Totally ordered set

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

reflexive — for all $𝑎 \in 𝑆$ , $(𝑎, 𝑎) \in 𝑅$
transitive — if $(𝑎, 𝑏) \in 𝑅$ and $(𝑏, 𝑐) \in 𝑅$ , then $(𝑎, 𝑐) \in 𝑅$
antisymmetric — if $(𝑎, 𝑏) \in 𝑅$ and $(𝑏, 𝑎) \in 𝑅$ , then $𝑎 = 𝑏$
total — for all $𝑎, 𝑏 \in 𝑆$ , $(𝑎, 𝑏) \in 𝑅$ or $(𝑏, 𝑎) \in 𝑅$

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

#state/tidy | #lang/en | #SemBr