Lattice-ordered group

$ l $-group

A group $ G $ on the set of elements of which a partial-order relation $ \leq $ is defined possessing the properties: 1) $ G $ is a lattice relative to $ \leq $, i.e. for any $ x, y \in G $ there are elements $ x \wedge y $, $ x \lor y $ such that $ x \wedge y \leq x, y $ and $ x \lor y \geq x, y $; for any $ z \in G $, $ z \leq x, y $ implies $ z \leq x \wedge y $, and for any $ t \in G $ and $ x, y \leq t $ one has $ x \lor y \leq t $; and 2) for any $ a, b, x, y \in G $ the inequality $ a \leq b $ implies $ xay \leq xby $. Similarly, a lattice-ordered group can be defined as an algebraic system of signature $ \langle \cdot , {} ^ {- 1} , e, \wedge, \lor \rangle $ that satisfies the axioms: 3) $ \langle G, \cdot , {} ^ {- 1} , e\rangle $ is a group; 4) $ \langle G, \lor , \wedge\rangle $ is a lattice; and 5) $ x( y \lor z) t = xyt \lor xzt $ and $ x( y \wedge z) t = xyt \wedge xzt $ for any $ x, y, z, t \in G $.

The lattice of elements of a lattice-ordered group is distributive (cf. Distributive lattice). The absolute value (respectively, the positive and the negative part) of an element $ x $ is the element $ | x | = x \lor x ^ {- 1} $ (respectively, $ x ^ {+} = x\lor e $ and $ x ^ {-} = x \wedge e $). In lattice-ordered groups, the following relations hold:

$$ x = x ^ {+} x ^ {-} ,\ \ | x | ^ {- 1} \leq x \leq | x | , $$

$$ | x | = x ^ {+} ( x ^ {-} ) ^ {- 1} ,\ x ^ {+} \wedge ( x ^ {-} ) ^ {- 1 } = e, $$

$$ ( x \lor y) ^ {- 1} = x ^ {- 1} \wedge y ^ {- 1} ,\ \ ( x \wedge y) ^ {- 1} = x ^ {- 1} \lor y ^ {- 1} . $$

Two elements $ x $ and $ y $ are called orthogonal if $ | x | \lor | y | = e $. Orthogonal elements commute.

A subset $ H $ of an $ l $-group $ G $ is called an $ l $-subgroup if $ H $ is a subgroup and a sublattice in $ G $; an $ l $-subgroup $ H $ is called an $ l $-ideal of $ G $ if it is normal and convex in $ G $. The set of $ l $-subgroups of a lattice-ordered group forms a sublattice of the lattice of all its subgroups. The lattice of $ l $-ideals of a lattice-ordered group is distributive. An $ l $-homomorphism of an $ l $-group $ G $ into an $ l $-group $ H $ is a homomorphism $ \phi $ of the group $ G $ into the group $ H $ such that

$$ \phi ( x \lor y) = \phi ( x) \lor \phi ( y) ,\ \ \phi ( x \wedge y) = \phi ( x) \wedge \phi ( y). $$

The kernels of $ l $-homomorphisms are precisely the $ l $-ideals of $ l $-groups. If $ G $ is an $ l $-group and $ M \subset G $, then the set $ M ^ \perp = \{ {x \in G } : {| x | \wedge | m | = e \textrm{ for every } m \in M } \} $ is a convex $ l $-subgroup in $ G $ (cf. Convex subgroup).

The group $ A( L) $ of one-to-one order-preserving mappings of a totally ordered set $ L $ onto itself is an $ l $-group (if for $ f, g \in A( L) $ one assumes that $ f \leq g $ if and only if $ f( \alpha ) \leq g( \alpha ) $ for all $ \alpha \in L $). Every $ l $-group is $ l $-isomorphic to an $ l $-subgroup of the lattice-ordered group $ A( L) $ for a suitable set $ L $.

The class of all lattice-ordered groups is a variety of signature $ \langle \cdot , {} ^ {- 1} , e, \wedge, \lor\rangle $ (cf. Variety of groups). Its most important subvariety is the class of lattice-ordered groups that can be approximated by totally ordered groups (the class of representable $ l $-groups, cf. also Totally ordered group).

References

[1]	G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973)
[2]	L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)
[a1]	M. Anderson, T. Feil, "Lattice-ordered groups. An introduction" , Reidel (1988)
[a2]	A.M.W. Glass (ed.) W.Ch. Holland (ed.) , Lattice-ordered groups. Advances and techniques , Kluwer (1989)
[a3]	J. Martinez (ed.) , Ordered algebraic structures , Kluwer (1989)