  We consider different classes of combinatory structures related to Krivine
realizability. We show, in the precise sense that they give rise to the same
class of triposes, that they are equivalent for the purpose of modeling
higher-order logic. We center our attentions in the role of a special kind of
Ordered Combinatory Algebras-- that we call the "Krivine ordered combinatory
algebras" ($\mathcal{KOCA}$s)-- that we propose as the foundational pillars for
the categorical perspective of Krivine's classical realizability as presented
by Streicher.
  Our procedure is the following: we show that each of the considered
combinatory structures gives rise to an indexed preorder, and describe a way to
transform the different structures into each other that preserves the
associated indexed preorders up to equivalence. Since all structures give rise
to the same indexed preorders, we only prove that they are triposes once: for
the class of $\mathcal{KOCA}$s.
  We finish showing that in $\mathcal{KOCA}$s, one can define realizability in
every higher-order language and in particular in higher-order arithmetic.
