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<p>We provide a complete classification of matrix semirings <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textbf{M}_n(S)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="bold">M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> over two-element additively idempotent semirings <i>S</i> with respect to the finite basis property. Our main theorem shows that for every integer <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n \ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, the semiring <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\textbf{M}_n(S)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="bold">M</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is finitely based if and only if <i>S</i> is distinct from a distributive lattice.</p>

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The finite basis problem for matrix semirings over a two-element additively idempotent semiring

  • 期刊论文

  • 发表日期：

    2026年8月5日

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  • Jun Jiao,
  • Miaomiao Ren

摘要

We provide a complete classification of matrix semirings \(\textbf{M}_n(S)\) M n ( S ) over two-element additively idempotent semirings S with respect to the finite basis property. Our main theorem shows that for every integer \(n \ge 2\) n ≥ 2 , the semiring \(\textbf{M}_n(S)\) M n ( S ) is finitely based if and only if S is distinct from a distributive lattice.