\begin{definition}[Definition:Antilexicographic Order/Family]
Let $\struct {I, \preceq}$ be an ordered set such that its dual $\struct {I, \succeq}$ is well-ordered.
For each $i \in I$, let $\struct {S_i, \preccurlyeq_i}$ be an ordered set.
Let $\ds D = \prod_{i \mathop \in I} S_i$ be the Cartesian product of the family $\family {\struct {S_i, \preccurlyeq_i} }_{i \mathop \in I}$ indexed by $I$.
Then the '''antilexicographic order''' on $D$ is defined as:
:$\ds \struct {D, \preccurlyeq_D} := {\bigotimes_{i \mathop \in I} }^a \struct {S_i, \preccurlyeq_i}$
where $\preccurlyeq_D$ is defined as:
:$\forall u, v \in D: u \preccurlyeq_D v \iff \begin {cases} u = v \\
\exists i \in I: \paren {\forall j > i: \map u j = \map v j \text { and } \map u i \preccurlyeq_i \map v i}
\end {cases}$
{{Proofread|I haven't a clue about the nature of $I$ -- I am just guessing based on intuition.}}
\end{definition}