Characteristic subgroups of lattice-ordered groups

Authors:
Richard D. Byrd, Paul Conrad and Justin T. Lloyd

Journal:
Trans. Amer. Math. Soc. **158** (1971), 339-371

MSC:
Primary 06.75

DOI:
https://doi.org/10.1090/S0002-9947-1971-0279014-7

MathSciNet review:
0279014

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Abstract | References | Similar Articles | Additional Information

Abstract: Characteristic subgroups of an -group are those convex -subgroups that are fixed by each -automorphism. Certain sublattices of the lattice of all convex -subgroups determine characteristic subgroups which we call socles. Various socles of an -group are constructed and this construction leads to some structure theorems. The concept of a shifting subgroup is introduced and yields results relating the structure of an -group to that of the lattice of characteristic subgroups. Interesting results are obtained when the -group is characteristically simple. We investigate the characteristic subgroups of the vector lattice of real-valued functions on a root system and determine those vector lattices in which every -ideal is characteristic. The automorphism group of the vector lattice of all continuous real-valued functions (almost finite real-valued functions) on a topological space (a Stone space) is shown to be a splitting extension of the polar preserving automorphisms by the ring automorphisms. This result allows us to construct characteristically simple vector lattices. We show that self-injective vector lattices exist and that an archimedean self-injective vector lattice is characteristically simple. It is proven that each -group can be embedded as an -subgroup of an algebraically simple -group. In addition, we prove that each representable (abelian) -group can be embedded as an -subgroup of a characteristically simple representable (abelian) -group.

**[1]**W. W. Babcock,*On linearly ordered topological spaces*, Dissertation, Tulane University, New Orleans, La., 1964.**[2]**K. A. Baker,*Topological methods in the algebraic theory of vector lattices*, Dissertation, Harvard University, Cambridge, Mass., 1964.**[3]**S. J. Bernau,*Unique representation of Archimedean lattice groups and normal Archimedean lattice rings*, Proc. London Math. Soc. (3)**15**(1965), 599–631. MR**0182661**, https://doi.org/10.1112/plms/s3-15.1.599**[4]**A. Bigard,*Contribution á la théorie des groupes réticulés*, Dissertation, Université de Paris, Paris, France, 1969.**[5]**Garrett Birkhoff,*Lattice theory*, Third edition. American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR**0227053****[6]**Richard D. Byrd,*Complete distributivity in lattice-ordered groups*, Pacific J. Math.**20**(1967), 423–432. MR**0207866****[7]**Richard D. Byrd and Justin T. Lloyd,*Closed subgroups and complete distributivity in lattice-ordered groups*, Math. Z.**101**(1967), 123–130. MR**0218284**, https://doi.org/10.1007/BF01136029**[8]**Paul Conrad,*The lattice of all convex 𝑙-subgroups of a lattice-ordered group*, Czechoslovak Math. J.**15 (90)**(1965), 101–123 (English, with Russian summary). MR**0173716****[9]**Paul Conrad,*Archimedean extensions of lattice-ordered groups*, J. Indian Math. Soc. (N.S.)**30**(1966), 131–160 (1967). MR**0224519****[10]**Paul Conrad,*Lex-subgroups of lattice-ordered groups*, Czechoslovak Math. J.**18 (93)**(1968), 86–103 (English, with Loose Russian summary). MR**0225697****[11]**-,*Introduction á la théorie des groupes réticulés*, Secrétariat mathématique, Paris, 1967. MR**37**#1289.**[12]**P. F. Conrad and J. E. Diem,*The ring of polar preserving endomorphisms of an abelian lattice-ordered group*, Illinois J. Math.**15**(1971), 222–240. MR**0285462****[13]**Paul Conrad and Donald McAlister,*The completion of a lattice ordered group*, J. Austral. Math. Soc.**9**(1969), 182–208. MR**0249340****[14]**Paul Conrad, John Harvey, and Charles Holland,*The Hahn embedding theorem for abelian lattice-ordered groups*, Trans. Amer. Math. Soc.**108**(1963), 143–169. MR**0151534**, https://doi.org/10.1090/S0002-9947-1963-0151534-0**[15]**L. Fuchs,*Partially ordered algebraic systems*, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London, 1963. MR**0171864****[16]**Leonard Gillman and Meyer Jerison,*Rings of continuous functions*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR**0116199****[17]**Graham Higman,*On infinite simple permutation groups*, Publ. Math. Debrecen**3**(1954), 221–226 (1955). MR**0072136****[18]**Charles Holland,*The lattice-ordered groups of automorphisms of an ordered set*, Michigan Math. J.**10**(1963), 399–408. MR**0158009****[19]**W. A. LaBach,*An interesting dual Galois correspondence*, Amer. Math. Monthly**74**(1967), 991–993. MR**0232713**, https://doi.org/10.2307/2315288**[20]**David M. Topping,*Some homological pathology in vector lattices*, Canad. J. Math.**17**(1965), 411–428. MR**0174499**, https://doi.org/10.4153/CJM-1965-042-2**[21]**Elliot Carl Weinberg,*Embedding in a divisible lattice-ordered group*, J. London Math. Soc.**42**(1967), 504–506. MR**0216996**, https://doi.org/10.1112/jlms/s1-42.1.504**[22]**B. Z. Vulikh,*Introduction to the theory of partially ordered spaces*, Translated from the Russian by Leo F. Boron, with the editorial collaboration of Adriaan C. Zaanen and Kiyoshi Iséki, Wolters-Noordhoff Scientific Publications, Ltd., Groningen, 1967. MR**0224522**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0279014-7

Keywords:
Socles of an -group,
-automorphism,
characteristic subgroup,
characteristically simple -group,
polar,
Boolean algebra,
completely reducible -group,
shifting subgroup,
-simple subgroup,
completely -reducible -group,
lex-subgroup,
prime subgroup,
closed subgroup,
cardinally indecomposable -group,
lex-extension,
basic element,
basis,
principal polar,
lex-kernel,
regular subgroup,
special element,
finite valued -group,
root system,
root,
vector lattice,
essential extension,
archimedean extension,
completely regular space,
real compact space,
splitting extension,
self-injective -group,
large subgroup,
hyperarchimedean -group,
radical,
ideal radical,
distributive radical,
singular element

Article copyright:
© Copyright 1971
American Mathematical Society