Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] R. D. Byrd and J. T. Lloyd, Closed subgroups and complete distributivity in lattice-ordered groups, Math. Z. 101 (1967), 123-130. MR 36 #1371. MR 0218284 (36:1371)
  • [2] R. D. Byrd, Complete distributivity in lattice-ordered groups, Pacific J. Math. 20 (1967), 423-432. MR 34 #7680. MR 0207866 (34:7680)
  • [3] P. Conrad, Some structure theorems for lattice-ordered groups, Trans. Amer. Math. Soc. 99 (1961), 212-240. MR 22 #12143. MR 0121405 (22:12143)
  • [4] -, The lattice of all convex l-subgroups of a lattice-ordered group, Czechoslovak Math. J. 15 (90) (1965), 101-123. MR 30 #3926. MR 0173716 (30:3926)
  • [5] -, A characterization of lattice-ordered groups by their convex l-subgroups, J. Austral. Math. Soc. 7 (1967), 145-159. MR 35 #5371. MR 0214521 (35:5371)
  • [6] -, Introduction à la théorie des groupes réticulés, Université de Paris, Faculté des Sciences de Paris, Secrétariat mathématique, Paris, 1967. MR 37 #1289.
  • [7] J. Martinez, Essential extensions of partial orders on groups, Trans. Amer. Math. Soc. 162 (1971), 35-61. MR 0295992 (45:5053)
  • [8] S. Wolfenstein, Valeurs normales dans un groupe réticulé, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 44 (1968), 337-342. MR 38 #3201. MR 0234887 (38:3201)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 06A55

Retrieve articles in all journals with MSC: 06A55

Additional Information

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

American Mathematical Society