# 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)