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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Hereditary properties and maximality conditions with respect to essential extensions of lattice group orders

Author: Jorge Martinez
Journal: Trans. Amer. Math. Soc. 166 (1972), 339-350
MSC: Primary 06A55
MathSciNet review: 0295993
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Abstract: An l-group will be denoted by the pair (G, P), where G is the group and P is the positive cone. The cone Q is an essential extension of P if every convex l-subgroup of (G, Q) is a convex l-subgroup of (G, P). Q is very essential over P if it is essential over P and for each $ 0 \ne x \in G$ and each Q-value D of x, there is a unique P-value C of x containing D. We seek conditions which are preserved by essential extensions; normal valuedness and the existence of a finite basis are so preserved.

We then investigate l-groups which have the property that their positive cone has no proper very essential extensions. Q is a c-essential extension of P if Q is essential over P and every closed convex l-subgroup of (G, Q) is closed in (G, P). We show that a wreath product of totally ordered groups has no proper very c-essential extensions. We derive a sufficient condition for the nonexistence of such extensions in case the l-group has property (F): no positive element exceeds an infinite set of pairwise disjoint elements.

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Keywords: Essential extensions of lattice group orders, c-essential extensions, hereditary properties, property (S), property (F), locally representable l-groups, laterally complete 1-groups
Article copyright: © Copyright 1972 American Mathematical Society

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