nLab lattice-ordered group

Contents

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 GG 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 ghg\leq h then gkhkg\cdot k\leq h\cdot k and also kgkhk\cdot g\leq k\cdot h for any g,h,kGg,h,k \in G.

Positive elements

The set, G +={gGg1}G^+=\{g\in G\mid g\geq 1\} is called the positive cone of (G,)(G,\leq). It completely determines the partial order since ghg\leq h if and only if hg 1G *{0}hg^{-1}\in G^*\cup \{0\}.

Lattice-ordered groups.

If G,)G,\leq) is a partially ordered group, and the partial order is a lattice, then we say GG is a lattice-ordered group, so for each pair , a,ba,b in GG has a join, aba \vee b and a meet, aba\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 ghg\le h, then h 1g 1h^{-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 (ab) 1=a 1b 1(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,yGx, y \in G, we can form x/y=xy 1x / y = x\cdot y^{-1}, and y\x=y 1xy\backslash x = y^{-1}\cdot x. In this case, if xyzx\cdot y \leq z, then, of course, yx\zy\leq x\backslash z and xz/yx\leq z / y. We then also have x 1=1/xx^{-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(1/x)=xx\cdot (1/x) = x.

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.

Last revised on February 23, 2024 at 19:59:18. See the history of this page for a list of all contributions to it.