Let $(A_n)_{n\in\mathbb N}$ be an increasing sequence of sets and let, for any $n\in\mathbb N$, $R_n$ be a relation on $A_n$ (i.e. a subset of $A_n\times A_n$, such that the following hold
- For any $n\in\mathbb N$, the relation $R_n$ is reflexive and symmetric on $A_n$,
- For any $n\le m\le k$, given $a_n\in A_n$, $a_m\in A_m$ and $a_k\in A_k$, such that $a_n R_m a_m$ and $a_m R_k a_k$, there exists some $l\ge k$ such that $a_n R_l a_k$.
That is to say, the relations $R_n$ themselves are not necessarily equivalence relations, but their union is an equivalence on $A_n$.
I was wondering if such a phenomenon has a commonly used name (e.g. pre-equivalence or graded equivalence relation). I came upon a sequence of relations of this sort recently, and would rather use the commonly used name, if such exists.
Thank you!