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<p>We define oriented and non-oriented genera for elements in a semigroup <i>S</i>. For a group <i>G</i> and an element <i>w</i> of <i>G</i>, we demonstrate that its oriented genus equals the fewest number of commutators in a factorization for it. And, we also demonstrate that its non-oriented genus equals the fewest number of squares in a factorization for it.</p>

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On genera for elements in a semigroup

  • 期刊论文

  • 发表日期：

    2025年8月22日

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  • Andrew Clifford,
  • Paul A. Cummings

摘要

We define oriented and non-oriented genera for elements in a semigroup S. For a group G and an element w of G, we demonstrate that its oriented genus equals the fewest number of commutators in a factorization for it. And, we also demonstrate that its non-oriented genus equals the fewest number of squares in a factorization for it.