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<p>In this paper we study the variety of the so-called flat nilpotent semiring. We establish a model for the free object in the ai-semiring variety <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textbf{FCN}_{k}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="bold">FCN</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> generated by all commutative flat <i>k</i>-nilpotent semiring over an alphabet <i>X</i>. Also, we provide a necessary and sufficient condition under which an identity is satisfied by the flat semiring <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(S_{c}(a_{1}\cdots a_{k})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>⋯</mo> <msub> <mi>a</mi> <mi>k</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

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On flat k-nilpotent semirings

  • 期刊论文

  • 发表日期：

    2025年7月13日

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  • Yanan Wu,
  • Jinxing Zhao

摘要

In this paper we study the variety of the so-called flat nilpotent semiring. We establish a model for the free object in the ai-semiring variety \(\textbf{FCN}_{k}\) FCN k generated by all commutative flat k-nilpotent semiring over an alphabet X. Also, we provide a necessary and sufficient condition under which an identity is satisfied by the flat semiring \(S_{c}(a_{1}\cdots a_{k})\) S c ( a 1 ⋯ a k ) .