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Register a new userOn Atomic Density of Numerical Semigroup Algebras
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A numerical semigroup $S$ is a cofinite, additively-closed subset of the nonnegative integers that contains $0$. In this paper, we initiate the study of atomic density, an asymptotic measure of the proportion of irreducible elements in a given ring or semigroup, for semigroup algebras. It is known that the atomic density of the polynomial ring $mathbb{F}_q[x]$ is zero for any finite field $mathbb{F}_q$; we prove that the numerical semigroup algebra $mathbb{F}_q[S]$ also has atomic density zero for any numerical semigroup~$S$. We also examine the particular algebra $mathbb{F}_2[x^2,x^3]$ in more detail, providing a bound on the rate of convergence of the atomic density as well as a counting formula for irreducible polynomials using M{o}bius inversion, comparable to the formula for irreducible polynomials over a finite field $mathbb{F}_q$.
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The enhanced power graph $mathcal P_e(S)$ of a semigroup $S$ is a simple graph whose vertex set is $S$ and two vertices $x,y in S$ are adjacent if and only if $x, y in langle z rangle$ for some $z in S$, where $langle z rangle$ is the subsemigroup generated by $z$. In this paper, first we described the structure of $mathcal P_e(S)$ for an arbitrary semigroup $S$. Consequently, we discussed the connectedness of $mathcal P_e(S)$. Further, we characterized the semigroup $S$ such that $mathcal P_e(S)$ is complete, bipartite, regular, tree and null graph, respectively. Also, we have investigated the planarity together with the minimum degree and independence number of $mathcal P_e(S)$. The chromatic number of a spanning subgraph, viz. the cyclic graph, of $mathcal P_e(S)$ is proved to be countable. At the final part of this paper, we construct an example of a semigroup $S$ such that the chromatic number of $mathcal P_e(S)$ need not be countable.
The aim of this paper is to investigate whether the class of automaton semigroups is closed under certain semigroup constructions. We prove that the free product of two automaton semigroups that contain left identities is again an automaton semigroup. We also show that the class of automaton semigroups is closed under the combined operation of free product followed by adjoining an identity. We present an example of a free product of finite semigroups that we conjecture is not an automaton semigroup. Turning to wreath products, we consider two slight generalizations of the concept of an automaton semigroup, and show that a wreath product of an automaton monoid and a finite monoid arises as a generalized automaton semigroup in both senses. We also suggest a potential counterexample that would show that a wreath product of an automaton monoid and a finite monoid is not a necessarily an automaton monoid in the usual sense.
In this paper we introduce the Schutzenberger category $mathbb D(S)$ of a semigroup $S$. It stands in relation to the Karoubi envelope (or Cauchy completion) of $S$ in the same way that Schutzenberger groups do to maximal subgroups and that the local divisors of Diekert do to the local monoids $eSe$ of $S$ with $ein E(S)$. In particular, the objects of $mathbb D(S)$ are the elements of $S$, two objects of $mathbb D(S)$ are isomorphic if and only if the corresponding semigroup elements are $mathscr D$-equivalent, the endomorphism monoid at $s$ is the local divisor in the sense of Diekert and the automorphism group at $s$ is the Schutzenberger group of the $mathscr H$-class of $S$. This makes transparent many well-known properties of Greens relations. The paper also establishes a number of technical results about the Karoubi envelope and Schutzenberger category that were used by the authors in a companion paper on syntactic invariants of flow equivalence of symbolic dynamical systems.
The cyclic graph $Gamma(S)$ of a semigroup $S$ is the simple graph whose vertex set is $S$ and two vertices $x, y$ are adjacent if the subsemigroup generated by $x$ and $y$ is monogenic. In this paper, we classify the semigroup $S$ such that whose cyclic graph $Gamma(S)$ is complete, bipartite, tree, regular and a null graph, respectively. Further, we determine the clique number of $Gamma(S)$ for an arbitrary semigroup $S$. We obtain the independence number of $Gamma(S)$ if $S$ is a finite monogenic semigroup. At the final part of this paper, we give bounds for independence number of $Gamma(S)$ if $S$ is a semigroup of bounded exponent and we also characterize the semigroups attaining the bounds.
We show that every periodic element of the free idempotent generated semigroup on an arbitrary biordered set belongs to a subgroup of the semigroup.
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