Mathematical Social Sciences 27 (1994) 73-89 0165-4896/94/$07.00 fQ 1994 - Elsevier Science B.V. All rights reserved 73 On permutation lattices Vincent Duquenne” URA, CNRS 1294, G&attique, Neurogt%tique et Comportement, 45 Rue des Saints P&es, 75006 Paris, France Ameziane Cherfouh VFR de Mathtmatiques, Vniversitt de Paris V, 12 Rue Cujas, 75005 Paris, France Communicated by M.F. Janowitz Received 3 February 1992 Revised 14 August 1993 Abstract The lattice perm(n) of permutations of an n-element set n = (1,. . , n}, ‘rooted’ at (1,. , n), is shown to be meet- and join-semidistributive, which implies known results such as the non-existence of M,-sublattices, and that the complementation defines a congruence relation in Perm(n) with 2”-’ classes. The meet-core of Perm(n) is shown to be the set of meet-irreducible elements together with all the elements that have two upper covers that moreover generate a covering sublattice isomorphic to Perm(3); this shows that the meet operation is completely expressible in terms of reversing adjacent pairs. A recursive construction of Perm(n) as a Galois lattice -via a kind of summation process - is given, which has been a key for obtaining clear drawings of Perm(4) and Perm(5) with our new PC/VGA graphic program GLAD (General Lattice Analysis and Design). Finally, the congruence lattice C(Perm(n)) is recursively characterized through the order of its meet-irreducible elements. Key words: Permutation; semidistributivity; Meet/ join core; Galois lattice representation; Heredity of collapsing; Lattices of congruences List of symbols n = (1,. , . ) n} a finite set a, P, Y . . f permutations p,(a) I-compatible pairs I I cardinal of. , . -<, <, i cover, order relations C, 5 u, n, u, n set operations v, A, v, A lattice operations * Corresponding author at: CNRS, 10-b rue A. Payen, F75015 Paris, France. SSDI 0165-4896(93)00733-B