L-variety

A class of -groups (cf. -group) that is distinguished within the class of all -groups by some system of -group identities: an -group belongs to if and only if for all ,

where are terms in the variables in the signature of , . (Cf. also Variety of groups.)

The class is defined by the following axiom system:

1) if , then is a group;

2) if , then is a lattice;

3) for all , ;

4) for all , .

Any -variety is closed under taking -subgroups, -homomorphisms, direct and Cartesian products, and is locally closed. If is an -group and is an -variety, then there exists in an -ideal such that for every convex -subgroup of , . For every -variety and set there exists an -group that is a free object in with set of generators , i.e., has the property: a mapping from into the -group , can be extended to an -homomorphism from into . There exists a description of the free -groups in terms of -groups and groups of order automorphisms of a suitable totally ordered set (cf. -group). The free -group on free generators has a faithful transitive representation in for some . It is a group with unique roots and orderable.

The most important -varieties are as follows: a) the class of Abelian -groups ; b) the class of the normal-valued -groups ; and c) the class of representable -groups .

The -variety is distinguished in by the identity

(here, ). An -group belongs to if and only if for any jump in the lattice of convex subgroups of one has: is an -ideal of and the quotient group is Abelian. If for an -variety , then .

The -variety is distinguished in by the identity . An -group belongs to if and only if is an -subgroup of a Cartesian product of -groups. If is a locally nilpotent -group, then .

The set of all -varieties is a complete distributive lattice. The power of this lattice is the continuum. For any -variety there exists an -variety such that covers in the lattice of -varieties. The set of all covers of 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)