Definition:Simple Order Product/Family
Jump to navigation
Jump to search
Definition
Let $I$ be an indexing set.
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 simple order product on $D$ is defined as:
- $\ds \struct {D, \preccurlyeq_D} := {\bigotimes_{i \mathop \in I} }^s \struct {S_i, \preccurlyeq_i}$
where $\preccurlyeq_D$ is defined as:
- $\forall u, v \in D: u \preccurlyeq_D v \iff \forall i \in I: \map u i \preccurlyeq_i \map v i$
Also see
- Results about order products can be found here.
Sources
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: Exercise $37$