Definition: A σ-automorphism of a σ-structure A is just a σ-isomorphism h : A → A. The identity map idA is a σ-automorphism of A, but there are typically many other σ-automorphisms.
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