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In this paper, the concepts of indecomposable matrices and fully indecomposable matrices over a distributive lattice $L$ are introduced, and some algebraic properties of them are obtained. Also, some characterizations of the set $F_n\left(L\right)$ of all $n\times n$ fully indecomposable matrices as a subsemigroup of the semigroup $H_n\left(L\right)$ of all $n\times n$ Hall matrices over the lattice $L$ are given.

Let $n$ be a positive integer, and $C_n\left(r\right)$ the set of all $n\times n$ $r$-circulant matrices over the Boolean algebra $B=\{0,1\}$, $G_n={\bigcup }_{r=0}^{n-1}{C}_n\left(r\right)$. For any fixed $r$-circulant matrix $C$ ($C\ne 0$) in $G_n$, we define an operation “$*$” in $G_n$ as follows: $A*B=ACB$ for any $A,B$ in $G_n$, where $ACB$ is the usual product of Boolean matrices. Then $(G_n,*)$ is a semigroup. We denote this semigroup by $G_n\left(C\right)$ and call it the sandwich semigroup of generalized circulant Boolean matrices with sandwich matrix $C$. Let $F$ be an idempotent element in $G_n\left(C\right)$ and $M\left(F\right)$ the maximal subgroup in $G_n\left(C\right)$ containing...