Abstract In the last four decades, researchers have introduced several algebraic structures, including $BCH$, $BZ$, $aBE$ and $Humberstone$-algebras. Iorgulescu (2016, J. Mult.-Valued Log. Soft Comput, 27, 353–406) generalized these algebras under the term $aRM$-algebras. This paper connects these algebras to their corresponding logical systems. Specifically, we define $aRM$-logic, demonstrate its strong soundness and completeness relative to $aRM$-algebras, and establish the independence of its axioms. We further explore connections between $aRM$-logic and $BCK/BCI$-logic, and introduce four schematic extensions of $aRM$-logic: $BCH$-logic, $BZ$-logic, $aBE$-logic and $H$-logic, each proven strongly sound and complete with respect to their associated algebras. We show that $aRM$-logic is algebraizable (in the sense of Blok and Pigozzi (1989, Algebraizable Logics, vol. 77. American Mathematical Society)) with the defining equation $p \rhd p \approx p$ and the equivalence formulas $\lbrace p \rhd q, q \rhd p \rbrace$. We also introduce $aR$-logic (a logic weaker than $aRM$-logic) and prove that it is the minimal algebraizable logic with the same defining equation $p \rhd p \approx p$ and equivalence formulas $\lbrace p \rhd q, q \rhd p \rbrace$.