Linearly ordered group

In mathematics, especially in abstract algebra, a linearly ordered or totally ordered group is a special kind of group that has a way to compare any two elements, just like numbers can be compared as smaller or larger. This comparison, called a total order, follows a special rule called translation-invariance.

There are three types of these orders. In a left-ordered group, if one element is smaller than another, multiplying both by the same element on the left keeps the order the same. In a right-ordered group, multiplying both elements by the same element on the right keeps the order. When both rules work together, it is called a bi-ordered group.

A group can be called left-orderable (or right-orderable, bi-orderable) if it is possible to find one of these special orders for it. One important clue that a group might be left-orderable is if none of its elements repeat after a certain number of multiplications, though this alone isn’t enough to guarantee it. While being left-orderable and right-orderable are the same for most groups, some groups can follow one rule but not the other.

Further definitions

In this part, we talk about a special kind of order in a group. Imagine a group with a rule that tells us when one thing is "less than" or "greater than" another. This rule has to follow some special patterns.

We call a number positive if it is bigger than the group's main element, called the identity. The collection of all positive numbers in the group is called the positive cone. This helps us understand the order in the group.

If the group is also "abelian" (a special kind of group where the order of operations doesn't matter), we can find a way to compare the "size" of any two numbers in the group using a rule called the triangle inequality. This rule helps us see how numbers work together in the group.

Examples

In math, a group that can be arranged in a straight line without any repeating elements is called torsion-free. Otto Hölder showed that a special type of group, called an Archimedean group, can be matched exactly to part of the number line.

Free groups and certain groups formed by weaving actions, like braid groups, can also be arranged in a straight line. However, not all groups can be arranged this way. For example, some groups made from moving in three dimensions cannot be placed in a straight line order.