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Distinguished extensions of a lattice-ordered group. (English) Zbl 0842.06012
Author’s abstract: “Any lattice-ordered group (l-group for short) is essentially extended by its lexicographic product with a totally ordered group. That is, an l-homomorphism (i.e., a group and lattice homomorphism) on the extension which is injective on the l-group must be injective on the extension as well. Thus no l-group has a maximal essential extension in the category IGp of l-groups with l-homomorphisms. However, an l-group is a distributive lattice, and so it has a maximal essential extension in the category \(\mathbf D\) of distributive lattices with lattice homomorphisms. A distinguished extension of one l-group by another is one which is essential in \(\mathbf D\). We characterize such extensions, and show that every l-group \(G\) has a maximal distinguished extension \(E(G)\) which is unique up to an l-isomorphism over \(G\). \(E(G)\) contains most other known completions in which \(G\) is order dense, and has most l-group completeness properties as a result. Finally, we show that if \(G\) is projectable then \(E(G)\) is the \(\alpha\)-completion of the projectable hull of \(G\)”.

MSC:
06F15 Ordered groups
06D05 Structure and representation theory of distributive lattices
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[1] Anderson, M. andFeil, T.,Lattice-Ordered Groups, an Introduction. Reidel Texts in the Mathematical Sciences. Reidel, Dordrecht, 1988. · Zbl 0636.06008
[2] Balbes, R.,Projective and injective distributive lattices. Pacific J. Math. (1967), 405-420. · Zbl 0157.34301
[3] Balbes, R. andDwinger, P.,Distributive Lattices. University of Missouri Press, Columbia, Missouri, 1974.
[4] Ball, R. N.,Convergence and Cauchy structures on lattice ordered groups. Trans. Amer. Math. Soc.259 (1980), 357-392. · Zbl 0441.06015
[5] Ball, R. N.,The distinguished completion of a lattice ordered group. In Algebra Carbondale 1980, Lecture Notes in Mathematics848, pages 208-217. Springer-Verlag, 1980.
[6] Ball, R. N.,The generalized orthocompletion and strongly projectable hull of a lattice ordered group. Can. J. Math.34 (1982), 621-661. · Zbl 0503.06016
[7] Ball, R. N.,Distributive Cauchy lattices. Algebra Universalis18 (1984), 134-174. · Zbl 0539.06008
[8] Ball, R. N.,Completions of l-groups. InLattice Ordered Groups, A. M. W. Glass and W. C. Holland, editors, chapter 7, pages 142-174. Kluwer, Dordrecht-Boston-London, 1989.
[9] Ball, R. N.,The structure of the ?-completion of a lattice ordered group. Houston J. Math.15 (1989), 481-515. · Zbl 0703.06009
[10] Ball, R. N. andDavis, G.,The ?-completion of a lattice ordered group. Czechoslovak Math. J.33 (108) (1983), 111-118. · Zbl 0517.06014
[11] Bernau, S. J.,Lateral and Dedekind completions of archimedean lattice groups. J. London Math. Soc.12 (1976), 320-322. · Zbl 0333.06008
[12] Bigard, A.,Keimel, K. andWolfenstein, S.,Groupes et Anneaux Reticules. Lecture Notes in Pure and Applied Mathematics 608. Springer-Verlag, 1977.
[13] Birkhoff, G.,Lattice Theory. Colloquium Publications XXV. American Mathematical Society, Providence, 3rd edition, 1967.
[14] Fuchs, L.,Partially Ordered Algebraic Systems. Pergamon Press, New York, 1963. · Zbl 0137.02001
[15] Glass, A. M. W.,Ordered Permutation Groups. London Mathematical Society Lecture Note Series 55. Cambridge University Press, Cambridge, 1981.
[16] Grätzer, G.,General Lattice Theory. Academic Press, New York, 1978. · Zbl 0385.06015
[17] Holland, W. C.,The lattice-ordered group of automorphisms of an ordered set. Mich. J. Math.10 (1963), 399-408. · Zbl 0116.02102
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