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archimedean lattice; joinfit coreflection; infinitesimals; fitness conditions
This paper extends the notion of an archimedean frame to frames which are not necessarily algebraic. The new notion is called {\it joinfitness\/} and is {\it Choice-free\/}. Assuming the Axiom of Choice and for compact normal algebraic frames, the new and the old coincide. There is a subfunctor from the category of compact normal frames with skeletal maps with joinfit values, which is almost a coreflection. Conditions making it so are briefly discussed. The concept of an {\it infinitesimal\/} element arises naturally, and the join of suitably chosen infinitesimals defines the joinfit nucleus. The paper concludes with mostly Choice-free applications of these ideas to commutative rings and their radical ideals.
[Ba96] Banaschewski B.: Radical ideals and coherent frames. Comment. Math. Univ. Carolin. 37 2 (1996), 349-370. MR 1399006 | Zbl 0853.06014
[Ba97] Banaschewski B.: Pointfree topology and the spectrum of $f$-rings. in Ordered Algebraic Structures, W.C. Holland & J. Martinez, Eds., Kluwer Acad. Publ., Dordrecht, 1997, pp.123-148. MR 1445110
[Ba02] Banaschewski B.: Functorial maximal spectra. J. Pure Appl. Algebra 168 (2002), 327-346. DOI 10.1016/S0022-4049(01)00101-3 | MR 1887162 | Zbl 0998.06007
[BaP96] Banaschewski B., Pultr A.: Booleanization. Cahiers Topologie Géom. Différentielle Catég. 37 (1996), 1 41-60. MR 1383446 | Zbl 0848.06010
[DM83] De Marco G.: Projectivity of pure ideals. Rend. Sem. Mat. Univ. Padova 69 (1983), 289-304. MR 0717003 | Zbl 0543.13004
[DPR81] Dickman R.F., Porter J.R., Rubin L.R.: Completely regular absolutes and projective objects. Pacific J. Math. 94 2 (1981), 277-295. DOI 10.2140/pjm.1981.94.277 | MR 0628580 | Zbl 0426.54005
[HM07] Hager A.W., Martínez J.: Patch-generated frames and projectable hulls. Appl. Categ. Structures 15 (2007), 49-80. DOI 10.1007/s10485-007-9062-y | MR 2306538 | Zbl 1122.06007
[HS68] Herrlich H., Strecker G.: $H$-closed spaces and reflective subcategories. Math. Ann. 177 (1968), 302-309. DOI 10.1007/BF01350722 | MR 0234427 | Zbl 0157.29104
[HS79] Herrlich H., Strecker G.: Category Theory. Sigma Series in Pure Mathematics 1, Heldermann Verlag, Berlin, 1979. MR 0571016 | Zbl 1125.18300
[Ho69] Hochster M.: Prime ideal structure in commutative rings. Trans. Amer. Math. Soc. 142 (1969), 43-60. DOI 10.1090/S0002-9947-1969-0251026-X | MR 0251026 | Zbl 0184.29401
[J82] Johnstone P.T.: Stone Spaces. Cambridge Studies in Advanced Mathematics 3, Cambridge Univ. Press, Cambridge, 1982. MR 0698074 | Zbl 0586.54001
[M73] Martínez J.: Archimedean lattices. Algebra Universalis 3 (1973), 247-260. DOI 10.1007/BF02945124 | MR 0349503
[M07] Martínez J.: Disjointifiable $\ell$-groups. Algebra Universalis, to appear.
[MZ03] Martínez J., Zenk E.R.: When an algebraic frame is regular. Algebra Universalis 50 (2003), 231-257. DOI 10.1007/s00012-003-1841-1 | MR 2037528 | Zbl 1092.06011
[MZ07a] Martínez J., Zenk E.R.: Regularity in algebraic frames. J. Pure Appl. Algebra 211 (2007), 566-580. DOI 10.1016/j.jpaa.2007.02.008 | MR 2341271 | Zbl 1121.06010
[MZ07b] Martínez J., Zenk E.R.: Epicompletion in frames with skeletal maps, I: Compact regular frames. Appl. Categ. Structures, to appear. MR 2421540
[MZ07c] Martínez J., Zenk E.R.: Epicompletion in frames with skeletal maps, II: Compact normal joinfit frames. Appl. Categ. Structures, to appear.
[MZ07d] Martínez J., Zenk E.R.: Epicompletion in frames with skeletal maps, III: Coherent normal Yosida frames. submitted.
[Mo54] Monteiro A.: L'arithmétique des filtres et les espaces topologiques. Segundo Symposium de Matemática, Villavicencio (Mendoza), 1954, pp.129-162. MR 0074805 | Zbl 0318.06019
[PT01] Pedicchio M.C., Tholen W.: Special Topics in Order, Topology, Algebra and Sheaf Theory. Cambridge Univ. Press, Cambridge, 2001. MR 2054273 | Zbl 1034.18001
[ST93] Snodgrass J.T., Tsinakis C.: Finite-valued algebraic lattices. Algebra Universalis 30 (1993), 311-318. DOI 10.1007/BF01190439 | MR 1225870 | Zbl 0806.06011
[U56] Utumi Y.: On quotient rings. Osaka Math. J. 8 (1956), 1-18. MR 0078966 | Zbl 0070.26601
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