Let ℕ represent the positive integers and ℕ₀ the non-negative integers. If b ∈ ℕ and Γ is a multiplicatively closed subset of ${\mathbb{Z}}_b=\mathbb{Z}/b\mathbb{Z}$, then the set $H_{\Gamma }=x\in \mathbb{N}|x+b\mathbb{Z}\in \Gamma \cup 1$ is a multiplicative submonoid of ℕ known as a congruence monoid. An arithmetical congruence monoid (or ACM) is a congruence monoid where Γ = ā consists of a single element. If $H_{\Gamma }$ is an ACM, then we represent it with the notation M(a,b) = (a + bℕ₀) ∪ 1, where a, b ∈ ℕ and a² ≡ a (mod b). A classical 1954 result of James and Niven implies that the only ACM...

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...

We present an algorithm for computing the greatest integer that is not a solution of the modular Diophantine inequality $axmodb\le x$, with complexity similar to the complexity of the Euclid algorithm for computing the greatest common divisor of two integers.

We study semigroups of labellings associated to a graph. These generalise the Jukes-Cantor model and phylogenetic toric varieties defined in [Buczynska W., Phylogenetic toric varieties on graphs, J. Algebraic Combin., 2012, 35(3), 421–460]. Our main theorem bounds the degree of the generators of the semigroup by g + 1 when the graph has first Betti number g. Also, we provide a series of examples where the bound is sharp.

All commutative semigroups $S$ are described such that the Jacobson radical is homogeneous in each ring graded by $S$.

In this study, we characterize all families of saturated numerical semigroups with multiplicity four. We also present some results about invariants of these semigroups.