nLab lattice-ordered group

Idea

A lattice-ordered group is a group which is also a lattice in a compatible way.

Partially ordered groups.

First we introduce partially ordered groups.

A group $G$ is said to be partially ordered if it is equipped with a partial order, $\le$, which is compatible with the group multiplication, $\cdot$, so, if $g\leq h$ then $g\cdot k\leq h\cdot k$ and also $k\cdot g\leq k\cdot h$ for any $g,h,k \in G$.

Positive elements

The set, $G^+=\{g\in G\mid g\geq 1\}$ is called the positive cone of $(G,\leq)$. It completely determines the partial order since $g\leq h$ if and only if $hg^{-1}\in G^*\cup \{0\}$.

Lattice-ordered groups.

If $G,\leq)$ is a partially ordered group, and the partial order is a lattice, then we say $G$ is a lattice-ordered group, so for each pair , $a,b$ in $G$ has a join, $a \vee b$ and a meet, $a\wedge b$.

Many sources abbreviate ‘lattice-ordered group’ to ‘$\ell$-group’.

Properties

• The group operation distibutes over both $\vee$ and $\wedge$.

• Inversion reverses the order so it $g\le h$, then $h^{-1}\leq g^{-1}$. (It is worth noting that this property is the opposite of that encountered with ordered groupoids.)

• A sort of de Morgan’s law holds, so $(a\vee b)^{-1}= a^{-1}\wedge b^{-1}$. This is easily proved once one notes that inversion reverses the order.

• The compatibility of the group multiplication with the lattice structure interpretas as saying that left and right multiplication by elements of the group give automorphisms of the lattice structure.

Lattice-ordered groups and residuated lattices

We note that given any $x, y \in G$, we can form $x / y = x\cdot y^{-1}$, and $y\backslash x = y^{-1}\cdot x$. In this case, if $x\cdot y \leq z$, then, of course, $y\leq x\backslash z$ and $x\leq z / y$. We then also have $x^{-1}= 1 / x$ - which is neat!

This gives that any lattice-ordered group gives a residuated lattice.

The lattice-ordered groups form a variet ub the category of residuated lattices that is axiomatised by the identity $x\cdot (1/x) = x$.

Related concepts

• ordered group

• lattice-ordered abelian group

• lattice-ordered ring

References

The original article that considered ordered groups in the non-commutative case is by Garrett Birkhoff:

• Garrett Birkhoff. Lattice-ordered groups, Annals of Math. 43, (1942) pp. 298 - 331.

He later considered Lattice-ordered Lie groups.

There are several monographs or books on the subject of lattice ordered groups.

• Marlow Anderson and Todd Feil, Lattice Ordered Groups: An Introduction, Reidel Texts in the Mathematical Sciences, vol 4, (1988), (doi:10.1007/978-94-009-2871-8).

• V. M. Kopytov, N. Ya. Medvedev, The Theory of Lattice-Ordered Groups, Mathematics and Its Applications, volume 307, 1994), Springer, doi.