Naïve set theory MOC

Totally ordered set

A totally ordered set or connex is a poset in which any two elements are in relation. order Hence it a set 𝑆 equipped with a Relation set 𝑅 that is

1. reflexive — for all 𝑎 ∈𝑆, (𝑎,𝑎) ∈𝑅
2. transitive — if (𝑎,𝑏) ∈𝑅 and (𝑏,𝑐) ∈𝑅, then (𝑎,𝑐) ∈𝑅
3. antisymmetric — if (𝑎,𝑏) ∈𝑅 and (𝑏,𝑎) ∈𝑅, then 𝑎 =𝑏
4. total — for all 𝑎,𝑏 ∈𝑆, (𝑎,𝑏) ∈𝑅 or (𝑏,𝑎) ∈𝑅

Viewing Posets as categories, a this is equivalent to a connex category. A subset of a poset that is total is called a chain.

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