Vol. 15, No. 4, 1965

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Representations of lattice-ordered groups having a basis

Justin Thomas Lloyd

Vol. 15 (1965), No. 4, 1313–1317

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.

Mathematical Subject Classification
Primary: 20.80
Secondary: 06.75
Received: 27 July 1964
Published: 1 December 1965
Justin Thomas Lloyd