# Order Theory/The Grothendieck axioms

AB3, AB3*, AB5 and AB5* are actually due to von Neumann, but Grothendieck's name stuck.

**Definition (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.

**Definition (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.

**Definition (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

**Definition (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

**Definition (AB6)**:

A class of preordered sets is said to satisfy the **sixth Grothendieck axiom AB6** if and only if for all