Order-Theory on emsenn.net

Monads and Comonads

Fri, 06 Mar 2026 00:00:00 +0000

¶Entry conditions

Use monads and comonads when you can:

Start from a category (or a poset viewed as a thin category).
Identify an endofunctor T : C -> C that “wraps” or “annotates” objects.
Ask whether the wrapping is coherent: can it be applied once, twice, or unwrapped, in ways that compose consistently?

¶Definitions

A monad on a category C is a triple (T, eta, mu) where:

Closure Operators and Fixed Points

Sun, 01 Mar 2026 00:00:00 +0000

¶Entry conditions

Use closure operators when you have:

A partially ordered set (or complete lattice) of structures.
A process that enlarges or completes a structure in a canonical way.
A need to find the smallest structure that is “closed” — stable under the process.

¶Definitions

A closure operator on a partially ordered set $(P, \leq)$ is a function $c : P \to P$ satisfying three properties [@davey_IntroductionLatticesOrder_2002; @erne_ClosureOperators_2009]:

Heyting Algebras

Sun, 01 Mar 2026 00:00:00 +0000

¶Entry conditions

Use Heyting algebras when you can:

Start from a lattice with top $\top$ and bottom $\bot$ .
Define an implication operation $a \Rightarrow b$ for every pair of elements.
Accept that some elements may lack complements — excluded middle may fail.

¶Definitions

A bounded lattice is a lattice with a greatest element $\top$ and a least element $\bot$ .

A Heyting algebra [@esakia_HeytingAlgebras_2019] is a bounded lattice $(H, \leq, \wedge, \vee, \bot, \top)$ equipped with a binary operation $\Rightarrow$ (called Heyting implication) satisfying:

Lattices

Fri, 26 Dec 2025 00:00:00 +0000

¶What this lesson covers

This lesson introduces lattices: partially ordered sets where every pair of elements has a greatest lower bound (meet) and a least upper bound (join). By the end, you will be able to determine whether a poset is a lattice, compute meets and joins, and recognize bounded and complete lattices.

¶Why it matters

A partial order lets you say “this is below that,” but it does not guarantee you can combine two elements. Given two topics in a library — say Modal Logic and Set Theory — a partial order under “is more specific than” tells you they are both below Mathematics. But is there a most specific topic that encompasses both? Is there a most general topic that both encompass?

Partial Orders

Fri, 26 Dec 2025 00:00:00 +0000

¶What this lesson covers

This lesson introduces partial orders: the mathematical structure that captures the idea of “this thing is at least as much as that thing” in a consistent way. By the end, you will be able to identify when a relation is a partial order, draw its Hasse diagram, and explain why partial orders matter for the structures that follow (lattices, Heyting algebras, closure operators).

¶Why it matters

Suppose you are organizing a library. You could sort books alphabetically — a total order where every pair of books has a definite ranking. But suppose instead you want to organize by topic. Introduction to Logic is more specific than Mathematics, and Modal Logic is more specific than Introduction to Logic. But how does Modal Logic compare to Set Theory? They are both more specific than Mathematics, but neither is more specific than the other. They are incomparable.