A367445 Number of discrete implications I:L_n^2-> L_n defined on the finite chain L_n={0,1,...n}, which satisfy the contrapositive symmetry with respect to some discrete negation N, i.e., I(x,y) = I(N(y), N(x)), for all x,y in L_n.

1, 8, 99, 2828, 152474

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 contrapositive symmetry with respect to some discrete negation N, 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(x,y) = I(N(y), N(x)), for all x,y in L_n (contrapositive symmetry with respect to a discrete negation N). A discrete negation N:L_n->L_n is a decreasing operator with N(0)=n and N(n)=0.

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.

When the discrete negation is N(x)=n-x, for all x in L_n, the enumeration is given in A366540.