Dave4Math / Courses / Ordered Sets

Ordered Sets

Master ordered sets and learn about partial orderings, mappings on ordered sets, chain conditions, order ideals, and much more. With this course, you get an in-depth look at ordered sets and an introduction to lattices. 


January 11, 2021

Created by Dave

This course begins Summer 2021.
Current Status
Not Enrolled
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Have you noticed that some courses have subject material that is difficult to learn upon first reading? Introducing Dave’s online math course: Ordered Sets. This course is very well developed and planned out. Dave does an outstanding job of explaining challenging topics. 

Who This Course is For

  1. Students in college who want to learn more mathematics.
  2. Anyone interested in ordered sets.
  3. Students who are enrolled in Introduction to Proofs and want to improve their grade.
  4. Anyone interested in majoring in mathematics, physics, or engineering.
  5. Anyone wanting to learn about partial orders, monotone mappings, residuated mappings, Galois connections, and lattices.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A calculator is not necessary.

What You’ll Learn in Ordered Sets

  • Partial Orderings
  • Up-Sets and Down-Sets
  • Closures
  • Spanning Sets
  • Bases
  • Chain Conditions
  • Order Ideals
  • Order-Preserving Mappings
  • Residuated Mappings
  • More on Closures
  • Isomorphisms
  • Max and Min with Galois Connections
  • Galois Connections
  • Closures and Dual Closures
  • Characterizations of Galois Connections
  • Semilattices and Lattices
  • Down-set Lattices
  • Sublattices
  • Lattice Morphisms
  • Complete Lattices

Course Description

In this course on ordered sets, we begin by detailing what a partial order relation is. We discuss up-sets and down-sets, linear independence, and bases for ordered sets. We also discuss order ideals and descending chain conditions on ordered sets. We then discuss monotone mappings, that is, mappings that preserve an ordering. Importantly, we then discuss residuated mappings and then Galois connections. We also discuss the computational aspects of Galois connections. We finish this part of the course by exploring closures and dual-closures.

In the latter part of the course, we study lattices, which are ordered sets with the meet and join operations defined in terms of the underlying ordering. We examine different kinds of lattices, including semi-lattices and complete lattices.

Recommended Prerequisites for Ordered Sets

I recommend the prerequisite course Elementary Set Theory. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free articles on introduction to proofs.

David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.