Let $\mathbb {B}_{k}$ be the general Boolean algebra and $T$ a linear operator on $M_{m,n}(\mathbb {B}_{k})$. If for any $A$ in $M_{m,n}(\mathbb {B}_{k})$ ($ M_{n}(\mathbb {B}_{k})$, respectively), $A$ is regular (invertible, respectively) if and only if $T(A)$ is regular (invertible, respectively), then $T$ is said to strongly preserve regular (invertible, respectively) matrices. In this paper, we will give complete characterizations of the linear operators that strongly preserve regular (invertible, respectively) matrices over $\mathbb {B}_{k}$. Meanwhile, noting that a general Boolean algebra $\mathbb {B}_{k}$ is isomorphic to a finite direct product of binary Boolean algebras, we also give some characterizations of linear operators that strongly preserve regular (invertible, respectively) matrices over $\mathbb {B}_{k}$ from another point of view.
We first investigate factorizations of elements of the semigroup $S$ of upper triangular matrices with nonnegative entries and nonzero determinant, provide a formula for $\rho (S)$, and, given $A\in S$, also provide formulas for $l(A)$, $L(A)$ and $\rho (A)$. As a consequence, open problem 2 and problem 4 presented in N. Baeth et al. (2011), are partly answered. Secondly, we study the semigroup of upper triangular matrices with only positive integral entries, compute some invariants of such semigroup, and also partly answer open Problem 1 and Problem 3 in N. Baeth et al. (2011).
In this paper we study some special residuated lattices, namely, idempotent residuated chains. After giving some properties of Green's relation $\mathcal D$ on the monoid reduct of an idempotent residuated chain, we establish a structure theorem for idempotent residuated chains. As an application, we give necessary and sufficient conditions for a band with an identity to be the monoid reduct of some idempotent residuated chain. Finally, based on the structure theorem for idempotent residuated chains, we obtain some characterizations of subdirectly irreducible, simple and strictly simple idempotent residuated chains.