Formally, a description of weak BCK-algebras can be obtained by replacing the first BCK axiom \ - \le z - y}\) by its weakening \. It is known that every weak BCK-algebra is completely determined by the structure of its initial segments. We consider weak BCK-algebras with De Morgan complemented, orthocomplemented and orthomodular sections, as well as those where sections satisfy a certain compatibility condition, and characterize each of these classes of algebras by an equation or quasi-equation. For instance, those (...) weak BCK-algebras in which all initial segments are De Morgan complemented are just commutative weak BCK-algebras. (shrink)

We present here, without proofs, some results from a paper which will appear in Latvijskij Matematiˇceckij Ezegodnik.

It was noted recently in [4] that the definitions in a certain system of protothetic cease to be creative if one extends the rule of substitution so that direct substitution of the so called incomplete expressions for functor variables becomes possible. However, precise formulations were omitted in [4]. Here, we describe the new rule in some details and formulate some relevant theorems. For more circumstantial discussion, see [5].