Reading part 4 of Category Theory Illustrated by Jencel P. on stream.

• Previous reading is 20251204190725
• This part is about "Orders", which is a way of ordering objects.
• An Order is composed of two things: An underlying set and a binary relation
• Orders follow the following laws:
  • Reflexivity - each object in the underlying set is either greater than or equal to itself
  • Transitivity - if a >= b and b >= c, then a >= c
  • Antisymmetry - The way you define the ordering cannot give contradictory results, such that there are no ties and that if x < y, then it is not true that y < x
  • Totality - For any two elements, one must always be bigger than the other. In other words, every element in the underlying set must be comparable to each other. Technically makes the reflexivity law redundant, since reflexivity is a special case of totality.
• Orders that do not follow the totality law (but follow reflexivity) are "partial orders" and linear orders are "total orders"
• In a partial order, each different linear order is known as a "chain".
• Greatest object - The object that is definitively greater than all the others in a partial or total order.
• Smallest object - The object that is definitively smaller than all the others in a partial or total order.
• I don't understand what the author means when they try to define what a join is. They seem to say conflicting things:
  • "[...] the join of a and b is defined as the smallest element that is bigger than both a and b (i.e. smallest c for which a ≤ c, and b ≤ c.)"
  • "Given any two elements in which one is bigger than the other (e.g. a ≤ b), the join is the bigger element (in this case b)."
  • So it seems like they're saying that the join is both the smallest element and the biggest element that is bigger than a and b.
• Meet - Given two elements a and b, a meet is the biggest element that is smaller than a and b
• Hasse Diagrams - A diagram in which bigger elements are positioned above smaller ones.
• It appears that poset is short for partially ordered set.
• An Inclusion order is one way of writing an order in which every element is written as a combination of the elements that are smaller than itself. For example, in a divides poset, 9 is written as (3, 3), 6 is written as (2, 3), and 10 is written as (5, 2). 2, 3, and 5 are written as themselves (since they are not divisible by anything other than 1).
• An order isomorphism is an isomorphism between two sets which does not change the ordering of the set.
• Birkhoff’s representation theorem
  • Finite partial orders that meet the following two criteria:
    1. All elements have joins and meets.
    2. Those meet and join operations distribute over one another, that is if we denote joins as meets as ∨ or ∧, then x ∨ (y∧z) = (x∨y) ∧ (x∨z).
  • Finite partial orders that satisfy the first criteria are known as lattices, those that satisfy both are known as distributive lattices.
• The "prime" elements used to construct other elements in an inclusion are known as join-irreducible elements.
• Lattices that have a least and greatest element are called bounded lattices
• Partial orders that just have joins and no meets (or vice versa) are called semilattices.
• Partial orders with meets for every pair of elements is known as a meet-semilattice.
• I should read that essay "A City is Not a Tree" by Christopher Alexander.
• They mention something called "Formal concept analysis" here and I don't really understand what the author is trying to say. Maybe it will be useful if i look into it more?
• Preorders follow two laws: transitivity and reflexivity.
  • If a preorder is symmetric, then it is a equivalence relation.
  • If a preorder is anti-symmetric, then it is a partial order. (I'm assuming it's possible that a preorder could be a total order if it follows totality, but they don't explicity say this)
  • equivalence classes - sets of elements in a preorder which are symmetric.
  • For every preorder, we can define a partial order of the equivalence classes.
• Like sets, an order is a type of category. Orders only allow for one morphism between two objects (that is, is one object less than or equal to another)
• Comparing orders to categories, "join" is to "coproduct" as "meet" is to "product"