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

Definition (AB3*):

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

Definition (AB5):

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

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

Definition (AB5*):

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

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

Definition (AB6):

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