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
KEYWORD
nonn,hard,more
AUTHOR
Marc Munar, Nov 18 2023
STATUS
approved