L-variety

A class $ {\mathcal X} $ of $ l $- groups (cf. $ l $- group) that is distinguished within the class $ {\mathcal L} $ of all $ l $- groups by some system $ X $ of $ l $- group identities: an $ l $- group $ G $ belongs to $ {\mathcal X} $ if and only if for all $ x _ {1} \dots x _ {n} \in G $,

$$ w _ {i} ( x _ {1} \dots x _ {n} ) = e \textrm{ for every } w _ {i} \in X, $$

where $ w _ {i} $ are terms in the variables $ x _ {1} \dots x _ {n} $ in the signature of $ {\mathcal L} $, $ l = \{ \cdot,e, ^ {- 1 } , \lor, \wedge \} $. (Cf. also Variety of groups.)

The class $ {\mathcal L} $ is defined by the following axiom system:

1) if $ G \in {\mathcal L} $, then $ \{ G, \cdot,e, ^ {- 1 } \} $ is a group;

2) if $ G \in {\mathcal L} $, then $ \{ G, \lor, \wedge \} $ is a lattice;

3) for all $ x,y,z,t $, $ x ( y \lor z ) t = xyt \lor xzt $;

4) for all $ x,y,z,t $, $ x ( y \wedge z ) t = xyt \wedge xzt $.

Any $ l $- variety is closed under taking $ l $- subgroups, $ l $- homomorphisms, direct and Cartesian products, and is locally closed. If $ G $ is an $ l $- group and $ {\mathcal X} $ is an $ l $- variety, then there exists in $ G $ an $ l $- ideal $ {\mathcal X} ( G ) \in {\mathcal X} $ such that $ H \subseteq {\mathcal X} ( G ) $ for every convex $ l $- subgroup $ H $ of $ G $, $ H \in {\mathcal X} $. For every $ l $- variety $ {\mathcal X} $ and set $ T $ there exists an $ l $- group $ F _ {\mathcal X} ( T ) \in {\mathcal X} $ that is a free object in $ {\mathcal X} $ with set of generators $ T $, i.e., $ F _ {\mathcal X} ( T ) $ has the property: a mapping $ \varphi $ from $ T $ into the $ l $- group $ G \in {\mathcal X} $, can be extended to an $ l $- homomorphism from $ F _ {\mathcal X} ( Y ) $ into $ G $. There exists a description of the free $ l $- groups $ F _ {\mathcal X} $ in terms of $ ro $- groups and groups $ { \mathop{\rm Aut} } ( X ) $ of order automorphisms of a suitable totally ordered set $ X $( cf. $ ro $- group). The free $ l $- group $ F = F _ {\mathcal L} $ on $ n \geq 2 $ free generators has a faithful transitive representation in $ { \mathop{\rm Aut} } ( X ) $ for some $ X $. It is a group with unique roots and orderable.

The most important $ l $- varieties are as follows: a) the class of Abelian $ l $- groups $ {\mathcal A} $; b) the class of the normal-valued $ l $- groups $ {\mathcal V} $; and c) the class of representable $ l $- groups $ {\mathcal R} $.

The $ l $- variety $ {\mathcal V} $ is distinguished in $ {\mathcal L} $ by the identity

$$ \left | x \right | \left | y \right | \wedge \left | y \right | ^ {2} \left | x \right | ^ {2} = \left | x \right | \left | y \right | $$

(here, $ | x | = x \lor x ^ {- 1 } $). An $ l $- group $ G $ belongs to $ {\mathcal V} $ if and only if for any jump $ A \subset B $ in the lattice $ {\mathcal C} ( G ) $ of convex subgroups of $ G $ one has: $ A $ is an $ l $- ideal of $ B $ and the quotient group $ B/A $ is Abelian. If $ {\mathcal X} \neq {\mathcal L} $ for an $ l $- variety $ {\mathcal X} $, then $ {\mathcal X} \subseteq {\mathcal V} $.

The $ l $- variety $ {\mathcal R} $ is distinguished in $ {\mathcal L} $ by the identity $ ( x \wedge y ^ {- 1 } xy ) \lor e = e $. An $ l $- group $ G $ belongs to $ {\mathcal R} $ if and only if $ G $ is an $ l $- subgroup of a Cartesian product of $ o $- groups. If $ G $ is a locally nilpotent $ l $- group, then $ G \in {\mathcal R} $.

The set of all $ l $- varieties is a complete distributive lattice. The power of this lattice is the continuum. For any $ l $- variety $ {\mathcal X} \neq {\mathcal L} $ there exists an $ l $- variety $ {\mathcal Y} $ such that $ {\mathcal Y} $ covers $ {\mathcal X} $ in the lattice of $ l $- varieties. The set of all covers of $ {\mathcal A} $ has been described.

References

[a1]	"Lattice-ordered groups: advances and techniques" A.M.W. Glass (ed.) W.Ch. Holland (ed.) , Kluwer Acad. Publ. (1989)
[a2]	V.M. Kopytov, N.Ya. Medvedev, "The theory of lattice-ordered groups" , Kluwer Acad. Publ. (1994) (In Russian)