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Register a new userGluing semigroups -- when and how
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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 $n=2$.
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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.
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 $Xsetminus H(X)$.
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.
The behaviour under coarsening functors of simple, entire, or reduced graded rings, of free graded modules over principal graded rings, of superfluous monomorphisms and of homological dimensions of graded modules, as well as adjoints of degree restriction functors, are investigated.
Let $R$ be a ring and $S$ a multiplicative subset of $R$. An $R$-module $P$ is called $S$-projective provided that the induced sequence $0rightarrow {rm Hom}_R(P,A)rightarrow {rm Hom}_R(P,B)rightarrow {rm Hom}_R(P,C)rightarrow 0$ is $S$-exact for any $S$-short exact sequence $0rightarrow Arightarrow Brightarrow Crightarrow 0$. Some characterizations and properties of $S$-projective modules are obtained. The notion of $S$-semisimple modules is also introduced. A ring $R$ is called an $S$-semisimple ring provided that every free $R$-module is $S$-semisimple. Several characterizations of $S$-semisimple rings are provided by using $S$-semisimple modules, $S$-projective modules, $S$-injective modules and $S$-split $S$-exact sequences.
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