Is there a good (comprehensive) ontology of abstract algebraic structures available anywhere online that shows their relationship with each other? For instance, magmas, setoids, rings, fields, groups, semirings, and lesser known structures mapped via their hierarchical relation with each other, as well as other, lesser known structures? Ideally, not just “substructureof” relationships will be mapped, but also compositional relationships, so vector spaces are shown to have a relationship with fields, whilst modules have a relation with rings (though that could be messy!).
It seems to me that you are asking for all canonical (whatever that means, which is problematical) forgetful functors among all possible equational categories (or maybe even more general than equational since fields cannot be equationally defined) and that doesn’t even seem possible in principle. For example, I have never heard of a setoid and I probably know about many equational categories you have never heard of. There is one whose objects are all modules, over all possible rings. Essentially an object is a pair (R,M) where R is a ring and M is an R-module. But that looks like a 2-sorted theory, although there is an equivalent single sorted theory.
A “setoid” is just a fancy name for an equivalence relation, Hari (perhaps carrying certain connotations regarding what one intends to do with this equivalence relation…).
I agree that it would be impossible to to have any “comprehensive” such chart (I will go further than Hari and say the task, no matter how it starts, will ultimately develop, it seems to me, into seeking a chart of all functors between all categories), but perhaps there is still a nice poster of some sort out there which nevertheless covers enough particular familiar relationships between enough particular familiar algebraic structures in line with the proposed archetypes as to please the OP. Though, that having been said… I don’t know of any.
(Also, tsk-tsk for leaving such a basic example as “categories” themselves out of your list (yeah, they’re not purely equational, but they’re defined by an “essentially algebraic” theory, which is better than fields can claim)…)
Well, I’m not asking for every algebraic structure, which of course would be impossible. I’m asking for a chart of the major ones used in everyday mathematics, and more esoteric minor structures that are hidden away in subfields of mathematics.
After searching, Wikipedia goes some way to providing what I need in its article on algebraic structure (for instance, I’d never even heard of bands and Moufang loops). Is there an even more comprehensive list than this?
Yes, it’s a set with an associated equivalence relation. You usually talk about setoids in the context of dependent type theory, where proof irrelevance matters. Your setoids there are then all proofs of a proposition with multistep \beta-reduction as the equivalence relation. This gives a hint as to why I’m asking the question 
Hm… That hint actually instills confusion in me which wasn’t there before. Why are you asking the question? Something about formalizing mathematics with dependent type theory?