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Register a new userPrincipal Matrices of Numerical Semigroups
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Principal matrices of a numerical semigroup of embedding dimension n are special types of $n times n$ matrices over integers of rank $leq n - 1$. We show that such matrices and even the pseudo principal matrices of size n must have rank $geq frac{n}{2}$ regardless of the embedding dimension. We give structure theorems for pseudo principal matrices for which at least one $n - 1 times n - 1$ principal minor vanish and thereby characterize the semigroups in embedding dimensions $4$ and $5$ in terms of their principal matrices. When the pseudo principal matrix is of rank $n - 1$, we give a sufficient condition for it to be principal.
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Given two semigroups $langle Arangle$ and $langle Brangle$ in ${mathbb N}^n$, we wonder when they can be glued, i.e., when there exists a semigroup $langle Crangle$ in ${mathbb N}^n$ such that the defining ideals of the corresponding semigroup rings satisfy that $I_C=I_A+I_B+langlerhorangle$ for some binomial $rho$. If $ngeq 2$ and $k[A]$ and $k[B]$ are Cohen-Macaulay, we prove that in order to glue them, one of the two semigroups must be degenerate. Then we study the two most degenerate cases: when one of the semigroups is generated by one single element (simple split) and the case where it is generated by at least two elements and all the elements of the semigroup lie on a line. In both cases we characterize the semigroups that can be glued and say how to glue them. Further, in these cases, we conclude that the glued $langle Crangle$ is Cohen-Macaulay if and only if both $langle Arangle$ and $langle Brangle$ are also Cohen-Macaulay. As an application, we characterize precisely the Cohen-Macaulay semigroups that can be glued when $n=2$.
This paper considers numerical semigroups $S$ that have a non-principal relative ideal $I$ such that $mu_S(I)mu_S(S-I)=mu_S(I+(S-I)) $. We show the existence of an infinite family of such which $I+(S-I)=Sbackslash{0}$. We also show examples of such pairs that are not members of this family. We discuss the computational process used to find these examples and present some open questions pertaining to them.
Let $mathcal C$ be a class of Hausdorff topological semigroups which contains all zero-dimensional Hausdorff topological semigroups. A semigroup $X$ is called $mathcal C$-$closed$ if $X$ is closed in each topological semigroup $Yin mathcal C$ containing $X$ as a discrete subsemigroup; $X$ is $projectively$ $mathcal C$-$closed$ if for each congruence $approx$ on $X$ the quotient semigroup $X/_approx$ is $mathcal C$-closed. A semigroup $X$ is called $chain$-$finite$ if for any infinite set $Isubseteq X$ there are elements $x,yin I$ such that $xy otin{x,y}$. We prove that a semigroup $X$ is $mathcal C$-closed if it admits a homomorphism $h:Xto E$ to a chain-finite semilattice $E$ such that for every $ein E$ the semigroup $h^{-1}(e)$ is $mathcal C$-closed. Applying this theorem, we prove that a commutative semigroup $X$ is $mathcal C$-closed if and only if $X$ is periodic, chain-finite, all subgroups of $X$ are bounded, and for any infinite set $Asubseteq X$ the product $AA$ is not a singleton. A commutative semigroup $X$ is projectively $mathcal C$-closed if and only if $X$ is chain-finite, all subgroups of $X$ are bounded and the union $H(X)$ of all subgroups in $X$ has finite complement $Xsetminus H(X)$.
Fix a poset $Q$ on ${x_1,ldots,x_n}$. A $Q$-Borel monomial ideal $I subseteq mathbb{K}[x_1,ldots,x_n]$ is a monomial ideal whose monomials are closed under the Borel-like moves induced by $Q$. A monomial ideal $I$ is a principal $Q$-Borel ideal, denoted $I=Q(m)$, if there is a monomial $m$ such that all the minimal generators of $I$ can be obtained via $Q$-Borel moves from $m$. In this paper we study powers of principal $Q$-Borel ideals. Among our results, we show that all powers of $Q(m)$ agree with their symbolic powers, and that the ideal $Q(m)$ satisfies the persistence property for associated primes. We also compute the analytic spread of $Q(m)$ in terms of the poset $Q$.
We survey some known results about operator semigroup generated by operator matrices with diagonal or coupled domain. These abstract results are applied to the characterization of well-/ill-posedness for a class of evolution equations with dynamic boundary conditions on domains or metric graphs. In particular, our ill-posedness results on the heat equation with general Wentzell-type boundary conditions complement those previously obtained by, among others, Bandle-von Below-Reichel and Vitillaro-Vazquez.
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