Lattice-ordered group
-group
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:
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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
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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).
References
[1] | G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973) |
[2] | L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963) |
Comments
References
[a1] | M. Anderson, T. Feil, "Lattice-ordered groups. An introduction" , Reidel (1988) |
[a2] | A.M.W. Glass (ed.) W.Ch. Holland (ed.) , Lattice-ordered groups. Advances and techniques , Kluwer (1989) |
[a3] | J. Martinez (ed.) , Ordered algebraic structures , Kluwer (1989) |
Lattice-ordered group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lattice-ordered_group&oldid=47589