Consider an sequence of unique values. Each value is composed of a surname and a first name in that order of significance. Based on that, I can define a complete ordering of values in the sequence.
However, I can also usefully define less fine-grained equivalence classes of items that have a "consistent" ordering, but where some values compare as equivalent (and have no particular ordering relative to each other) even though in the "full" ordering they compare as greater and lesser.
For example, sometimes, I may want to search an index for a particular surname, not caring about the first name.
In the most general case, there are finer-grain and coarser-grain but otherwise consistent ways to divide a set of values into equivalence classes and order those classes. There are generally many (mutually incompatible) finer grained orderings for any one coarser-grained ordering. It is possible to have a partitioning that is neither finer-grain nor coarser-grain than another different partitioning.
The trouble is that when I use the terminology of being "consistent", people get confused. A lot of people won't accept that an ordering in which $a=b$ is consistent with (but coarser grained) than one in which $a<b$.
I imagine that there are likely to be official terms to describe these relationships in abstract algebra, but my abstract algebra is pretty limited.
Am I right? If so, which is the relevant field of abstract algebra, and what are the relevant terms?