Correction to: Davenport constant for semigroups
- PDF / 1,212,846 Bytes
- 9 Pages / 439.37 x 666.142 pts Page_size
- 12 Downloads / 179 Views
Correction to: Davenport constant for semigroups Guoqing Wang1
© Springer Science+Business Media, LLC, part of Springer Nature 2020
Correction to: Semigroup Forum 76:234–238 (2008) https://doi.org/10.1007/s00233-007-9019-3
Introduction The Davenport constant of a finite abelian group G, denoted D(G) , is defined as the smallest positive integer 𝓁 such that every sequence of terms from G of length at least 𝓁 contains one or more terms whose product is the identity element of G. This invariant was popularized by H. Davenport in the 1960’s, notably for its significance in algebraic number theory (as mentioned in [5]). Though attributed to H. Davenport, who proposed the study of this constant in 1965, Rogers [6] in 1963 pioneered the investigation of this combinatorial invariant. This invariant together with the celebrated EGZ Theorem obtained by Erdős et al. [2] in 1961 led to the emergence of a branch in additive group theory, called zero-sum theory (see [3] for a survey). In 2008, the author and Gao [9] formulated the definition of the Davenport constant for finite commutative semigroups. Definition A [9] Let S be a finite commutative semigroup with identity. Let T be a sequence of terms from the semigroup S . We call T reducible if T contains a proper subsequence T ′ ( T ′ ≠ T ) such that the sum of all terms of T ′ equals the sum of all terms of T. Define the Davenport constant of the semigroup S , denoted D(S) , to be the smallest positive integer 𝓁 such that every sequence T of length at least 𝓁 of terms from S is reducible.
Communicated by Laszlo Marki. The original article can be found online at https://doi.org/10.1007/s00233-007-9019-3. * Guoqing Wang [email protected] 1
School of Mathematics Science, Tiangong University, Tianjin 300387, People’s Republic of China
13
Vol.:(0123456789)
G. Wang
The above definition can be also applied to infinite commutative semigroups with little modification, see [8]. Regretfully, one result in [9] (Theorem B below) on the Davenport constant for the multiplicative semigroup of a direct product of residue class rings has turned out to be incorrect. On the other hand, since the definition was proposed, several results (see [1, 7, 8, 10, 11]) have been obtained on the Davenport constant of commutative semigroups. In the present paper, we will fix this mistake by proving a weaker result (Theorem 1) by a different method. Theorem B [9] For integers r ≥ 1 , n1 , … , nr > 1 , let R = ℤn1 ⊕ ⋯ ⊕ ℤnr . Let SR be the multiplicative semigroup of the ring R and U(R) be the group of units of R. Then D(SR ) − D(U(R)) = #{i ∈ [1, r] ∶ 2 ∥ ni }. Theorem 1 For integers r ≥ 1 , n1 , … , nr > 1 , let R = ℤn1 ⊕ ⋯ ⊕ ℤnr . Then
#{i ∈ [1, r] ∶ 2 ∥ ni } ≤ D(SR ) − D(U(R)) ≤ #{i ∈ [1, r] ∶ 2 ∣ ni }.
Preliminaries For integers m, n ∈ ℤ , we set [m, n] = {d ∈ ℤ ∶ m ≤ d ≤ n} . For a prime p and an integer n, let 𝜈p (n) be the p-adic valuation of n, i.e., the largest integer k such that pk divides n if n ≠ 0 , and 𝜈p (0) = ∞. • In the rest of this paper, we always let S be a f
Data Loading...
