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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 the semigroups isomorphic to principal ideals of finitely generated commutative monoids. We define the concept of finite presentation for this kind of semigroups. Furthermore, we show how to obtain information on these semigroups from their presentations.

Let $S$ be a numerical semigroup. We say that $h\in \mathbb{N}\setminus S$ is an isolated gap of $S$ if $\{h-1,h+1\}\subseteq S.$ A numerical semigroup without isolated gaps is called a perfect numerical semigroup. Denote by $\mathrm{m}\left(S\right)$ the multiplicity of a numerical semigroup $S$. A covariety is a nonempty family $𝒞$ of numerical semigroups that fulfills the following conditions: there exists the minimum of $𝒞,$ the intersection of two elements of $𝒞$ is again an element of $𝒞$, and $S\setminus \left\{\mathrm{m}\right(S\left)\right\}\in 𝒞$ for all $S\in 𝒞$ such that $S\ne min\left(𝒞\right).$ We prove that the set $𝒫\left(F\right)=\{S:S$ is a perfect numerical semigroup with...

We study numerical semigroups $S$ with the property that if $m$ is the multiplicity of $S$ and $w\left(i\right)$ is the least element of $S$ congruent with $i$ modulo $m$, then $0<w\left(1\right)<\cdots <w(m-1)$. The set of numerical semigroups with this property and fixed multiplicity is bijective with an affine semigroup and consequently it can be described by a finite set of parameters. Invariants like the gender, type, embedding dimension and Frobenius number are computed for several families of this kind of numerical semigroups.