The Layered Catalan Monoids: Structure and Determinants

In this paper, we introduce and study a class of monoids, called Layered Catalan Monoids (\( {LC}_n \)), which satisfy the structural conditions for $\ll$-smoothness as defined in~\cite{Sha-Det2}. These monoids are defined by specific identities inspired by Catalan monoids. We establish their canonical forms and compute their determinant, proving that it is non-zero for \(1 \leq n \leq 7\) but vanishes for \(n \geq 8\).