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Keywords:
lattice-ordered group; minimal prime subgroup; maximal $d$-subgroup; archimedean $l$-group; $\bold {W}$
lattice-ordered group; minimal prime subgroup; maximal $d$-subgroup; archimedean $l$-group; $\bold {W}$
Summary:
It is our aim to contribute to the flourishing collection of knowledge centered on the space of minimal prime subgroups of a given lattice-ordered group. Specifically, we are interested in the inverse topology. In general, this space is compact and $T_1$, but need not be Hausdorff. In 2006, W. Wm. McGovern showed that this space is a boolean space (i.e. a compact zero-dimensional and Hausdorff space) if and only if the $l$-group in question is weakly complemented. A slightly weaker topological property than having a base of clopen subsets is having a clopen $\pi $-base. Recall that a $\pi $-base is a collection of nonempty open subsets such that every nonempty open subset of the space contains a member of the $\pi $-base; obviously, a base is a $\pi $-base. In what follows we classify when the inverse topology on the space of prime subgroups has a clopen $\pi $-base.
References:
[1] Azarpanah, F., Karamzadeh, O. A. S., Aliabad, A. Rezai: On $z^0$-ideals in $C(X)$. Fund. Math. 160 (1999), 15-25. DOI 10.4064/fm_1999_160_1_1_15_25 | MR 1694400 | Zbl 0991.54014
[2] Bhattacharjee, P., McGovern, W. Wm.: When $ Min(A)^{-1}$ is Hausdorff. Commun. Algebra 41 (2013), 99-108. DOI 10.1080/00927872.2011.617228 | MR 3010524 | Zbl 1264.13004
[3] Bhattacharjee, P., McGovern, W. Wm.: Lamron $\ell$-groups. Quaest. Math. 41 (2018), 81-98. DOI 10.2989/16073606.2017.1372529 | MR 3761490 | Zbl 07117253
[4] Bhattacharjee, P., McGovern, W. Wm.: Maximal $d$-subgroups and ultrafilters. Rend. Circ. Mat. Palermo, Series 2 67 (2018), 421-440. DOI 10.1007/s12215-017-0323-9 | MR 3911999 | Zbl 06992778
[5] Conrad, P.: Lattice Ordered Groups. Tulane Lecture Notes. Tulane University, New Orleans (1970). Zbl 0258.06011
[6] Conrad, P., Martínez, J.: 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
[7] Darnel, M. R.: Theory of Lattice-Ordered Groups. Pure and Applied Mathematics 187. Marcel Dekker, New York (1995). MR 1304052 | Zbl 0810.06016
[8] Ghashghaei, E., McGovern, W. Wm.: Fusible rings. Commun. Algebra 45 (2017), 1151-1165. DOI 10.1080/00927872.2016.1206347 | MR 3573366 | Zbl 1386.16021
[9] Huijsmans, C. B., Pagter, B. de: On $z$-ideals and $d$-ideals in Riesz spaces. II. Indag. Math. 42 (1980), 391-408. DOI 10.1016/1385-7258(80)90040-2 | MR 0597997 | Zbl 0451.46003
[10] Huijsmans, C. B., Pagter, B. de: Maximal $d$-ideals in a Riesz space. Can. J. Math. 35 (1983), 1010-1029. DOI 10.4153/CJM-1983-056-6 | MR 0738841 | Zbl 0505.46004
[11] Kist, J.: Compact spaces of minimal prime ideals. Math. Z. 111 (1969), 151-158. DOI 10.1007/BF01111196 | MR 0245566 | Zbl 0177.06404
[12] Knox, M. L., McGovern, W. Wm.: Feebly projectable $\ell$-groups. Algebra Univers. 62 (2009), 91-112. DOI 10.1007/s00012-010-0041-z | MR 2645226 | Zbl 1192.06014
[13] Levy, R.: Almost-$P$-spaces. Can. J. Math. 29 (1977), 284-288. DOI 10.4153/CJM-1977-030-7 | MR 0464203 | Zbl 0342.54032
[14] Martínez, J., Zenk, E. R.: When an algebraic frame is regular. Algebra Univers. 50 (2003), 231-257. DOI 10.1007/s00012-003-1841-1 | MR 2037528 | Zbl 1092.06011
[15] Martínez, J., Zenk, E. R.: Epicompletion in frames with skeletal maps. I.: Compact regular frames. Appl. Categ. Struct. 16 (2008), 521-533. DOI 10.1007/s10485-007-9110-7 | MR 2421540 | Zbl 1156.06004
[16] McGovern, W. Wm.: Neat rings. J. Pure Appl. Algebra 205 (2006), 243-265. DOI 10.1016/j.jpaa.2005.07.012 | MR 2203615 | Zbl 1095.13025
[17] Speed, T. P.: Spaces of ideals of distributive lattices. II: Minimal prime ideals. J. Aust. Math. Soc. 18 (1974), 54-72. DOI 10.1017/S144678870001911X | MR 0354476 | Zbl 0294.06009
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