Subalgebra-related congruences

The algebraic structures studied in classical abstract algebra: groups, rings, fields, vector spaces, modules, etc allow theorists to brush aside congruences and focus on subalgebras instead. Most abstract algebra textbooks describe how normal subgroups, ideals, submodules, etc produce the congruence relations. This can be formalized with a mapping between the two universal algebra lattices $Sub(A)$ and $Con(A)$. \[ f: (K \subseteq Sub(A)) \to Con(A) \] I call this the congruencization mapping, which is a term which probably hasn't been used before. The idea is that congruences are comparatively more complicated, so if we can construct them from simpler objects that can save us a bit of work. Indeed, this is the case with groups and related structures where $Con(A)$ is fully determined by the sublattice of normal subgroups. In semigroups, separate mappings to $Con(A)$ need to be considered. The obvious one that comes to mind is that in a Rees congruence semigroup there is a mapping from ideals to congruences. \[ f: (Ideals(S) \subseteq Sub(S)) \to Con(A) \] Once we have formally defined these mappings to $Con(A)$ we can separately consider their properties. It is clear that in both cases, the larger the subalgebra the bigger the congruence, so the mapping is monotone. In particular, the larger the ideal, the larger the Rees congruence associated with it.