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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.
0
1, 9, 519, 150120, 202728377
OFFSET
1,2
COMMENTS
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).
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).
LINKS
M. Munar, S. Massanet and D. Ruiz-Aguilera, A review on logical connectives defined on finite chains, Fuzzy Sets and Systems, Volume 462, 2023.
CROSSREFS
Particular case of the enumeration of discrete implications in general, enumerated in A360612.
The enumeration of discrete negations in general is given in A001700.
Sequence in context: A367552 A230671 A281443 * A003398 A015508 A281800
KEYWORD
nonn,hard,more
AUTHOR
Marc Munar, Nov 18 2023
STATUS
approved