The lattice of -group varieties

Author:
J. E. Smith

Journal:
Trans. Amer. Math. Soc. **257** (1980), 347-357

MSC:
Primary 06F15; Secondary 06B20

DOI:
https://doi.org/10.1090/S0002-9947-1980-0552262-X

MathSciNet review:
552262

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For any type of abstract algebra, a variety is an equationally defined class of such algebras. Recently, attempts have been made to study varieties of lattice-ordered groups (*l*-groups). Martinez has shown that the set **L** of all *l*-group varieties forms a lattice under set inclusion with a compatible associative multiplication. Certain varieties (*p* prime) have been proved by Scrimger to be minimal nonabelian varieties in **L**. In the present paper, it is shown that these varieties can be used to produce varieties minimal with respect to properly containing various other varieties in **L**. Also discussed are the relations among the , and it is established that all infinite collections of the have the same least upper bound in **L**. Martinez has also classified *l*-groups using torsion classes, a generalization of the idea of varieties. It is proved here that **L** is not a sublattice of **T**, the lattice of torsion classes.

**[1]**S. J. Bernau,*Varieties of lattice groups are closed under*-*completion*, Sympos. Math.**21**(1977).**[2]**Garrett Birkhoff,*Lattice Theory*, American Mathematical Society, New York, 1940. MR**0001959****[3]**G. Grätzer,*Lattice theory*, Freeman, San Francisco, Calif., 1971.**[4]**George Grätzer,*Universal algebra*, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1968. MR**0248066****[5]**Charles Holland,*The lattice-ordered groups of automorphisms of an ordered set*, Michigan Math. J.**10**(1963), 399–408. MR**0158009****[6]**W. Charles Holland,*The largest proper variety of lattice ordered groups*, Proc. Amer. Math. Soc.**57**(1976), no. 1, 25–28. MR**0406902**, https://doi.org/10.1090/S0002-9939-1976-0406902-0**[7]**W. Charles Holland,*Varieties of 𝑙-groups are torsion classes*, Czechoslovak Math. J.**29(104)**(1979), no. 1, 11–12. MR**518135****[8]**J. T. Lloyd,*Lattice-ordered groups and o-permutation groups*, Dissertation, Tulane University, 1964.**[9]**Jorge Martinez,*Free products in varieties of lattice-ordered groups*, Czechoslovak Math. J.**22(97)**(1972), 535–553. MR**0311536****[10]**Jorge Martinez,*Varieties of lattice-ordered groups*, Math. Z.**137**(1974), 265–284. MR**0354483**, https://doi.org/10.1007/BF01214370**[11]**Jorge Martinez,*Torsion theory for lattice-ordered groups*, Czechoslovak Math. J.**25(100)**(1975), 284–299. MR**0389705****[12]**Hanna Neumann,*Varieties of groups*, Springer-Verlag New York, Inc., New York, 1967. MR**0215899****[13]**E. B. Scrimger,*A large class of small varieties of lattice-ordered groups*, Proc. Amer. Math. Soc.**51**(1975), 301–306. MR**0384644**, https://doi.org/10.1090/S0002-9939-1975-0384644-7**[14]**Elliot Carl Weinberg,*Free lattice-ordered abelian groups. II*, Math. Ann.**159**(1965), 217–222. MR**0181668**, https://doi.org/10.1007/BF01362439

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
06F15,
06B20

Retrieve articles in all journals with MSC: 06F15, 06B20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1980-0552262-X

Keywords:
Lattice-ordered group,
variety,
lattice,
Scrimger variety,
torsion class

Article copyright:
© Copyright 1980
American Mathematical Society