Extensions of linear involutory semigroups (Q1177856)

A linear involutory semigroup is a totally ordered set \((S,<)\) with a binary operation \(+\) and a unary operation \(*\) so that \((S,+)\) is an Abelian semigroup with zero, and for all \(a\), \(b\), \(c\) in \(S\), \(a^{**}=a\), \(a<b\) implies \(b^*<a^*\) and \(a+c<b+c\), and \(a^*<(a+b)^*+b\). \textit{L. Berg} [Rostocker Math. Kolloq. 33, 49-56 (1988; Zbl 0612.06011)] gave all linear involutory semigroups of order 5, and \textit{W. Peters} [ibid. 35, 45-56 (1988; Zbl 0676.20040)] computed those of orders 6 and 7. Furthermore, Peters gave algorithms for producing examples of orders 8 and 9. The purpose of this paper is to formalize and verify algorithms for constructing linear involutory semigroups of higher orders.