The projectable hull of an archimedean $\ell$-group with weak unit

Document Type : Research Paper


1 Department of Mathematics and CS, Wesleyan University, Middletown, CT 06459.

2 H. L. Wilkes Honors College, Florida Atlantic University, Jupiter, FL 33458.


The much-studied projectable hull of an $\ell$-group $G\leq pG$ is an essential extension, so that, in the case that $G$ is  archimedean with weak unit, ``$G\in {\bf W}$", we have for the Yosida representation spaces a ``covering map" $YG \leftarrow YpG$. We have earlier \cite{hkm2} shown that (1) this cover has a characteristic minimality property, and that (2) knowing $YpG$, one can write down $pG$. We now show directly that for $\mathscr{A}$, the boolean algebra in the power set of the minimal prime spectrum $Min(G)$, generated by the sets $U(g)=\{P\in Min(G):g\notin P\}$ ($g\in G$), the Stone space $\mathcal{A}\mathscr{A}$ is a cover of $YG$ with the minimal property of (1); this extends the result from \cite{bmmp} for the strong unit case. Then, applying (2) gives the pre-existing description of $pG$, which includes the strong unit description of \cite{bmmp}. The present methods are largely topological, involving details of covering maps and Stone duality.


Dedicated to Bernhard Banaschewski on the Occasion of his 90th Birthday, Communicated by Themba Dub


[1] Ball, R.N., Marra, V., McNeill, D., and Pedrini, A., From Freudenthal's Spectral Theorem to projectable hulls of unital archimedean lattice-groups, through compactification of minimal spectra, arXiv: 1406-1352 V2.
[2] Bigard, A., Keimel, K., and Wolfenstein, S., "Groupes et Anneaux R'eticul'es", Lecture Notes in Math. 608, Springer-Verlag, Berlin-New York, 1977.
[3] Carral, M. and Coste, M., Normal spectral spaces and their dimensions, J. Pure Appl. Algebra 30(3) (1983), 227-235.
[4] Darnel, M., "Theory of Lattice-Ordered Groups", Monographs and Textbooks in Pure and Applied Mathematics 187, Marcel Dekker, Inc., New York, 1995.
[5] Engelking, R., "General Topology". Second edition. Sigma Series in Pure Mathematics 6, Heldermann Verlag, Berlin, 1989.
[6] Fine, N.J., Gillman, L., and Lambek, J., "Rings of Quotients of Rings of Functions", McGill Univ. Press, Montreal, 1966.
[7] Hager, A.W., Minimal covers of topological spaces, Papers on general topology and related category theory and topological algebra (New York, 1985/1987), 44-59, Ann. New York Acad. Sci. 552, New York Acad. Sci., New York, 1989.
[8] Hager, A.W., Kimber, C.M., and McGovern, W.Wm., Weakly least integer closed groups, Rend. Circ. Mat. Palermo (2), 52(3) (2003), 453-480.
[9] Hager, A.W. and McGovern, W.Wm., The Yosida space and representation of the projectable hull of an archimedean `-group with weak unit, Quaest. Math., 40(1) (2017), 57-63.
[10] Hager, A.W. and Robertson, L., Representing and ringifying a Riesz space, Symp. Math 21 (1977), 411-431.
[11] Hager, A.W. and L. Robertson, On the embedding into a ring of an archimedean-group, Canad. J. Math. 31 (1979), 1-8.
[12] Johnson, D.G. and Kist, J.E., Prime ideals in vector lattices, Canad. J. Math. 14 (1962), 517-528.
[13] Luxemburg, W.A.J. and Zaanen, A.C., "Riesz Spaces". Vol. I., North-Holland Publishing Co., Amsterdam-London, 1971.
[14] Mart'inez, J., Hull classes of Archimedean lattice-ordered groups with unit: a survey, Ordered algebraic structures, 89-121, Dev. Math. 7, Kluwer Acad. Publ., Dordrecht, 2002.
[15] Porter, J. and Woods, R.G., "Extensions and Absolutes of Hausdorf Spaces", Springer-Verlag, New York, 1988.
[16] Sikorski, R., "Boolean Algebras", Third edition. 25 Springer-Verlag New York Inc., New York, 1969.
[17] Veksler, A.I. and Gev{i}ler, V.A., Order and disjoint completeness of linear partially ordered spaces, Sib. Math. J. 13 (1972), 30-35. (Plenum translation).