**Commutative Deductive Systems of Pseudo-M Algebras**

Andrzej Walendziak

We investigate the property of commutativity for various generalizations of pseudo-BCK algebras (pseudo-M, pseudo-RM, pseudo-RML, pseudo-aRML** algebras and many others). We give an axiom system for commutative pseudo-aRML** algebras and show that every such algebra (𝐴,→, ⇝, 1) is a join-semilattice with respect to the associated join operation ∨ given by 𝑥 ∨ 𝑦 = (𝑥 → 𝑦) ⇝ y. We define the commutative deductive systems of pseudo-M algebras and prove that a pseudo-aRML** algebra with the additional condition (pD) is commutative if and only if each of its deductive systems is commutative. We introduce the notion of BB-deductive system and then we construct the quotient algebra 𝒜/𝐷 of a pseudo-RM algebra 𝒜 via a closed BBdeductive system 𝐷 of 𝒜. Finally, we show that a BB-deductive system 𝐷 of a pseudo-RML algebra 𝒜 with (pD) is commutative if and only if 𝒜/𝐷 is a commutative pseudo-aRML** algebra.

*Keywords*: Pseudo-M, pseudo-CI, pseudo-BCH, pseudo-BCK algebra, commutative pseudo-M algebra, (commutative) deductive system, quotient algebra

*2020 Mathematics Subject Classification:* 03G25, 06A06, 06F35