If $a$ and $b$ are positive integers with $a\le b$ and $a^2\equiv a\mathrm{mod}b$, then the set$$M_{a,b}=\{x\in \mathbb{N}:x\equiv a\mathrm{mod}b\text{or}x=1\}$$is a multiplicative monoid known as an arithmetical congruence monoid (or ACM). For any monoid $M$ with units $M^{\times }$ and any $x\in M\setminus M^{\times }$ we say that $t\in \mathbb{N}$ is a factorization length of $x$ if and only if there exist irreducible elements $y_1,...,y_t$ of $M$ and $x=y_1\cdots y_t$. Let $ℒ\left(x\right)=\{t_1,...,t_j\}$ be the set of all such lengths (where $t_i\<t_{i+1}$ whenever $i\<j$). The Delta-set of the element $x$ is defined as the set of gaps in $ℒ\left(x\right)$: $\Delta \left(x\right)=\{t_{i+1}-t_i:1\le i\<k\}$ and the Delta-set of the monoid $M$ is given by ${\bigcup }_{x\in M\setminus M^{\times }}\Delta \left(x\right)$. We consider the $\Delta \left(M\right)$ when $M=M_{a,b}$ is an ACM with...

Let $𝔞$, $I$, $J$ be ideals of a Noetherian local ring $(R,𝔪,k)$. Let $M$ and $N$ be finitely generated $R$-modules. We give a generalized version of the Duality Theorem for Cohen-Macaulay rings using local cohomology defined by a pair of ideals. We study the behavior of the endomorphism rings of $H_{I,J}^t\left(M\right)$ and $D\left(H_{I,J}^t\left(M\right)\right)$, where $t$ is the smallest integer such that the local cohomology with respect to a pair of ideals is nonzero and $D(-):={\mathrm{Hom}}_R(-,E_R\left(k\right))$ is the Matlis dual functor. We show that if $R$ is a $d$-dimensional complete Cohen-Macaulay ring and $H_{I,J}^i\left(R\right)=0$...