Ordered Algebraic Structures pp 245-260 | Cite as

# Least Integer Closed Groups

Chapter

## Abstract

An a-closure of a lattice-ordered group is an extension which is maximal with respect to preserving the lattice of convex *l*-subgroups under contraction. We describe the a-closures of some local singular archimedean lattice-ordered groups with designated weak unit. In particular, we provide explicit descriptions of all of the a-closures of groups that are singularly convex, such as the group C(X, ℤ) of continuous integer-valued functions on a zero-dimensional space.

## Keywords

Vector Lattice Singular Part Riesz Space Prime Subgroup Weak Unit
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## References

- [1]M. Anderson,
*Locally flat vector lattices*, Canad. J. Math.**32**no. 4, (1980), 924–936.MathSciNetzbMATHCrossRefGoogle Scholar - [2]E. Aron,
*Embedding lattice-ordered algebras in uniformly closed algebras.*(1971) University of Rochester Dissertation.Google Scholar - [3]E. Aron and A. Hager,
*Convex vector lattices and t-algebras.*Topology and Appl.**12**(1981), 1–10.MathSciNetzbMATHCrossRefGoogle Scholar - [4]R. Ball and A. Hager,
*One the localic Yosida representation of an archimedean lattice-ordered group with weak order unit.*J. Pure and Applied Algebra,**70**(1991), 17–43.MathSciNetzbMATHCrossRefGoogle Scholar - [5]R. Ball and A. Hager,
*Algebraic extensions of an archimedean lattice-ordered group, I.*J. Pure and Applied Algebra,**85**(1993), 1–20.MathSciNetzbMATHCrossRefGoogle Scholar - [6]P. Conrad,
*Archimedean extensions of lattice-ordered groups.*J. Indian Math. Soc.**30**(1966), 131–160.MathSciNetzbMATHGoogle Scholar - [7]
- [8]P. Conrad, M. Darnel and D. G. Nelson,
*Valuations of lattice-ordered groups.*J. Algebra**192**no. 1, (1997), 380–411.MathSciNetzbMATHCrossRefGoogle Scholar - [9]M. Darnel,
*Theory of Lattice-Ordered Groups.*Pure and Appl. Math.**187**(1995) Marcel Dekker, New York.Google Scholar - [10]L. Gillman and M. Jerison,
*Rings of Continuous Functions.*(1960) D. Van Nos-trand Publ. Co.zbMATHGoogle Scholar - [11]A. Hager and C. Kimber,
*Some examples of hyperarchimedean lattice-ordered groups.*Submitted.Google Scholar - [12]A. Hager, C. Kimber and W. McGovern
*Pseudo-least integer closed groups.*In progress.Google Scholar - [13]A. Hager and J. Martinez,
*Archimedean singular lattice-ordered groups.*Alg. Univer.**40**(1998), 119–147.MathSciNetzbMATHCrossRefGoogle Scholar - [14]A. Hager and L. Robertson,
*Extremal units in an archimedean Riesz space.*Rend. Sem. Mat. Univ. Padova**59**(1978), 97–115.MathSciNetzbMATHGoogle Scholar - [15]A. Hager and L. Robertson,
*Representing and ringifying a Riesz space.*Symp. Math.**21**(1977), 411–431.MathSciNetGoogle Scholar - [16]M. Henriksen, J. Isbell, and D. Johnson,
*Residue class fields of lattice-ordered algebras.*Fund. Math.**50**(1961), 107–117.MathSciNetzbMATHGoogle Scholar - [17]M. Henriksen and D. Johnson,
*On the structure of a class of Archimedean lattice-ordered algebras.*Fund. Math.**50**(1961), 73–94.MathSciNetzbMATHGoogle Scholar - [18]C. Kimber and W. McGovern,
*Bounded away lattice-ordered groups.*(1998) Manuscript.Google Scholar - [19]J. Martinez,
*The hyper-archimedean kernel sequence of a lattice-ordered group.*Bull. Austral. Math. Soc.**10**(1974), 337–340.MathSciNetzbMATHCrossRefGoogle Scholar - [20]K. Yosida,
*On the representation of the vector lattice.*Proc. Imp. Acad. Tokyo**18**(1942), 339–343.MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Science+Business Media Dordrecht 2002