Order Theory/The Grothendieck axioms

AB3, AB3*, AB5 and AB5* are actually due to von Neumann, but Grothendieck's name stuck. AB3 A preordered class is said to satisfy the third Grothedieck axiom AB3 iff for all families indexed over a set , a least upper bound exists. AB3* A preordered class is said to satisfy the dual third Grothedieck axiom AB3* iff for all families indexed over a set , a greatest lower bound exists. AB5 A class of preordered sets is said to satisfy the fifth Grothendieck axiom AB5 if and only if for all and all families of elements of elements of and each element , we have AB5* A class of preordered sets is said to satisfy the dual fifth Grothendieck axiom AB5* if and only if for all and all families of elements of elements of and each element , we have AB6 A class of preordered sets is said to satisfy the sixth Grothendieck axiom AB6 if and only if for all