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

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.

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.

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Two comments: I didn't find a good definition for a left-right translation at the moment. Could you make clear what you are thinking of? In case an article is not accessible to you, you might ask the author to provide a preprint. –  Keinstein May 13 '14 at 17:00
@Keinstein Given a binary operation $*$ a left translation is a function $L_a$ such that $L_a(x):=a*x$. I'll make it more clear in the question. About the preprint request... I'm an amateurish mathematician.. maybe is not good to ask for a preprint to some professional mathematicians. –  MphLee May 13 '14 at 17:04

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.

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Very interesting! The concept of monounary algebras (A,f) is really interesting and is one of the reason of this question... even if I can't understand exatly how Clone theory is related to this problem (I was not even able to get what is a Clone). You know some good ("easy") introduction to the topic and some related Keywords? –  MphLee May 15 '14 at 17:25
A short introduction is available at algebra.uni-linz.ac.at/Students/UniversalAlgebra/s11/… while the fundamental book is „Funktionen- und Relationenalgebren” by Pöschel/Kalužnin” available from Springer: springer.com/new+%26+forthcoming+titles+%28default%29/book/… . This gives two other keywords “function(al) algebras“ and “relation(al) algebras“ as mathematical objects which have nearly nothing in common with the branches functional algbra and relational algebra from mathematics/computer science. –  Keinstein May 16 '14 at 5:13

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.

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Thanks for the answer, by tree order i mean a partial order where for every $a\in G$ the set $\{b:c\le a\}$ is linear ordered by $\le$. I don't know if is a semilattice but thanks for the hint. At least I've some keywords to search better. –  MphLee May 15 '14 at 17:07
If a set is tree-ordered by your definition and has an infimum (smallest element) then it will be a meet semilattice. (Using graph theory terminology, this means it has to be a "tree" and not a "forest".) The converse isn't true though. –  Xodarap May 15 '14 at 19:57
@Xodarp ah maybe I understand. By the way the point is that everything I find about trees is always inside a graph theory framework...and I am searching for something from order theory with link to abstract algebra. The meet semilattice seems exatly this contact. even if, as you said, It has to be rooted and I'm interested in roteless "trees" too. –  MphLee May 15 '14 at 20:01
A tree order is always a semilattice if it is connected. Even if it doesn't have a smallest element. On the other hand only tree orders whose order ideals (downsets: $\downarrow a:=\{x\mid x≤a\}$) are suborders of $\mathbb Z$ are graph theoretical trees. Otherwise the underlying undirected graph has circles. –  Keinstein May 16 '14 at 5:18
@Keinstein: Really? Asked here: math.stackexchange.com/questions/804516/… –  Xodarap May 21 '14 at 20:00