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 $(S,\le )$ is said to satisfy the third Grothedieck axiom AB3 iff for all families $(x_{\alpha }{)}_{\alpha \in A}$ indexed over a set $A$, a least upper bound $\underset{\alpha \in A}{\bigvee }{x}_{\alpha }$ exists.

Definition (AB3*):

A preordered class $(S,\le )$ is said to satisfy the dual third Grothedieck axiom AB3* iff for all families $(x_{\alpha }{)}_{\alpha \in A}$ indexed over a set $A$, a greatest lower bound $\underset{\alpha \in A}{\bigwedge }{x}_{\alpha }$ exists.

Definition (AB5):

A class $ℬ$ of preordered sets is said to satisfy the fifth Grothendieck axiom AB5 if and only if for all $(S,\le )\in ℬ$ and all families of elements $(x_{\alpha }{)}_{\alpha \in A}$ of elements of $S$ and each element $y\in S$, we have

$\left(\underset{\alpha \in A}{\bigvee }{x}_{\alpha }\right)\wedge y=\underset{\alpha \in A}{\bigvee }(x_{\alpha }\wedge y)$

Definition (AB5*):

A class $ℬ$ of preordered sets is said to satisfy the dual fifth Grothendieck axiom AB5* if and only if for all $(S,\le )\in ℬ$ and all families of elements $(x_{\alpha }{)}_{\alpha \in A}$ of elements of $S$ and each element $y\in S$, we have

$\left(\underset{\alpha \in A}{\bigwedge }{x}_{\alpha }\right)\vee y=\underset{\alpha \in A}{\bigwedge }(x_{\alpha }\vee y)$

Definition (AB6):

A class $ℬ$ of preordered sets is said to satisfy the sixth Grothendieck axiom AB6 if and only if for all $(S,\le )\in ℬ$