Institute of Mathematics

of the Czech Academy of Sciences
 
 
 
 

Mathematica Bohemica, first online, pp. 1-10

When ${\rm Max}_d(G)$ is zero-dimensional

Ricardo CARRERA, Ramiro H. LAFUENTE-RODRIGUEZ, Warren Wm. McGOVERN

Received May 18, 2025.   Published online March 26, 2026.
Abstract:  Our setting here is W. A W-object is an archimedean lattice-ordered group $G$ together with distinguished weak order unit $u\in G^+$. R. H. Lafuente-Rodriguez, W. Wm. McGovern (2021) classified when the space of maximal $d$-subgroups under the hull-kernel topology of such an object has a clopen $\pi$-base. In this article, we characterize when the space has the stronger property of zero-dimensionality.
Keywords:  archimedean lattice-ordered group; $d$-subgroups; zero-dimensional compact Hausdorff space
Classification MSC:  54H12, 06F15, 06F20
DOI:  10.21136/MB.2026.0070-25
PDF available at:  Institute of Mathematics CAS
References:
[1] R. N. Ball, V. Marra, D. McNeill, A. Pedrini: From Freudenthal's spectral theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra. Forum Math. 30 (2018), 513-526. DOI 10.1515/forum-2017-0044 | MR 3770007 | Zbl 1528.06024
[2] B. Banaschewski: On Wallman's method of compactification. Math. Nachr. 27 (1963), 105-114. DOI 10.1002/mana.19630270107 | MR 0160186 | Zbl 0113.37701
[3] G. Bezhanishvili, S. D. Melzer: Algebraic frames in Priestley duality. Algebra Univers. 86 (2025), Article ID 6, 25 pages. DOI 10.1007/s00012-024-00876-2 | MR 4844969 | Zbl 07962022
[4] P. Bhattacharjee: Maximal $d$-elements of an algebraic frame. Order 36 (2019), 377-390. DOI 10.1007/s11083-018-9472-5 | MR 3983485 | Zbl 1435.06007
[5] P. Bhattacharjee: $Max(dL)$ revisited. Topology Appl. 357 (2024), Article ID 109057, 13 pages. DOI 10.1016/j.topol.2024.109057 | MR 4795721 | Zbl 07939819
[6] P. Bhattacharjee, W. Wm. McGovern: Lamron $\ell$-groups. Quaest. Math. 41 (2018), 81-98. DOI 10.2989/16073606.2017.1372529 | MR 3761490 | Zbl 1436.06043
[7] P. Bhattacharjee, W. Wm. McGovern: Maximal $d$-subgroups and ultrafilters. Rend. Circ. Mat. Palermo (2) 67 (2018), 421-440. DOI 10.1007/s12215-017-0323-9 | MR 3911999 | Zbl 1528.06023
[8] A. Bigard, K. Keimel, S. Wolfenstein: Groupes et anneaux réticulés. Lecture Notes in Mathematics 608. Springer, Berlin (1977). (In French.) DOI 10.1007/BFb0067004 | MR 0552653 | Zbl 0384.06022
[9] P. Conrad, J. Martinez: Complemented lattice-ordered groups. Indag. Math., New Ser. 1 (1990), 281-297. DOI 10.1016/0019-3577(90)90019-J | MR 1075880 | Zbl 0735.06006
[10] M. R. Darnel: Theory of Lattice-Ordered Groups. Pure and Applied Mathematics, Marcel Dekker 187. Marcel Dekker, New York (1995). MR 1304052 | Zbl 0810.06016
[11] A. W. Hager, C. M. Kimber, W. Wm. McGovern: Weakly least integer closed groups. Rend. Circ. Mat. Palermo, II. Ser. 52 (2003), 453-480. DOI 10.1007/BF02872765 | MR 2029554 | Zbl 1072.06013
[12] A. W. Hager, W. Wm. McGovern: The projectable hull of an Archimedean $\ell$-group with weak unit. Categ. Gen. Algebr. Struct. Appl. 7 (2017), 165-179. MR 3681523 | Zbl 1423.06057
[13] M. Henriksen, M. Jerison: The space of minimal prime ideals of a commutative ring. Trans. Am. Math. Soc. 115 (1965), 110-130. DOI 10.1090/S0002-9947-1965-0194880-9 | MR 0194880 | Zbl 0147.29105
[14] M. Henriksen, J. Vermeer, R. G. Woods: Quasi $F$-covers of Tychonoff spaces. Trans. Am. Math. Soc. 303 (1987), 779-803. DOI 10.1090/S0002-9947-1987-0902798-0 | MR 0902798 | Zbl 0653.54025
[15] M. Henriksen, J. Vermeer, R. G. Woods: Wallman covers of compact spaces. Diss. Math. 280 (1989), 5-34. MR 0997373 | Zbl 0719.54033
[16] R. Lafuente-Rodriguez, W. Wm. McGovern: When $ Min(G)^{-1}$ has a clopen $\pi$-base. Math. Bohem. 146 (2021), 69-89. DOI 10.21136/MB.2020.0114-18 | MR 4227312 | Zbl 1499.54143
[17] H. Wallman: Lattices and topological spaces. Ann. Math. (2) 39 (1938), 112-126. DOI 10.2307/1968717 | MR 1503392 | Zbl 0018.33202
Affiliations:   Ricardo Carrera, Department of Mathematics, Halmos College of Arts and Sciences, Nova Southeastern University, Fort Lauderdale, FL 33328, USA; Ramiro H. Lafuente-Rodriguez, University of South Dakota, Department of Mathematical Sciences, Vermillion, SD 57069, USA; Warren Wm. McGovern (corresponding author), Wilkes Honors College, Florida Atlantic University, Jupiter, FL 33458, USA, e-mail: warren.mcgovern@fau.edu
[List of online first articles] [Contents of Mathematica Bohemica] [Full text of the older issues of Mathematica Bohemica at DML-CZ]
© Institute of Mathematics CAS, 2026



 
PDF available at:


Institute of Mathematics CAS
free access