Partial, Total, and Lattice Orders in Group Theory

The algebraic structure of groups is familiar to anyone who has studied abstract algebra. Familiar to anyone who has studied any mathematics is the concept of ordering, for example the standard less than or equal to with the integers. We can combine these structures to create a new hybrid algebraic structure. In the following, we will define exactly how order structures and group structures interact, and then survey the results. We will examine how the order operation affects the elements of the group for various types of orders. We will generally explore partially ordered groups and their properties. We will also discuss lattice ordered groups in detail, exploring general properties such as the triangle inequality and absolute value in lattice ordered groups. Finally we will explore the theory of permutations, isomorphisms, and homomorphisms in lattice ordered groups. 1 Groups and Orders Recall that a relation on a set X is a subset of X ×X. A relation on X is a partial order if it satisfies the following: 1. For all x ∈ X, x x (reflexive) 2. If x y and y x, then x = y, for all x, y ∈ X (antisymmetric) 3. If x y and y z, then x z, for all x, y, z ∈ X (transitive) Furthermore, is a total order when x y and y x, then x = y, for all x, y ∈ X. This property is called totality, and if has it we say is dichotomous.