Introductions to posets on algerbaic structures (Everything I need to know about them)

Question

I need a good and complete introduction to Tree-like orders and partial orders on algebraic structures with one operations. I accept basic texts too.

I'm looking for free online texts mostly because I can't buy nothing at the moment.

My interest are:

Terminology and basic results about partial orders on infinite sets, tree-orders on infinite sets, rootless trees and definition on these posets structures of "compatible" binary operations (or hyperoperations a.k.a set valued operations).

Or from the opposite point of view:

Definition of "compatible" tree-like orders and and partial orders from binary operations/algebraic structures with one operation.

To make a pragmatic example

I'm interested of when and how the transitive closures $\le_x$ of left-right translations $l_x(y)=x*y$ in an algebraic structure $(G,*)$ are a family of partial orders relations $\{\le_x\}_{x \in G}$ on $G$ and everything linked with these topics.

Note: I'm not mainly interested on orders on rings, fields and lattice theory.

Answer 1

Just as you didn't mention groups: There has been done much work on po-groups as well.

As far as I know, some people are working on quasiorders (preorders) on algeraic structures. Clone theory tells us that it is sufficient to evaluate the compatible quasiorders on monounary algebras. An arbitrary article about this topic is: The Lattice of Compatible Quasiorders of Acyclic Monounary Algebras, Order, November 2011, Volume 28, Issue 3, pp 481-497 by Danica Jakubíková-Studenovská,Reinhard Pöschel,Sándor Radeleczki.

Answer 2

To add to Keinstein's answer, I liked Glass' Partially Ordered Groups. Despite the name, it has a lot on lattice- and right-ordered groups, and is a pretty wide-ranging reference.

(I assume by "tree-ordered" you mean what I call a "semilattice", which are somewhat covered in Glass.)

You may also be interested in reference request for ordered groups question.