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I'm interested in general heuristics where, for specific algebraic structures, we introduce new maps that are "almost" homomorphisms (or "almost" isomorphisms) but not quite so. Here are some that I have encountered in group theory (and may also be used in ring theory and commutative/noncommutative algebra):

1. A "pseudo-homomorphism" (sometimes also called "quasi-homomorphism") which is a set map for groups whose restriction to any abelian subgroup is a homomorphism. In other words, if two elements commute, then the image of the product is the product of the images. General idea: require the composition with certain kinds of injective maps to be homomorphisms.
2. A "1-homomorphism" which is a set map for groups whose restriction to any cyclic subgroup is a homomorphism. General idea: require that the restriction to any subalgebra generated by at most $k$ elements is a homomorphism. Note that for algebras defined using at most 2-ary operations, the only interesting case is $k = 1$.
3. An element map that sends subgroups to subgroups. The induced map on the lattice of subgroups is termed a "projectivity". General idea: Require the map to induce a map on some derivative structure (e.g., the lattice of subalgebras) that looks like it could have come from a homomorphism.

My main interest is from a group theory perspective but I'd also be interested in constructions for other algebraic structures.

ADDED LATER: There have been a lot of interesting examples here. My original focus was to look at properties of maps that could be considered, at least in principle, between two arbitrary objects. Preferably something that could be composed to give a new category-of-sorts. But there've been some interesting examples of maps that go to fixed target groups and whose definition uses additional information about the structure of those target groups. These could also be of potential interest, so please feel free to give such examples too.

2. A "1-homomorphism" which is a set map for groups whose restriction to any cyclic subgroup is a homomorphism. General idea: require that the restriction to any subalgebra generated by at most $k$ elements is a homomorphism. Note that for algebras defined using at most 2-ary operations, the only interesting case is $k = 1$.