Learn Partial Orders
What you will be able to do
- Given a relation on a finite set, determine whether it is a partial order by checking reflexivity, antisymmetry, and transitivity.
- Draw a Hasse diagram for a finite partial order.
- Given a finite lattice, compute the meet and join of two elements.
- Identify whether a poset is a lattice, a bounded lattice, or a complete lattice.
Prerequisites
None. This skill is a starting point. It requires comfort with sets and relations (what a set is, what a binary relation is) but no formal training.
Lessons
[Gap: no lesson on partial orders exists yet that follows the lesson design principles (backward design, concrete-before-abstract, exercises). The existing heyting-algebras lesson in mathematics/objects/posets/curricula/ covers Heyting algebras but assumes partial orders are already known. A dedicated partial orders lesson needs to be written.]
When that lesson exists, it should cover: the definition via three axioms checked against concrete examples (divisibility, subset inclusion, ancestry), Hasse diagrams as a tool for visualization, and meets/joins as greatest lower bound and least upper bound.
Scope
This skill covers partial orders, lattices, and basic lattice operations. It does not cover:
- Heyting algebras or Boolean algebras (covered by learn-heyting-algebras, which depends on this skill)
- Closure operators (covered by a separate skill, also depending on this one)
- Order-preserving maps, Galois connections, or domain theory
- Total orders, well-orders, or ordinals
Verification
Given the divisibility relation on {1, 2, 3, 4, 6, 12}: draw its Hasse diagram, compute the meet and join of 4 and 6, and determine whether this poset is a lattice.