Let T^S = {f| f:S->T} be all functions from a source set S to a target structure (set + operations + relations + axioms) T.
You can lift all the operations and relations from T to T^S, and you'll get a structure with the same type signature.
Universal equations involving operations remain true when lifted. Therefore if T is a variety[0], T^S is a variety of the same type.
So for example if S a set with 2 elements, then T^S is TxT + lifted properties. If T is an Abelian group then TxT is also an Abelian group. If T is a ring, TxT is also a ring. If T is a field, TxT is not a field since (0,1) has no inverse.
What about the relations, what types of identities remain true when lifted form T to T^S?
You can lift all the operations and relations from T to T^S, and you'll get a structure with the same type signature.
Universal equations involving operations remain true when lifted. Therefore if T is a variety[0], T^S is a variety of the same type.
So for example if S a set with 2 elements, then T^S is TxT + lifted properties. If T is an Abelian group then TxT is also an Abelian group. If T is a ring, TxT is also a ring. If T is a field, TxT is not a field since (0,1) has no inverse.
What about the relations, what types of identities remain true when lifted form T to T^S?
[0] : https://en.wikipedia.org/wiki/Variety_(universal_algebra)