A class of bisimple inverse semigroups
R. J. Warne · Pacific Journal of Mathematics · 1966
studied by Rees [6, p. 108, Theorem 3.3] may be substituted as a class of P in the above construction (here, P/J*f ~ (P, +), [8, p. 1118, Equation 2.9]).By [2, p. 553, Theorem 3.1], this substitution will yield the multiplication for the class of bisimple (inverse with identity) semigroups such that E s is integrally ordered in terms of ordered quadruples.We carry out the indicated calculations, which are routine, in detail here to yield Equation 3.4, which with the equality definition ((g, n), (h, m)) = {{g u n,), (h u m,)) if ggT 1 -hhr 1 , n^n, and m = m u is the structure theorem in terms of ordered quadruples.(The author was aware of this result in the spring of 1963.)N. R. Reilly informed us he had a multiplication for these semigroups (*, p. 572) in terms of ordered triples.His elegant formulation follows from our quadruple formulation by an application of [2, p. 548, Equation 1.2].A still more convenient formulation is S ~ U x C with a suitable multiplication.