A precedence theorem for semigroups (Q1911746)

In the enumeration of algebraic structures of a given type on a set \(S\), to avoid ``duplications'', one often employs an ordering of the possible operations on \(S\). This idea is made explicit in this paper for the case of semigroups. Let \(<\) be a total order on \(S\). Consider two binary operations \(m\) and \(m'\) on \(S\). Then define \(m<m'\) lexicographically, that is, \(m<m'\) if and only if there are \(a,b\in S\) with the following properties: \(m(x,y)=m'(x,y)\) for all \(y\) and all \(x<a\); \(m(a,y)=m'(a,y)\) for all \(y<b\); \(m(a,b)<m'(a,b)\). For a permutation \(\sigma\) of \(S\) let \(m_\sigma\) be the binary operation induced by \(\sigma\), that is, \(m_\sigma(x,y)=\sigma^{-1}m(\sigma x,\sigma y)\); similarly, let \(m^*_\sigma(x,y)=\sigma^{-1}m(\sigma y,\sigma x)\). Then \(\sigma\) is an isomorphism of \((S,m)\) onto \((S,m_\sigma)\) and an anti-isomorphism onto \((S,m^*_\sigma)\). A precedence order on \((S,m)\) is a total order \(<\) on \(S\) such that \(m\leq m_\sigma\) and \(m\leq m^*_\sigma\) for every permutation \(\sigma\) of \(S\). In enumerations of finite algebras, semigroups in particular, one usually sets out with a fixed total order \(<\) on \(S\) and then keeps only those operations for which \(<\) is a precedence order. It is proved here that, in a finite semigroup, the least element under any precedence order is an idempotent in the kernel.