Without the finiteness restriction a posetal category with all products is automatically cocomplete, and dually, by a theorem about complete lattices.
32.
*PM : complete lattice homomorphism, id = 9241 new !-- WP guess : complete lattice homomorphism-- Status:
33.
*PM : complete lattice homomorphism, id = 9241 new !-- WP guess : complete lattice homomorphism-- Status:
34.
More generally, these definitions make sense in any partially ordered set, provided the infima exist, such as in a complete lattice.
35.
More special versions of both are continuous and algebraic continuous lattices and algebraic lattices, which are just complete lattices with the respective properties.
36.
Complete distributivity is a self-dual property, i . e . dualizing the above statement yields the same class of complete lattices.
37.
Would there be a free construction of complete lattices from posets, then both constructions could be composed, which contradicts the negative result above.
38.
Let ( L, \ leq ) be a complete lattice, with infimum and supremum symbolized by \ wedge and \ vee, respectively.
39.
Complete distributivity is again a self-dual property, i . e . dualizing the above statement yields the same class of complete lattices.
40.
If is a partially ordered set, a " completion " of means a complete lattice with an order-embedding of into.
