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Short maximal chains in the lattice of clones over a finite set
Author(s) -
Szendrei Ágnes
Publication year - 1983
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19831100105
Subject(s) - uncountable set , mathematics , finitary , combinatorics , lattice (music) , prime (order theory) , maximal element , element (criminal law) , discrete mathematics , set (abstract data type) , physics , countable set , computer science , acoustics , political science , law , programming language
Let ℒ k (3 ≦ k < ℵ 0 ) denote the lattice of clones of finitary operations on a k ‐element set. One interesting characteristic of the uncountable lattice ℒ k is the minimum l k of the cardinalities of maximal chains in ℒ k . It is known that for k prime l k = 5. In this paper the 5‐element maximal chains contained in ℒ k are investigated. It is proved that for k composite l k ≧ 6 while for k prime there are exactly ( k –2)! ( k + 4) 5‐element maximal chains in ℒ k , each closely related to a group operation.