Trivial temperament
A trivial temperament is something that fits the mathematical definition of regular temperament, but is a unique, extreme case that people might be uncomfortable calling a "temperament". There are two types of trivial temperaments: just intonation, which leaves all intervals untempered, and single-pitch tuning, which tempers out all intervals.
Just intonation
The mapping for a just intonation subgroup of rank n is an n×n identity matrix, and transforms said subgroup to itself. In musical terms, this means that nothing is tempered. The set of commas that are tempered out is {1/1}, but that is still a valid set, so just intonation still counts as valid regular temperaments.
There is an infinite family of these temperaments, one for each subgroup of JI. The 2-limit version is equivalent to 1et. The 3-limit version, or pythagorean tuning, is a rank-2 temperament, which has all the properties of any other rank-2 temperament except that it tempers out no commas. 5-limit JI is rank-3, 7-limit JI is rank-4, etc.
Vector proposes the name identity temperament[idiosyncratic term] for this family of temperaments.
Single-pitch tuning
The single-pitch tuning is the rank-0 temperament, in which every interval is a comma. Thus all notes are represented by the same note. This is different from 1edo because not even octaves exist. The mapping for this is the 0-val, ⟨0 0 … 0], and its wedgie is a single entry.
As with JI, there is technically a temperament of a single pitch for every subgroup.
Gene Ward Smith proposes the name unison temperament for this family of temperaments[1], as all intervals are equated to the unison. Keenan Pepper proposes the name Om temperament[idiosyncratic term]. Om is a reference to that syllable's use in Hindu meditation practices, for there is only one temperament-distinct pitch in the whole system, in the same way that Om in the meditation sense is the only word you need to create the whole universe.