This monograph gathers the author's results obtained since April 2019, which have been published in seven papers. Starting from theframe-work centered on BCK algebras, m-BCK algebras are defi ned ('m' coming from 'magma') (algebras including the MV and theBoolean algebras) and other more general algebras, thus obtaining an analogous new frame-work, centered on m-BCK algebras.'Bridge theorems' connecting the two frameworks are obtained, in the involutive case.Incorporated within this new frame-work are the quantum structures: the bounded involutive lattices, the De Morgan algebras, the ortholattices, the quantum MV algebras, the orthomodular lattices, with important new results and many examples obtained by usingthe computer program Prover9/Mace4. Also included in the book are some new results, never published before, on L algebras, on quantum Balgebras, pseudo-BCI/pseudo-BCK algebras and prealgebras (in the non-commutative case) and important final results on quantum structures, with many examples.This monograph has 17 chapters, divided into three parts: Part I focuses on the 'world' of BCK algebras and of pseudo-BCK algebras, Part II concentrates on the 'world' of m-BCK algebras and Part III is about 'bridge theorems' between the two 'worlds'.