2012/08/23 by Gábor Czédli, Gabor Czedli, Czedli, Gabor +3
Computer Science · Mathematics · #06C10 (Primary) 20E15 (Secondary) #Advanced Algebra and Logic #FOS: Mathematics #Rings and Algebras (math.RA) #Rough Sets and Fuzzy Logic #math.RA #msc:06C10 #msc:20E15
paper · pdf · doi:10.48550/arxiv.1208.4749
16 pages, 1 figure
openalex publication_date 2012/08/23 · arxiv created 2013/01/09 · arxiv updated 2013/01/10 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
Let H and K be finite composition series of a group G. The intersections Hi∩ Kj of their members form a lattice CSL(H,\K) under set inclusion. Improving the Jordan-Hölder theorem, G. Grätzer, J.B. Nation and the present authors have recently shown that H and K determine a unique permutation pi such that, for all i, the i-th factor of H is "down-and-up projective" to the pi(i)-th factor of K. Equivalent definitions of pi were earlier given by R.P. Stanley and H. Abels. We prove that pi determines the lattice CLS(H,K). More generally, we describe slim semimodular lattices, up to isomorphism, by permutations, up to an equivalence relation called "sectionally inverted or equal". As a consequence, we prove that the abstract class of all CSL(H,K) coincides with the class of duals of all slim semimodular lattices.