Order, lattices, ordered algebraic structures Definition

Product order

Builds on Partial order Cartesian product

Definition

Let $I$ be a set and let $(P_i, \le_i)$ be a partially ordered set for each $i \in I$ . The product order on the Cartesian product $\prod_{i \in I} P_i$ is the relation $\le$ given by $x \le y \iff x_i \le_i y_i \text{ for all } i \in I.$

Remarks

It is a partial order: reflexivity, antisymmetry and transitivity hold in each coordinate, hence for tuples. For two factors, $(a, b) \le (a', b')$ means $a \le a'$ and $b \le b'$ . Example: in $\N \times \N$ , $(1, 2) \le (3, 2)$ , since $1 \le 3$ and $2 \le 2$ . Non-example: $(1, 2)$ and $(2, 1)$ are incomparable, since the first coordinate rises and the second falls. For finitely many factors, the product order is contained in the lexicographic order on the same set, which is total when all factors are total. The power set of a set $I$ , ordered by inclusion, is isomorphic to the product order on $\set{0, 1}^I$ .

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Cartesian product concept
Partial order concept
Product order definition

Tag	`YT45QJKPNK`
URI	`m2://YT45QJKPNK`
Kind	definition
Writing	W0 stub
MSC	06A06
Defines	Product order
Uses	Partial order, Cartesian product

Product order

Definition

Remarks

Reading path

Further reading

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