# Is there a name for posets in which every nonempty finite subset with a lower bound has an infimum?

A poset in which every nonempty finite subset has an infimum is called a semilattice. I am wondering if there is a name coined for posets in which every nonempty finite lower-bounded subset has an infimum.

• I might be wrong, but for a nonempty finite subset to have an infimum, it must be lower-bounded, since the infimum is the greatest element (if it exists) of the set of lower bounds. So both definitions amount to the same. – Zamu Feb 27 '16 at 17:57
• Let me point out that by "associativity" reasons such property is equivalent to requiring that every finite subset of cardinal 2 which is lower-bounded has an infimum. But I am not familiar with any name for such notion. – boumol Feb 27 '16 at 18:29
• This is very close to the dual notion of a dcpo (except for only looking at finite subsets). Take a look at the notion of "filtered complete partial order" in en.wikipedia.org/wiki/Directed_complete_partial_order – boumol Mar 2 '16 at 9:30
• @Zamu no this is not the same, consider e.g. the poset made of two incomparable elements: this is not a semilattice, whule it satisfies the condition that every nonempty finite lower-boynded subset has an infimum. – polmath Mar 30 '16 at 16:13
• @boumol I dont think this is equivalent to the same condition where one replaces "finite" by "with two elements", though i dont have an immediare counterexample. – polmath Mar 30 '16 at 16:16