Strong Pseudo-De Morgan Algebras and Pseudo-Involutive Psuedo-BCK Algebras

The well-known R₀ implication is developed to pseudo-De Morgan algebras, which is called generalized pseudo-R₀ implication. The notion of strong pseudo-De Morgan algebras is introduced, and its elementary properties are discussed. Secondly, two necessary and sufficient conditions are proved as follows: (1) A pseudo-De Morgan algebra A with generalized pseudo-R₀ implication becomes a pseudo-involutive pseudo-BCK algebra if and only if A is a strong pseudo-De Morgan algebra. (2) A pseudo-De Morgan algebra A with generalized pseudo-R₀ implication and corresponding ⊕ operator becomes a pseudo-regular residuated lattice if and only if A is a strong pseudo-De Morgan algebra. Finally, all pseudo-De Morgan algebras, strong pseudo-De Morgan algebras and proper pseudo-involutive pseudo-BCK algebras are obtained by MATLAB software when the order number is smaller than or equal to 8. Furthermore, starting with bounded distributive lattices, we discussed the classification problem of lower-order pseudo-involutive pseudo-BCK algebras.