On Free Products of Semigroups and a New Proof of Isbellʹs Zigzag Theorem

J. R. Isbell proved his famous zigzag theorem for semigroups using essentially topological methods in [Epimorphisms and dominions, in “Proceedings of the Conference on Categorical Algebra, La Jolla, 1965,” pp. 232–246]. Since then a number of authors have proved this result using a variety of different techniques. We present in this paper a description of the free product of a special amalgam of monoids using the “homological” techniques introduced by the author in [Proc. London Math. Soc. (3)52 (1986), 119–141] and from this derive a short proof of the zigzag theorem. This is the first proof which makes direct use of the amalgamated free product.