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algebraic frame; dimension; $d$-elements; $z$-elements; lattice-ordered group; $f$-ring
In an algebraic frame $L$ the dimension, $\dim (L)$, is defined, as in classical ideal theory, to be the maximum of the lengths $n$ of chains of primes $p_0<p_1<\cdots <p_n$, if such a maximum exists, and $\infty $ otherwise. A notion of “dominance” is then defined among the compact elements of $L$, which affords one a primefree way to compute dimension. Various subordinate dimensions are considered on a number of frame quotients of $L$, including the frames $dL$ and $zL$ of $d$-elements and $z$-elements, respectively. The more concrete illustrations regarding the frame convex $\ell $-subgroups of a lattice-ordered group and its various natural frame quotients occupy the second half of this exposition. For example, it is shown that if $A$ is a commutative semiprime $f$-ring with finite $\ell $-dimension then $A$ must be hyperarchimedean. The $d$-dimension of an $\ell $-group is invariant under formation of direct products, whereas $\ell $-dimension is not. $r$-dimension of a commutative semiprime $f$-ring is either 0 or infinite, but this fails if nilpotent elements are present. $sp$-dimension coincides with classical Krull dimension in commutative semiprime $f$-rings with bounded inversion.
[1] M. F. Atiyah and I. G. MacDonald: Introduction to Commutative Algebra. Addison-Wesley, 1969. MR 0242802
[2] B. Banaschewski: Pointfree topology and the spectrum of $f$-rings. Ordered Algebraic Structures. W. C. Holland and J. Martínez, Eds., Kluwer Acad. Publ., 1997, pp. 123–148. MR 1445110
[3] A. Bigard, K. Keimel and S. Wolfenstein: Groupes et Anneaux Réticulés. Lecture Notes in Mathematics 608, Springer-Verlag, Berlin-Heidelberg-New York, 1977. MR 0552653
[4] G. Birkhoff: Lattice Theory (3rd Ed.). AMS Colloq. Publ. XXV, Providence, 1967. MR 0227053
[5] P. F. Conrad: Epi-archimedean groups. Czechoslovak Math. J. 24 (1974), 192–218. MR 0347701 | Zbl 0319.06009
[6] P. Conrad and J. Martínez: Complemented lattice-ordered groups. Indag. Math. (N.S.) 1 (1990), 281–297. DOI 10.1016/0019-3577(90)90019-J | MR 1075880
[7] M. Darnel: Theory of Lattice-Ordered Groups. Marcel Dekker, New York, 1995. MR 1304052 | Zbl 0810.06016
[8] L. Gillman and M. Jerison: Rings of Continuous Functions. Graduate Texts Math. 43, Springer Verlag, Berlin-Heidelberg-New York, 1976. MR 0407579
[9] A. W. Hager, C. M. Kimber and W. Wm. McGovern: Least-integer closed groups. Proc. Conf. Ord. Alg. Struc. (Univ. of Florida, March 2001); J. Martínez, Ed.; Kluwer Acad. Publ. (2002), 245–260. MR 2083043
[10] M. Henriksen, J. Martínez and R. G. Woods: Spaces $X$ in which all prime $z$-ideals of $C(X)$ are either minimal or maximal. Comment. Math. Univ. Carolinae. 44 (2003), 261–294. MR 2026163
[11] C. B. Huijsmans and B. de Pagter: On $z$-ideals and $d$-ideals in Riesz spaces, I. Indag. Math. 42, Fasc. 2 (1980), 183–195. MR 0577573
[12] C. B. Huijsmans, B. de Pagter: On $z$-ideals and $d$-ideals in Riesz spaces, II. Indag. Math. 42, Fasc. 4 (1980), 391–408. MR 0597997
[13] C. B. Huijsmans and de Pagter: Ideal theory in $f$-algebras. Trans AMS 269 (January, 1982), 225–245. MR 0637036
[14] P. J. Johnstone: Stone Spaces. Cambridge Studies in Adv. Math, Cambridge Univ. Press. 3 (1982). MR 0698074 | Zbl 0499.54001
[15] S. Larson: Convexity conditions on $f$-rings. Canad. J. Math. XXXVIII (1986), 48–64. MR 0835035 | Zbl 0588.06011
[16] W. A. J. Luxemburg and A. C. Zaanen: Riesz Spaces, I. North Holland, Amsterdam-London, 1971.
[17] J. Martínez: Archimedean lattices. Alg. Universalis 3 (1973), 247–260. DOI 10.1007/BF02945124 | MR 0349503
[18] J. Martínez: Archimedean-like classes of lattice-ordered groups. Trans. AMS 186 (1973), 33–49. DOI 10.1090/S0002-9947-1973-0332614-X | MR 0332614
[19] J. Martínez: The hyper-archimedean kernel sequence of a lattice-ordered group. Bull. Austral. Math. Soc. 10 (1974), 337–349. DOI 10.1017/S0004972700041022 | MR 0349524
[20] J. Martínez: The $z$-dimension of an archimedean $f$-ring. Work in progress.
[21] J. Martínez: The regular top of an algebraic frame. Work in progress.
[22] J. Martínez and S. D. Woodward: Bézout and Prüfer $f$-rings. Comm. in Alg. 20 (1992), 2975–2989.
[23] J. Martínez and E. R. Zenk: When an algebraic frame is regular. Alg. Universalis 50 (2003), 231–257. DOI 10.1007/s00012-003-1841-1 | MR 2037528
[24] A. Monteiro: L’Arithmétique des filtres et les espaces topologiques. Segundo Symposium de Matemática; Villavicencio (Mendoza), 1954, pp. 129–162. MR 0074805 | Zbl 0058.38503
[25] J. T. Snodgrass and C. Tsinakis: Finite-valued algebraic lattices. Alg. Universalis 30 (1993), 311–318. DOI 10.1007/BF01190439 | MR 1225870
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