The Double Infinite Chain Condition, and Generalized Deviations of Posets and Modules
by
Mark L. Teply
University of Wisconsin-Milwaukee
Coauthors: Toma Albu

An arbitrary poset P is said to have the double infinite chain condition (DICC) if each chain of elements of P indexed by the order type \zeta of the integers stabilizes either to the left or to the right or to both sides. We investigate the DICC and its Krull-like dimension extension, which we call the balanced Krull dimension . We show that P has balanced Krull dimension bk(P) if and only if P has  \zeta-deviation  k_\zeta(P)  in the sense of Pouzet and Zaguia if and only if P has Krull dimension  k(P)  (or dual Krull dimension  k⁰(P) ); when these dimensions exist, then

k\zeta (P) <= bk(P) <= min { k(P), k0(P) } .

Since  \zeta  is a symmetric order type, we generalize our results to the case of symmetric order types, as well as sets of order types. We give applications to Grothendieck categories and torsion theories in module categories.

Date received: January 17, 1999