login
A367447
Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n}, which satisfy the law of importation with respect to a discrete t-norm T, i.e., I(T(x,y),z)=I(x,I(y,z)), for all x,y in L_n.
0
1, 9, 102, 1529, 28702
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 law of importation 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 I(T(x,y),z)=I(x,I(y,z)), for all x,y,z in L_n (law of importation 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).
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.
Sequence in context: A339643 A083452 A356244 * A081461 A231646 A110698
KEYWORD
nonn,hard,more
AUTHOR
Marc Munar, Nov 18 2023
STATUS
approved