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Assume you have an algebraic problem that outputs a list of (finite) lattices on $n$ points for a given number $n$.

Question 1: Is there a way to search the internet/literature to see what exactly those lattices are?

Maybe there is an existing search engine for such kind of problems (and maybe for related combinatorial structures) already.

So the input is a list of lattice on $n$ vertices and the output might be where those lattices appeared before or at least a list of properties of such lattices have (so that they might be a subclass of a class of lattices with a name).

In my concrete case those lattices seem to contain the class of extremal and the class of semidistributive lattices (see for example https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/posets/lattices.html for the definitions).

Question 2: What are known classes (with a name) of lattices that contain the extremal and semidistributive lattices?

Maybe a common generalisation of those two classes has been studied already in the literature.

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I suggest the following directions, although I am not sure they may help in your specific case.

  1. As you have a list for each $n$, you may consider the sequence defined by their length, and query The On-Line Encyclopedia of Integer Sequences (OEIS).

  2. If your list of lattices for a given $n$ is ordered, you may build an integer sequence from each of them (for instance, the size of their covering relation, or their height) and query OEIS again.

  3. In the paper Sandpile models and lattices: a comprehensive survey, we show that some lattice classes (in particular, distributive and upper locally distributive lattices) are strongly related to various kinds of games; see Figure 17 and the preceding paragraphs. Maybe such an approach could be used in your case? It may for instance be a way to obtain each element of the lattice list from the previous one; or a way to obtain the lattice list for $n+1$ from the one for $n$.

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