Abstract
Within archimedean \(\ell \)-groups, “\(H \in W^{*}\)” means H contains a strong unit, and “\(G \in SW^{*}\) (respectively, \(ESW^{*}\))” means there are \(H \in W^{*}\) and an embedding (respectively, essential embedding) \(G \le H\). This paper continues earlier work in which we developed methods of attacking the question “\(G \in SW^{*}\)?” and gave many examples with answer “No” and “Yes”. Here, we take up the question “\(G \in ESW^{*}\)?”—focusing on the differences between \(SW^{*}\) and \(ESW^{*}\) and the different kinds of minimal adjunctions of strong unit (essential and not)—and again give many examples.
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References
Anderson, M., Feil, T.: Lattice-Ordered Groups. Reidel, Dordrecht (1988)
Bigard, A., Keimel, K., Wolfenstein, S.: Groupes et Anneaux Réticulés. Lecture Notes in Mathematics, vol. 608. Springer, Berlin (1977). (French)
Conrad, P.: The essential closure of an archimedean lattice-ordered group. Duke Math. J. 38, 151–160 (1971)
Conrad, P., Martinez, J.: Settling a number of questions about hyper-archimedean lattice-ordered groups. Proc. Am. Math. Soc. 109, 291–296 (1990)
Darnel, M.R.: Theory of Lattice-Ordered Groups. Pure and Applied Mathematics, vol. 187. Marcel Dekker, New York (1995)
Engelking, R.: General Topology. Sigma Series in Pure Mathematics, vol. 6, 2nd edn. Heldermann Verlag, Berlin (1989)
Gillman, L., Jerison, M.: Rings of Continuous Functions. The University Series in Higher Mathematics. Van Nostrand, Princeton (1960)
Hager, A.W.: Minimal covers of topological spaces. Ann. N. Y. Acad. Sci. 552, 44–59 (1989)
Hager, A.W., Johnson, D.G.: Some comments and examples on generation of (hyper-)archimedean \(\ell \)-groups and \(f\)-rings. Ann. Fac. Sci. Toulouse Math. 19 (Fascicule Special), 75–100 (2010)
Hager, A., van Mill, J.: Egoroff, , and convergence properties in some archimedean vector lattices. Stud. Math. 231, 269–285 (2015)
Hager, A.W., Robertson, L.C.: Representing and Ringifying a Riesz Space. Symposia Mathematica, Vol. XXI (Convengo sulle Misure su Gruppi e su Spazi Vettoriali, Convengo sui Gruppi e Anelli Ordinati, INDAM, Rome, 1975), pp. 411–431. Academic Press, London (1977)
Hager, A.W., Robertson, L.C.: On the embedding into a ring of an Archimedean \(\ell \)-group. Can. J. Math. 31, 1–8 (1979)
Hager, A., Scowcroft, P.: Adjoining a strong unit to an archimedean lattice-ordered group (to appear in Order)
Henriksen, M., Johnson, D.G.: On the structure of a class of archimedean lattice-ordered algebras. Fund. Math. 50, 73–94 (1961/1962)
Herrlich, H., Strecker, G.E.: Category Theory. An Introduction. Sigma Series in Pure Mathematics, vol. 1, 3rd edn. Heldermann Verlag, Lemgo (2007)
Porter, J.R., Woods, R.G.: Extensions and Absolutes of Hausdorff Spaces. Springer, New York (1988)
Tamano, K.: Cardinal Functions, Part II. In: Hart, K.P., Nagata, J., Vaughan, J. (eds.) Encyclopedia of General Topology, pp. 15–17. Elsevier, Amsterdam (2004)
van Douwen, E.: The integers and topology. In: Kunen, K., Vaughan, J.E. (eds.) Handbook of Set-Theoretic Topology, pp. 111–167. North-Holland, Amsterdam (1984)
Acknowledgements
We thank the referee for careful reading which led to improvements in our paper, including a new example in Section 5.
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Presented by W. Wm. McGovern.
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Hager, A.W., Scowcroft, P. Essential adjunction of a strong unit to an archimedean lattice-ordered group. Algebra Univers. 81, 37 (2020). https://doi.org/10.1007/s00012-020-00665-7
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DOI: https://doi.org/10.1007/s00012-020-00665-7
Keywords
- Archimedean lattice-ordered group
- Archimedean order
- Strong unit
- Essential extension
- C(X)
- Extremally disconnected space and cover