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When studying random geometries and related mathematical/physical stuff conflicting naming convention pops up regarding the naming of the different ensemble types of triangulations (in general dimensions, but let's say d = 2,3,4).

I can differentiate between some, let's say combinatorial triangulations, where each simplex can be connected to another one via a (d-1) dimensional sub simplex, and a (d-1) dimensional sub simplex shares precisely 2 simplices. One can allow for multiple adjacencies of the simplices without degenerating the sub-simplices (so each (k)-simplex shares a given number of (k+1)-simplices), typically in physics we refer to such things as "self energies". However, one can say, that the endpoints of an edge can be the same vertex, and in that case one introduces tadpoles in the dual-graph of the triangulations, creating degenerate triangulations. Some calls already the previous one degenerate.

Could someone clarify the proper "naming convention" of the type of triangulations (together with an example, if it differs from the three classes of mine).

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    $\begingroup$ In math, the first kind of triangulation you describe are just called simplicial compelxes, and the second kind are called simplicial posets (this was mentioned in comments to another question you asked here). The third "degenerate" kind I'm not so sure, it may be close to the notion of $\Delta$-complex or simplicial set. $\endgroup$
    – Sam Hopkins
    Commented Dec 15, 2023 at 13:03
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    $\begingroup$ In arxiv.org/abs/1805.10186 the authors consider $\Delta$-complexes where the simplices may be glued to themselves along a permutation of their vertices, and they call these objects "symmetric $\Delta$-complexes." $\endgroup$
    – Sam Hopkins
    Commented Dec 15, 2023 at 13:24

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