  We study the residuated basic logic ($\mathsf{RBL}$) of residuated basic
algebra in which the basic implication of Visser's basic propositional logic
($\mathsf{BPL}$) is interpreted as the right residual of a non-associative
binary operator $\cdot$ (product). We develop an algebraic system
$\mathsf{S_{RBL}}$ of residuated basic algebra by which we show that
$\mathsf{RBL}$ is a conservative extension of $\mathsf{BPL}$. We present the
sequent formalization $\mathsf{L_{RBL}}$ of $\mathsf{S_{RBL}}$ which is an
extension of distributive full non-associative Lambek calculus
($\mathsf{DFNL}$), and show that the cut elimination and subformula property
hold for it.
