A group on the set of elements of which a partial-order relation is defined possessing the properties: 1) is a lattice relative to , i.e. for any there are elements , such that and ; for any , implies , and for any and one has ; and 2) for any the inequality implies . Similarly, a lattice-ordered group can be defined as an algebraic system of signature that satisfies the axioms: 3) is a group; 4) is a lattice; and 5) and for any .
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 is the element (respectively, and ). In lattice-ordered groups, the following relations hold:
Two elements and are called orthogonal if . Orthogonal elements commute.
A subset of an -group is called an -subgroup if is a subgroup and a sublattice in ; an -subgroup is called an -ideal of if it is normal and convex in . The set of -subgroups of a lattice-ordered group forms a sublattice of the lattice of all its subgroups. The lattice of -ideals of a lattice-ordered group is distributive. An -homomorphism of an -group into an -group is a homomorphism of the group into the group such that
The kernels of -homomorphisms are precisely the -ideals of -groups. If is an -group and , then the set is a convex -subgroup in (cf. Convex subgroup).
The group of one-to-one order-preserving mappings of a totally ordered set onto itself is an -group (if for one assumes that if and only if for all ). Every -group is -isomorphic to an -subgroup of the lattice-ordered group for a suitable set .
The class of all lattice-ordered groups is a variety of signature (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 -groups, cf. also Totally ordered group).
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Lattice-ordered group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lattice-ordered_group&oldid=15070