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<p>A monoid <i>S</i> is said to be weakly right coherent if every finitely generated right ideal of <i>S</i> is finitely presented as a right <i>S</i>-act. It is known that <i>S</i> is weakly right coherent if and only if it satisfies the following conditions: <i>S</i> is right ideal Howson, meaning that the intersection of any two finitely generated right ideals of <i>S</i> is finitely generated; and the right annihilator congruences of <i>S</i> are finitely generated as right congruences. We examine the behaviour of these two conditions (in the more general setting of semigroups) under certain algebraic constructions and deduce closure results for the class of weakly right coherent monoids. In particular, we show that the class of weakly right coherent monoids is closed under taking direct products and monoid free products.</p>

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Weakly right coherent monoids

  • 期刊论文

  • 发表日期：

    2025年6月2日

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  • Levent Michael Dasar,
  • Victoria Gould,
  • Craig Miller

摘要

A monoid S is said to be weakly right coherent if every finitely generated right ideal of S is finitely presented as a right S-act. It is known that S is weakly right coherent if and only if it satisfies the following conditions: S is right ideal Howson, meaning that the intersection of any two finitely generated right ideals of S is finitely generated; and the right annihilator congruences of S are finitely generated as right congruences. We examine the behaviour of these two conditions (in the more general setting of semigroups) under certain algebraic constructions and deduce closure results for the class of weakly right coherent monoids. In particular, we show that the class of weakly right coherent monoids is closed under taking direct products and monoid free products.