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.

1, 9, 102, 1529, 28702

Offset: 1

Author

Marc Munar, Nov 18 2023

Keywords

• nonn
• hard
• more

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.