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Image:Monotone Boolean functions 0,1,2,3.svg|400px|thumb|right| The free distributive lattices of monotonic Boolean functions on 0, 1, 2, and 3 arguments, with 2, 3, 6, and 20 elements respectively (move mouse over right diagram to see description) circle 659 623 30 contradiction circle 658 552 35 A and B and C circle 587 480 35 A and B circle 659 481 35 A and C circle 729 481 35 B and C circle 588 410 35 (A and B) or (A and C) circle 658 410 35 (A and B) or (B and C) circle 729 410 35 (A and C) or (B and C) circle 548 339 30 A circle 604 339 30 B circle 758 339 30 C circle 661 339 35 (A or B) and (A or C) and (B or C) <====> (A and B) or (A and C) or (B and C) circle 588 268 35 (A or B) and (A or C) circle 659 267 35 (A or B) and (B or C) circle 729 268 35 (A or C) and (B or C) circle 588 197 35 A or B circle 658 197 35 A or C circle 729 197 35 B or C circle 658 126 35 A or B or C circle 659 56 30 tautology desc bottom-left In mathematics, the Dedekind numbers are a rapidly growing sequence of integers named after Richard Dedekind, who defined them in 1897. The Dedekind number ''M''(''n'') counts the number of monotonic Boolean functions of ''n'' variables. Equivalently, it counts the number of antichains of subsets of an ''n''-element set, the number of elements in a free distributive lattice with ''n'' generators, or the number of abstract simplicial complexes with ''n'' elements. Accurate asymptotic estimates of ''M''(''n'')〔; ; .〕 and an exact expression as a summation,〔.〕 are known. However Dedekind's problem of computing the values of ''M''(''n'') remains difficult: no closed-form expression for ''M''(''n'') is known, and exact values of ''M''(''n'') have been found only for ''n'' ≤ 8.〔.〕 ==Definitions== A Boolean function is a function that takes as input ''n'' Boolean variables (that is, values that can be either false or true, or equivalently binary values that can be either 0 or 1), and produces as output another Boolean variable. It is monotonic if, for every combination of inputs, switching one of the inputs from false to true can only cause the output to switch from false to true and not from true to false. The Dedekind number ''M''(''n'') is the number of different monotonic Boolean functions on ''n'' variables. An antichain of sets (also known as a Sperner family) is a family of sets, none of which is contained in any other set. If ''V'' is a set of ''n'' Boolean variables, an antichain ''A'' of subsets of ''V'' defines a monotone Boolean function ''f'', where the value of ''f'' is true for a given set of inputs if some subset of the true inputs to ''f'' belongs to ''A'' and false otherwise. Conversely every monotone Boolean function defines in this way an antichain, of the minimal subsets of Boolean variables that can force the function value to be true. Therefore, the Dedekind number ''M''(''n'') equals the number of different antichains of subsets of an ''n''-element set.〔.〕 A third, equivalent way of describing the same class of objects uses lattice theory. From any two monotone Boolean functions ''f'' and ''g'' we can find two other monotone Boolean functions ''f'' ∧ ''g'' and ''f'' ∨ ''g'', their logical conjunction and logical disjunction respectively. The family of all monotone Boolean functions on ''n'' inputs, together with these two operations, forms a distributive lattice, the lattice given by Birkhoff's representation theorem from the partially ordered set of subsets of the ''n'' variables with set inclusion as the partial order. This construction produces the free distributive lattice with ''n'' generators.〔The definition of free distributive lattices used here allows as lattice operations any finite meet and join, including the empty meet and empty join. For the free distributive lattice in which only pairwise meets and joins are allowed, one should eliminate the top and bottom lattice elements and subtract two from the Dedekind numbers.〕 Thus, the Dedekind numbers count the number of elements in free distributive lattices.〔; ; .〕 The Dedekind numbers also count the number of abstract simplicial complexes on ''n'' elements, families of sets with the property that any subset of a set in the family also belongs to the family. Any antichain determines a simplicial complex, the family of subsets of antichain members, and conversely the maximal simplices in a complex form an antichain.〔.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dedekind number」の詳細全文を読む スポンサード リンク
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