Direct product of posets

Definition

The direct product of posets is a construction that combines two or more partially ordered sets (posets) into a new poset, where the elements are tuples formed from the elements of the original posets, and the order is defined component-wise. This means that for two tuples to be comparable in the direct product, each corresponding component must also be comparable in their respective posets. Understanding this concept is essential for exploring relationships and structures within different posets.

5 Must Know Facts For Your Next Test

1. In the direct product of two posets, say P and Q, an element (a, b) is less than or equal to (c, d) if and only if a ≤P c and b ≤Q d.
2. The direct product of finite posets results in a finite poset, and if the original posets are finite chains, the resulting product will also be a finite chain.
3. The direct product preserves certain properties such as being a bounded poset; if both posets have least and greatest elements, so does their direct product.
4. The direct product can be extended to any finite number of posets, leading to tuples of higher dimensions with similar component-wise comparison rules.
5. The direct product of two totally ordered sets results in a poset that is not totally ordered unless one of the sets is a singleton.

Review Questions

• How does the direct product of posets relate to the individual ordering properties of each poset?
  • The direct product of posets relies on the individual ordering properties of each original poset to establish its own order. In this construction, for two tuples to be ordered, each corresponding element must follow the specific order defined in its respective poset. This means that understanding how elements are compared in their original contexts is crucial for comprehending their relationships in the direct product.
• Discuss how the direct product affects the boundedness properties of the original posets and provide an example.
  • The direct product preserves boundedness properties from the original posets. If both constituent posets have least and greatest elements, these will also exist in the direct product. For example, if P has a least element p_0 and Q has a least element q_0, then (p_0, q_0) will serve as the least element in their direct product. Similarly, if P has a greatest element p_max and Q has a greatest element q_max, then (p_max, q_max) will be the greatest element in the product.
• Evaluate the implications of forming a direct product of posets when both are totally ordered sets versus when one is not; what can be concluded about their structure?
  • When forming a direct product of two totally ordered sets, the resulting structure retains some order but typically becomes more complex. If both sets are totally ordered, every pair will be comparable in the product. However, if one set is not totally ordered while the other is, then there will be elements in the resulting structure that are not comparable. This illustrates how introducing non-totally ordered components adds layers to order complexity within the combined structure.

Related terms

Partially Ordered Set (Poset): A set equipped with a binary relation that is reflexive, antisymmetric, and transitive, which allows for the comparison of some elements.

Product Order: An ordering on a product set where two elements are comparable if each corresponding component meets the order relation defined in their respective posets.

Cartesian Product: The Cartesian product of sets is the set of all ordered pairs (or tuples) from two or more sets, forming the basis for constructing products of posets.