Scholarlink

<p>There is strong evidence for the belief that ‘almost all’ finite semigroups, whether we consider multiplication operations on a fixed set or their isomorphism classes, are nilpotent of index 3 (3-nilpotent for short). The only known method for counting all semigroups of given order is exhaustive testing, but formulæ&#xa0; exist for the numbers of 3-nilpotent ones, and it is also known that ‘almost all’ of these are rigid (have only trivial automorphism). Here we express the number of distinct 3-nilpotent semigroup operations on a fixed set of cardinality <i>n</i> as a sum of Stirling numbers, and provide a new expression for the number of isomorphism classes of 3-nilpotent semigroups of cardinality <i>n</i>. We introduce a notion of semirigidity for semigroups (as a generalization of rigidity) and find computationally tractable formulæ&#xa0; giving an upper bound for the number of pairwise non-isomorphic semirigid 3-nilpotent semigroups, and thus an improved lower bound for the number of all 3-nilpotent semigroups up to isomorphism. Analogous formulæ&#xa0; are also developed for isomorphism classes such as commutative and self-dual semigroups, and for equivalence classes (isomorphic or anti-isomorphic). The method relies on an application of the theory of orbit counting in permutation group actions. Our main results are accompanied by tables containing values of these numbers and bounds up to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n=10\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>10</mn> </mrow> </math></EquationSource> </InlineEquation> with computations carried out in GAP (but perfectly feasible well beyond this value of <i>n</i>).</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Semirigidity and the enumeration of nilpotent semigroups of index three

  • 期刊论文

  • 发表日期：

    2025年12月24日

查看全文
  • Igor Dolinka,
  • D. G. FitzGerald,
  • James D. Mitchell

摘要

There is strong evidence for the belief that ‘almost all’ finite semigroups, whether we consider multiplication operations on a fixed set or their isomorphism classes, are nilpotent of index 3 (3-nilpotent for short). The only known method for counting all semigroups of given order is exhaustive testing, but formulæ  exist for the numbers of 3-nilpotent ones, and it is also known that ‘almost all’ of these are rigid (have only trivial automorphism). Here we express the number of distinct 3-nilpotent semigroup operations on a fixed set of cardinality n as a sum of Stirling numbers, and provide a new expression for the number of isomorphism classes of 3-nilpotent semigroups of cardinality n. We introduce a notion of semirigidity for semigroups (as a generalization of rigidity) and find computationally tractable formulæ  giving an upper bound for the number of pairwise non-isomorphic semirigid 3-nilpotent semigroups, and thus an improved lower bound for the number of all 3-nilpotent semigroups up to isomorphism. Analogous formulæ  are also developed for isomorphism classes such as commutative and self-dual semigroups, and for equivalence classes (isomorphic or anti-isomorphic). The method relies on an application of the theory of orbit counting in permutation group actions. Our main results are accompanied by tables containing values of these numbers and bounds up to \(n=10\) n = 10 with computations carried out in GAP (but perfectly feasible well beyond this value of n).