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%I A367446 #7 Nov 19 2023 10:33:47
%S A367446 1,9,519,150120,202728377
%N A367446 Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n}, which satisfy the generalized modus ponens with respect to a discrete t-norm T, i.e., T(x,I(x,y))<=y, for all x,y in L_n.
%C A367446 Number of discrete implications I:L_n^2->L_n defined on the finite chain L_n={0,1,...,n} satisfying the generalized modus ponens with respect to a discrete t-norm T, i.e., the number of binary functions I:L_n^2->L_n such that I is decreasing in the first argument, increasing in the second argument, I(0,0)=I(n,n)=n and I(n,0)=0 (discrete implication), and T(x,I(x,y))<=y, for all x,y in L_n (generalized modus ponens with respect to a discrete t-norm T). A discrete t-norm T is a binary operator T:L_n^2->L_n such that T is increasing in each argument, commutative (T(x,y)=T(y,x) for all x,y in L_n), associative (T(x,T(y,z))=T(T(x,y),z) for all x,y,z in L_n) and has neutral element n (T(x,n)=x for all x in L_n).
%C A367446 Also, the number of discrete implications I satisfying the generalized modus tollens with respect to a discrete t-norm T and the classical discrete negation N_C, given by N_C(x)=n-x for all x in L_n, i.e., T(N(y),I(x,y)) <= N(x) for all x,y in L_n (generalized modus tollens with respect to a discrete t-norm T and a discrete negation N).
%H A367446 M. Munar, S. Massanet and D. Ruiz-Aguilera, <a href="https://doi.org/10.1016/j.fss.2023.01.004">A review on logical connectives defined on finite chains</a>, Fuzzy Sets and Systems, Volume 462, 2023.
%Y A367446 Particular case of the enumeration of discrete implications in general, enumerated in A360612.
%Y A367446 The enumeration of discrete negations in general is given in A001700.
%K A367446 nonn,hard,more
%O A367446 1,2
%A A367446 _Marc Munar_, Nov 18 2023

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