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A semigroup is a set $S$ together with a binary operation that is associative. Examples of semigroups are the set of finite strings over a fixed alphabet (under concatenation) and the positive integers (under addition, maximum, or minimum). A monoid is a semigroup with a neutral element. Of course, any group is also a monoid/semigroup.

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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")
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