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  • 1 Department of Mathematics and Statistics, Sam Houston State University, Box 2206, Huntsville, TX 77341-2206, USA
  • 2 Department of Mathematics Phillips Hall, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA
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Abstract  

Let ℤn be the finite cyclic group of order n and S ⊆ ℤn. We examine the factorization properties of the Block Monoid B(ℤn, S) when S is constructed using a method inspired by a 1990 paper of Erdős and Zaks. For such a set S, we develop an algorithm in Section 2 to produce and order a set {Mi}i=1n−1 which contains all the non-primary irreducible Blocks (or atoms) of B(ℤn, S). This construction yields a weakly half-factorial Block Monoid (see [9]). After developing some basic properties of the set {Mi}i=1n−1, we examine in Section 3 the connection between these irreducible blocks and the Erdős-Zaks notion of “splittable sets.” In particular, the Erdős-Zaks notion of “irreducible” does not match the classic notion of “irreducible” for the commutative cancellative monoids B(ℤn, S). We close in Sections 4 and 5 with a detailed discussion of the special properties of the blocks M1 with an emphasis on the case where the exponents of M1 take on extreme values. The work of Section 5 allows us to offer alternate arguments for two of the main results of the original paper by Erdős and Zaks.

  • Impact Factor (2019): 0.693
  • Scimago Journal Rank (2019): 0.412
  • SJR Hirsch-Index (2019): 20
  • SJR Quartile Score (2019): Q3 Mathematics (miscellaneous)
  • Impact Factor (2018): 0.664
  • Scimago Journal Rank (2018): 0.412
  • SJR Hirsch-Index (2018): 19
  • SJR Quartile Score (2018): Q2 Mathematics (miscellaneous)

Manuscript Submission: HERE