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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 | )abbrev category PERMCAT PermutationCategory
++ Authors: Holger Gollan, Johannes Grabmeier, Gerhard Schneider
++ Date Created: 27 July 1989
++ Date Last Updated: 29 March 1990
++ Description:
++ PermutationCategory provides a categorial environment
++ for subgroups of bijections of a set (that is, permutations)
PermutationCategory(S) : Category == SIG where
S : SetCategory
SIG ==> Group with
cycle : List S -> %
++ cycle(ls) coerces a cycle ls, that is, a list with not
++ repetitions to a permutation, which maps ls.i to
++ ls.i+1, indices modulo the length of the list.
++ Error: if repetitions occur.
cycles : List List S -> %
++ cycles(lls) coerces a list list of cycles lls
++ to a permutation, each cycle being a list with not
++ repetitions, is coerced to the permutation, which maps
++ ls.i to ls.i+1, indices modulo the length of the list,
++ then these permutations are mutiplied.
++ Error: if repetitions occur in one cycle.
eval : (%,S) -> S
++ eval(p, el) returns the image of el under the
++ permutation p.
elt : (%,S) -> S
++ elt(p, el) returns the image of el under the
++ permutation p.
orbit : (%,S) -> Set S
++ orbit(p, el) returns the orbit of el under the
++ permutation p, that is, the set which is given by applications of
++ the powers of p to el.
"<" : (%,%) -> Boolean
++ p < q is an order relation on permutations.
++ Note that this order is only total if and only if S is totally ordered
++ or S is finite.
if S has OrderedSet then OrderedSet
if S has Finite then OrderedSet
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