Determine if the following functions are homomorphisms / embeddings / isomorphisms. You must only answer: "not a homomorphism", "homomorphism buffnot an embedding", "embedding but not an isomorphism", "isomorphism". If not specified, the interpretations of the symbols in the signatures are the standard ones. a. n ↦ -n from (ℤ, 0, +) to (ℤ, 0, +). b. n ↦ n from (ℕ+, ≤) to A := (ℕ+, ≤A), where ≤A is the divisibility relation, i.e. n ≤A m exactly when n divides m. c. n ↦ n from A := (ℕ+, ≤A) to (ℕ+, ≤), where ≤A is the divisibility relation, i.e. n ≤A m exactly when n divides m.
Determine if the following functions are homomorphisms / embeddings / isomorphisms.
You must only answer: "not a homomorphism", "homomorphism buffnot an embedding", "embedding but not an isomorphism", "isomorphism".
If not specified, the interpretations of the symbols in the signatures are the standard ones.
a. n ↦ -n from (ℤ, 0, +) to (ℤ, 0, +).
b. n ↦ n from (ℕ+, ≤) to A := (ℕ+, ≤A), where ≤A is the divisibility relation, i.e. n ≤A m exactly when n divides m.
c. n ↦ n from A := (ℕ+, ≤A) to (ℕ+, ≤), where ≤A is the divisibility relation, i.e. n ≤A m exactly when n divides m.
The concepts of homomorphisms, embeddings, and isomorphisms play crucial roles in understanding the relationships between different structures. These concepts provide a way to map elements and operations from one algebraic structure to another while preserving certain algebraic properties.
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