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Definition

A meet-semilattice or finitely complete (0,1)-category is a poset or (0,1)-category $(P, \leq)$ with

• a term $\top:P$

• a family of dependent terms

  $$a:P \vdash t(a):a \leq \top$$

  representing that $\top$ is terminal in the poset.

• a binary operation $(-)\wedge(-):P \times P \to P$

• two families of dependent terms

  $$a:P, b:P \vdash p_a(a, b):a \wedge b \leq a$$
  $$a:P, b:P \vdash p_b(a, b):a \wedge b \leq b$$

• a family of dependent terms

  $$a:P, b:P, (-)\otimes(-):P \times P \to P, p_a(a, b):a \leq a \otimes b, p_b(a, b):b \leq a \otimes b \vdash p(a,b):(a \otimes b) \leq (a \wedge b)$$

  representing that $\wedge$ is a product in the poset.

If $P$ is only a (0,1)-precategory, then it is called a finitely complete (0,1)-precategory

See also

• poset

• join-semilattice

• lattice