Previous |  Up |  Next


dimension of a frame; $z$-ideals; scattered space; natural typing of open sets
This paper continues the investigation into Krull-style dimensions in algebraic frames. Let $L$ be an algebraic frame. $\operatorname{dim}(L)$ is the supremum of the lengths $k$ of sequences $p_0< p_1< \cdots <p_k$ of (proper) prime elements of $L$. Recently, Th. Coquand, H. Lombardi and M.-F. Roy have formulated a characterization which describes the dimension of $L$ in terms of the dimensions of certain boundary quotients of $L$. This paper gives a purely frame-theoretic proof of this result, at once generalizing it to frames which are not necessarily compact. This result applies to the frame $\Cal C_z(X)$ of all $z$-ideals of $C(X)$, provided the underlying Tychonoff space $X$ is Lindelöf. If the space $X$ is compact, then it is shown that the dimension of $\Cal C_z(X)$ is at most $n$ if and only if $X$ is scattered of Cantor-Bendixson index at most $n+1$. If $X$ is the topological union of spaces $X_i$, then the dimension of $\Cal C_z(X)$ is the supremum of the dimensions of the $\Cal C_z(X_i)$. This and other results apply to the frame of all $d$-ideals $\Cal C_d(X)$ of $C(X)$, however, not the characterization in terms of boundaries. An explanation of this is given within, thus marking some of the differences between these two frames and their dimensions.
[AB91] Adams M.E., Beazer R.: Congruence properties of distributive double $p$-algebras. Czechoslovak Math. J. 41 (1991), 395-404. MR 1117792 | Zbl 0758.06008
[BP04] Ball R.N., Pultr A.: Forbidden forests in Priestley spaces. Cah. Topol. Géom. Différ. Catég. 45 1 (2004), 2-22. MR 2040660 | Zbl 1062.06020
[BKW77] Bigard A., Keimel K., Wolfenstein S.: Groupes et anneaux réticulés. Lecture Notes in Mathematics 608, Springer, Berlin-Heidelberg-New York, 1977. MR 0552653 | Zbl 0384.06022
[Bl76] Blair R.L.: Spaces in which special sets are $z$-embedded. Canad. J. Math. 28 (1976), 673-690. MR 0420542 | Zbl 0359.54009
[BlH74] Blair R.L., Hager A.W.: Extensions of zerosets and of real valued functions. Math. Z. 136 (1974), 41-57. MR 0385793
[CL02] Coquand Th., Lombardi H.: Hidden constructions in abstract algebra: Krull dimension of distributive lattices and commutative rings. Commutative Ring Theory and Applications (M. Fontana, S.-E. Kabbaj, S. Wiegand, Eds.), pp.477-499; Lecture Notes in Pure and Appl. Math., 231, Marcel Dekker, New York, 2003. MR 2029845
[CLR03] Coquand Th., Lombardi H., Roy M.-F.: Une caractérisation élémentaire de la dimension de Krull. preprint.
[D95] Darnel M.: Theory of Lattice-Ordered Groups. Marcel Dekker, New York, 1995. MR 1304052 | Zbl 0810.06016
[En89] Engelking R.: General Topology. Sigma Series in Pure Math. 6, Heldermann, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[Es98] Escardó M.H.: Properly injective spaces and function spaces. Topology Appl. 89 (1998), 75-120. MR 1641443
[GJ76] Gillman L., Jerison M.: Rings of Continuous Functions. Graduate Texts in Mathematics 43, Springer, Berlin-Heidelberg-New York, 1976. MR 0407579 | Zbl 0327.46040
[HJ61] Henriksen M., Johnson D.G.: On the structure of a class of archimedean lattice-ordered algebras. Fund. Math. 50 (1961), 73-94. MR 0133698 | Zbl 0099.10101
[MLMW94] Henriksen M., Larson S., Martínez J., Woods R.G.: Lattice-ordered algebras that are subdirect products of valuation domains Trans. Amer. Math. Soc. 345 (1994), 1 195-221. MR 1239640
[HMW03] Henriksen M., Martínez J., Woods R.G.: Spaces $X$ in which all prime $z$-ideals of $C(X)$ are either minimal or maximal. Comment. Math. Univ. Carolinae 44 2 (2003), 261-294. MR 2026163
[HW04] Henriksen M., Woods R.G.: Cozero complemented spaces: when the space of minimal prime ideals of a $C(X)$ is compact. Topology Appl. 141 (2004), 147-170. MR 2058685 | Zbl 1067.54015
[HuP80a] Huijsmans C.B., de Pagter B.: On $z$-ideals and $d$-ideals in Riesz spaces, I. Indag. Math. 42 2 (1980), 183-195. MR 0577573 | Zbl 0442.46022
[HuP80b] Huijsmans C.B., de Pagter B.: On $z$-ideals and $d$-ideals in Riesz spaces, II. Indag. Math. 42 4 (1980), 391-408. MR 0597997 | Zbl 0451.46003
[J82] Johnstone P.J.: Stone Spaces. Cambridge Studies in Advanced Mathematics, vol. 3, Cambridge Univ. Press, Cambridge, 1982. MR 0698074 | Zbl 0586.54001
[JT84] Joyal A., Tierney M.: An extension of the Galois theory of Grothendieck. Mem. Amer. Math. Soc. 51 309 (1984), 71 pp. MR 0756176 | Zbl 0541.18002
[Ko89] Koppelberg S.: Handbook of Boolean Algebras, I. J.D. Monk, Ed., with R. Bonnet; North Holland, Amsterdam-New York-Oxford-Tokyo, 1989. MR 0991565
[M73a] Martínez J.: Archimedean lattices. Algebra Universalis 3 (1973), 247-260. MR 0349503
[M04a] Martínez J.: Dimension in algebraic frames. Czechoslovak Math. J., to appear. MR 2291748
[M04b] Martínez J.: Unit and kernel systems in algebraic frames. Algebra Universalis, to appear. MR 2217275
[MZ03] Martínez J., Zenk E.R.: When an algebraic frame is regular. Algebra Universalis 50 (2003), 231-257. MR 2037528 | Zbl 1092.06011
[MZ06] Martínez J., Zenk E.R.: Dimension in algebraic frames, III: dimension theories. in preparation.
[Mr70] Mrowka S.: Some comments on the author's example of a non-$R$-compact space. Bull. Acad. Polon. Sci., Ser. Math. Astronom. Phys. 18 (1970), 443-448. MR 0268852
[Se59] Semadeni Z.: Sur les ensembles clairsemés. Rozprawy Mat. 19 (1959), 39 pp. MR 0107849 | Zbl 0137.16002
[Se71] Semadeni Z.: Banach Spaces of Continuous Functions. Polish Scientific Publishers, Warsaw, 1971. MR 0296671 | Zbl 0478.46014
Partner of
EuDML logo