Connected relation in the context of "Total order"

In set theory a serial relation is a homogeneous relation expressing the connection of an element of a sequence to the following element. The successor function used by Peano to define natural numbers is the prototype for a serial relation.

Bertrand Russell used serial relations in The Principles of Mathematics (1903) as he explored the foundations of order theory and its applications. The term serial relation was also used by B. A. Bernstein for an article showing that particular common axioms in order theory are nearly incompatible: connectedness, irreflexivity, and transitivity.

In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation ${\displaystyle \leq }$ on some set ${\displaystyle X}$, which satisfies the following for all ${\displaystyle a,b}$ and ${\displaystyle c}$ in ${\displaystyle X}$:

1. ${\displaystyle a\leq a}$ (reflexive).
2. If ${\displaystyle a\leq b}$ and ${\displaystyle b\leq c}$ then ${\displaystyle a\leq c}$ (transitive).
3. If ${\displaystyle a\leq b}$ and ${\displaystyle b\leq a}$ then ${\displaystyle a=b}$ (antisymmetric).
4. ${\displaystyle a\leq b}$ or ${\displaystyle b\leq a}$ (strongly connected, formerly called totality).

Requirements 1. to 3. just make up the definition of a partial order.Reflexivity (1.) already follows from strong connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders.Total orders are sometimes also called simple, connex, or full orders.