# 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)
How to Cite This Entry:
L-variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L-variety&oldid=47548
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article