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Register a new userOn The Enhanced Power Graph of a Semigroup
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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.
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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.
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.
For a finite group $G$ with a normal subgroup $H$, the enhanced quotient graph of $G/H$, denoted by $mathcal{G}_{H}(G),$ is the graph with vertex set $V=(Gbackslash H)cup {e}$ and two vertices $x$ and $y$ are edge connected if $xH = yH$ or $xH,yHin langle zHrangle$ for some $zin G$. In this article, we characterize the enhanced quotient graph of $G/H$. The graph $mathcal{G}_{H}(G)$ is complete if and only if $G/H$ is cyclic, and $mathcal{G}_{H}(G)$ is Eulerian if and only if $|G/H|$ is odd. We show some relation between the graph $mathcal{G}_{H}(G)$ and the enhanced power graph $mathcal{G}(G/H)$ that was introduced by Sudip Bera and A.K. Bhuniya (2016). The graph $mathcal{G}_H(G)$ is complete if and only if $G/H$ is cyclic if and only if $mathcal{G}(G/H)$ is complete. The graph $mathcal{G}_H(G)$ is Eulerian if and only if $|G|$ is odd if and only if $mathcal{G}(G)$ is Eulerian, i.e., the property of being Eulerian does not depend on the normal subgroup $H$.
We show that every periodic element of the free idempotent generated semigroup on an arbitrary biordered set belongs to a subgroup of the semigroup.
In this paper we present a new embedding of a semigroup into a semiband (idempotent-generated semigroup) of depth 4 (every element is the product of 4 idempotents) using a semidirect product construction. Our embedding does not assume that S is a monoid (although it assumes a weaker condition), and works also for (non-monoid) regular semigroups. In fact, this semidirect product is particularly useful for regular semigroups since we can defined another embedding for these semigroups into a smaller semiband of depth 2. We shall then compare our construction with other known embeddings, and we shall see that some properties of S are preserved by our embedding.
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