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Register a new userStructure of long idempotent-sum free sequences over finite cyclic semigroups
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Guoqing Wang
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Let $mathcal{S}$ be a finite cyclic semigroup written additively. An element $e$ of $mathcal{S}$ is said to be idempotent if $e+e=e$. A sequence $T$ over $mathcal{S}$ is called {sl idempotent-sum free} provided that no idempotent of $mathcal{S}$ can be represented as a sum of one or more terms from $T$. We prove that an idempotent-sum free sequence over $mathcal{S}$ of length over approximately a half of the size of $mathcal{S}$ is well-structured. This result generalizes the Savchev-Chen Structure Theorem for zero-sum free sequences over finite cyclic groups.
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Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ over $G$ can be written in the form $S=(n_1g)cdotldotscdot(n_lg)$ where $gin G$ and $n_1, ldots, n_lin[1, ord(g)]$, and the index $ind(S)$ of $S$ is defined to be the minimum of $(n_1+cdots+n_l)/ord(g)$ over all possible $gin G$ such that $langle g rangle =G$. In this paper, we determine the index of any minimal zero-sum sequence $S$ of length 5 when $G=langle grangle$ is a cyclic group of a prime order and $S$ has the form $S=g^2(n_2g)(n_3g)(n_4g)$. It is shown that if $G=langle grangle$ is a cyclic group of prime order $p geq 31$, then every minimal zero-sum sequence $S$ of the above mentioned form has index 1 except in the case that $S=g^2(frac{p-1}{2}g)(frac{p+3}{2}g)((p-3)g)$.
Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)cdotldotscdot(n_lg)$ where $gin G$ and $n_1, ldots, n_lin[1, ord(g)]$, and the index $ind(S)$ of $S$ is defined to be the minimum of $(n_1+cdots+n_l)/ord(g)$ over all possible $gin G$ such that $langle g rangle =G$. An open problem on the index of length four sequences asks whether or not every minimal zero-sum sequence of length 4 over a finite cyclic group $G$ with $gcd(|G|, 6)=1$ has index 1. In this paper, we show that if $G=langle grangle$ is a cyclic group with order of a product of two prime powers and $gcd(|G|, 6)=1$, then every minimal zero-sum sequence $S$ of the form $S=(g)(n_2g)(n_3g)(n_4g)$ has index 1. In particular, our result confirms that the above problem has an affirmative answer when the order of $G$ is a product of two different prime numbers or a prime power, extending a recent result by the first author, Plyley, Yuan and Zeng.
Let $G$ be a finite cyclic group of order $n ge 2$. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)cdot ... cdot (n_lg)$ where $gin G$ and $n_1,..., n_l in [1,ord(g)]$, and the index $ind (S)$ of $S$ is defined as the minimum of $(n_1+ ... + n_l)/ord (g)$ over all $g in G$ with $ord (g) = n$. In this paper we prove that a sequence $S$ over $G$ of length $|S| = n$ having an element with multiplicity at least $frac{n}{2}$ has a subsequence $T$ with $ind (T) = 1$, and if the group order $n$ is a prime, then the assumption on the multiplicity can be relaxed to $frac{n-2}{10}$. On the other hand, if $n=4k+2$ with $k ge 5$, we provide an example of a sequence $S$ having length $|S| > n$ and an element with multiplicity $frac{n}{2}-1$ which has no subsequence $T$ with $ind (T) = 1$. This disproves a conjecture given twenty years ago by Lemke and Kleitman.
Let $p > 155$ be a prime and let $G$ be a cyclic group of order $p$. Let $S$ be a minimal zero-sum sequence with elements over $G$, i.e., the sum of elements in $S$ is zero, but no proper nontrivial subsequence of $S$ has sum zero. We call $S$ is unsplittable, if there do not exist $g$ in $S$ and $x,y in G$ such that $g=x+y$ and $Sg^{-1}xy$ is also a minimal zero-sum sequence. In this paper we show that if $S$ is an unsplittable minimal zero-sum sequence of length $|S|= frac{p-1}{2}$, then $S=g^{frac{p-11}{2}}(frac{p+3}{2}g)^4(frac{p-1}{2}g)$ or $g^{frac{p-7}{2}}(frac{p+5}{2}g)^2(frac{p-3}{2}g)$. Furthermore, if $S$ is a minimal zero-sum sequence with $|S| ge frac{p-1}{2}$, then $ind(S) leq 2$.
We study some sum-product problems over matrix rings. Firstly, for $A, B, Csubseteq M_n(mathbb{F}_q)$, we have $$ |A+BC|gtrsim q^{n^2}, $$ whenever $|A||B||C|gtrsim q^{3n^2-frac{n+1}{2}}$. Secondly, if a set $A$ in $M_n(mathbb{F}_q)$ satisfies $|A|geq C(n)q^{n^2-1}$ for some sufficiently large $C(n)$, then we have $$ max{|A+A|, |AA|}gtrsim minleft{frac{|A|^2}{q^{n^2-frac{n+1}{4}}}, q^{n^2/3}|A|^{2/3}right}. $$ These improve the results due to The and Vinh (2020), and generalize the results due to Mohammadi, Pham, and Wang (2021). We also give a new proof for a recent result due to The and Vinh (2020). Our method is based on spectral graph theory and linear algebra.
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