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Let and be algebraic structures of the same species. A homomorphism is a function that preserves the structure of the species.
If and are partially ordered sets, a homomorphism from to is a function such that for all , if , then .
If and are groups, with group law of , then a homomorphism is a function such that for all , Similarly, if and are fields or rings, a homomorphism from to is a function such that for all , In other words, distributes over addition and multiplication.