This monograph gathers the author's results obtained since April 2019, which have been published in seven papers. Starting from the

frame-work centered on BCK algebras, m-BCK algebras are defi ned ('m' coming from 'magma') (algebras including the MV and the

Boolean 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 using

the computer program Prover9/Mace4. Also included in the book are some new results, never published before, on L algebras, on quantum B

algebras, 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'.