notes-math-relationalHomomorphism

i expect there is already a name for this, but i don't know what it is (if you do, pls email me, i'd like to learn about it). i chose the name 'relational homomorphism' in analogy to relational bayes nets.

an example of what i am calling a 'relational homomorphism': say each person likes others from their country and dislikes others from elsewhere. Consider the relationship between (a) a model with the set of all people and the 'likes' relation, and (b) a model with a single country and two people, one of whom is a citizen of that country and one who isn't. This isn't a homomorhism but it seems to be something that should have a name.

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 what mathematical construct describes the followingsituation. The thing i'm after reminds me of relational bayes nets and of homomorphisms, hence the title of this email, although it is not a homomorphism.

Say there is a set of many people and the people are members of various clubs (there are many clubs, too). And say that two people are 'affiliates' if they are members of the same club. So we have a model with a set of people, a set of clubs, a membership relation between people and clubs, and an 'affiliate' relation between people. But the 'affiliate' relation is determined once you know the membership relation.

To be more concrete, let's say that "Structure A" has two clubs, and four people, with each club having two members, and no one is a member of more than one club.

Now consider another structure, "Structure B". In Structure B there is also a set of people, a set of clubs, a membership relation between people and clubs, and an 'affiliate' relation between people. But Structure B only has two people, and one club. One of the people is a member of the club, and one is not. The people are not affiliates of each other, but the first person is an affiliate of themself.

What is the relation between Structure A and Structure B? They are not homomorphic, because if you map both of the clubs from Structure A into the one club of Structure B, you'll get that everyone is affiliated with everyone else, which is not true in Structure A, so this map doesn't preserve the 'affiliate' relation (and similar arguments apply for other possible mappings).

But it seems to me that there is a special relationship between Structure A and Structure B, because Structure A is sort of a 'tiled' or 'iterated' version of Structure B, in that the rule for the determination of the affiliate relation from the membership relation is per-club, and Structure B captures this rule.

So what's it called?

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it is also reminiscent of Exchangable random structures.

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