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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 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 September 10, 2021 at 06:16:44. See the history of this page for a list of all contributions to it.