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Lattices in the sense of discrete subgroups of Euclidean spaces, as used in number theory, discrete geometry, Lie groups, etc. (Not to be confused with lattice theory or lattices as used in physics! For lattices (ordered sets), use the tag: [lattice-theory])
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Is there existing terminology for this technical condition on semilattices?
in lattice theoretic terms, the condition is "finite breadth"
(more on this in the comment above and with search engine requests for "breadth lattice")