Computing finite commutative semigroups
The problem is how to list all distinct (not isomorphic or antiisomorphic) semigroups of fixed order \(n\). Let \(S\) be a finite totally ordered set. All binary operations on \(S\) can be totally ordered lexicographically. Let \(m':S\times S\to S\) and \(m'':S\times S\to S\). Define \(m'<m''\) if there is \(a,b\in S\) satisfying the following: \(m'(x,y)=m''(x,y)\) if \(x<a\); \(m'(a,y)=m''(a,y)\) for all \(y<b\) and \(m'(a,b)<m''(a,b)\). For any permutation \(\sigma\) of \(S\), define \(m^\sigma:S\times S\to S\) by \(m^\sigma(x,y)=\sigma^{-1}m(\sigma x,\sigma y)\) for all \(x,y\in S\). If \(m^\sigma\geq m\) for all permutations \(\sigma\) of \(S\), then \(S\) is said to have precedence. If \(m^r\geq m\) only for all transpositions \(r\) of \(S\), then \(S\) is said to have weak precedence. If \(S\) has precedence, equivalently, there is no permutation \(\sigma\) with \(m^\sigma<m\), then we add a semigroup \((S,m)\) to the list since it is not isomorphic to any of the previous semigroups. Assume that \(S\) has weak precedence. When does \(S\setminus\{a\}\) having precedence imply \(S\) having precedence? The criteria are given in the various cases, and also the improved actual algorithm is given. Finally as an application the author classifies 11,545,843 commutative semigroups of order 9.
scientific article; zbMATH DE number 936962