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The finite basis theorem for relatively normal lattices. (English) Zbl 0819.06009
From the authors’ introduction: A lower-bounded, distributive lattice is called relatively normal provided its set of prime ideals is a root- system under set-inclusion. Our primary objective in this paper is to provide an in depth study of relatively normal lattices, isolate some of their remarkable properties and lay the foundation for future research in this area. Special instances of our results appear in the literature on lattice-ordered groups, and related areas. The present study demonstrates, in particular, that many fundamental results in these fields admit a purely lattice-theoretic development. An example in point is the following generalization of P. Conrad’s Finite Basis Theorem [Mich. Math. J. 7, 171-180 (1960; Zbl 0103.015)]: For a nontrivial lower- bounded, distributive lattice \(D\), the following statements are equivalent: (1) \(D\) is relatively normal and every orthogonal subset of \(D\) is finite. (2) The ideal lattice \(I(D)\) of \(D\) is isomorphic to the lattice \(O(R)\) of lower sets of some root-system \(R\) with finitely many roots. (3) There exist a nonnegative integer \(r\) and an ascending chain \(Z_ 0\subset Z_ 1\subset \dots \subset Z_ r =D\) of ideals of \(D\) satisfying the following conditions: (a) For \(0\leq i\leq r\), \(Z_ i\) is the join in \(I(D)\) of a set \(L(i)= \{A_ 1^{(i)}, \dots, A_{n_ i}^{(i)}\}\) of ideals of \(D\), where (b) \(L(0)\) is a finite orthogonal set of ideals of \(D\) which are chains; (c) For \(0<i\leq r\), each ideal in \(L(i)\) is either in \(L(i-1)\) or else a proper ordinal extension of the join in \(I(D)\) of two or more of the ideals in \(L(i-1)\).

MSC:
06D05 Structure and representation theory of distributive lattices
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