### Pseudo-BCK Algebras Derived from Directoids

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**Authors:**A. REZAEI

**DOI:**10.46793/KgJMat2201.125R

**Abstract:**

The aim of this paper is to derive pseudo-BCK algebras from directoids and vice versa. We generalize some results proved by Ivan Chajda et al. in the case of BCK-algebras. We assign to an arbitrary pseudo-BCK algebra a semilattice-like structure and observe that this is the point where directoids are diﬀerent from the semilattice-like structures. Finally, the relation between commutative deductive systems and derive directoids from a bounded pseudo-BCK(pDN) algebras and a characterization of commutative deductive systems of a bounded pseudo-BCK(pDN) algebra in terms of directoids is discussed.

**Keywords:**

(Commutative) pseudo-BCK algebra, pseudo-BCK lattice, (commutative) deductive system, directoid, semilattice.

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