A convex l-subgroup C of a
lattice-ordered group G is said to be a prime subgroup provided the collection L(C)
of left cosets of G by C is totally-ordered by the relation: xC ≦ yC if and only if
there exists c ∈ C such that xc ≦ y. A collection C of prime subgroups of G is called
a representation for G if ⋂C contains no proper l-ideal of G. A representation C is
said to be irreducible if the intersection of any proper subcollection of C does
contain a proper l-ideal of G. C is a minimal representation if each element
of C is a minimal prime subgroup. A representation C is ∗-irreducible if
⋂C= {1} while ⋂(C−{C})≠{1} for every C ∈C. In this paper it is shown that
an l-group with a basis admits a minimal irreducible representation and
that such a representation can be chosen in essentially only one way. In
particular, an l-group with a normal basis has a unique minimal irreducible
representation. In addition, two properties equivalent to the existence of
a basis are derived; namely the existence of a representation C such that
each element of C has a nontrivial polar and the existence of a ∗-irreducible
representation.
