New Generalizations of BCI, BCK and Hilbert Algebras — Part I
Hilbert algebras are particular cases of BCK algebras, while BCK algebras are particular cases of BCI algebras. In Part I, starting with a list of properties of BCK algebras and with the algebras BCH, BCC, BZ, BE, pre-BCK, old generalizations of BCI algebras or of BCK algebras, we introduce new generalizations of BCI or of BCK algebras and, consequently, new generalizations of Hilbert algebras. Namely, we found thirty one new distinct generalizations of BCI or of BCK algebras and twenty new distinct generalizations of Hilbert algebras. We show the hierarchies existing between all these algebras, old or new ones.
In Part II, we present 83 examples of proper finite algebras, at least one example of each of the 59 (= 7 (old) + 1 (Hilbert) + 31 + 20) old or new algebras.
Keywords: BCI algebra, BCK algebra, Hilbert algebra, BCH algebra, BCC algebra, BZ algebra, BE algebra, pre-BCK algebra, RM algebra, RML algebra, weak BCK-algebra, (generalized) Tarski algebra
AMS classification (2010): 06F35, 03G25, 06A06