What's the definition of "is less than" and how is it different from "strict total ordering"? Strict total ordering on set $A$ means a relation $<:{\forall}x,y\,{\in}\,A:x≠y:(x<y)∨(y<x)$ which is antireflexive, asymmetric and transitive.
In this form, both "is greater than" and "is lesser than" are strict total ordering. So how to expand the above definition to define the relation "is lesser than" on $A$?
 A: To answer your question you should ask yourself another question: what does "x is lesser than y" mean? To say so you should firstly decide the order you want to impose on your set: decided the order you also decided who is lesser than who (provided the order is total, or you could have two elements that are neither equal, bigger or smaller one than another). The answer of your question is then: you can't "expand" the definition of strict total ordering to define the relation "being lesser than"; you can just define what "being lesser than" means by imposing an order. For axample consider $\mathbb{N}$. You have the usual order (that is strict and total descending from the Peano axioms):
$$x < y \iff \exists 0\neq a\in\mathbb{N}:x+a=y$$
and you say "x is lesser than y" if $<$ is satisfied. But you could order $\mathbb{N}$ with another relation and get a different definition of "being lesser than". Hope I've been clear enough.
EDIT maybe this phrase sums up everything clearly: you can CALL the relation $<$ that you decide with the name of "being lesser than".
