ON AMALGAMATION OF PARTIALLY ORDERED MONOIDS

Abstract

In 1927 O. Schreier proved in his article: "Die Untergruppen der freien Gruppen" (published in Abh. Math. Sem. Univ. Hamburg, 5, pp. 161–183) that any amalgam of groups can be embedded in a group (see for example Categorical Algebraic Prop- erties. A Compendium on Amalgamation, Congruence Extension, Epimorphisms, Residual Smallness, and Injectivity by E.W. Kiss et al., in Studia Sci. Math. Hun- garica, 18 (1983), pp. 79-141). That semigroup amalgams are not always embeddable was ...rst shown in 1957 by N. Kimura in his doctoral thesis. Kimura’s example can be found in the text The Algebraic Theory of Semigroups, Vol. II, by A.H. Cli¤ord and G.B. Preston. This example may be termed the ...rst motivator for an independent study of semigroup amalgams. The study was begun by J.M. Howie in the early 1960s. Howie’s initial work was also reproduced by Cli¤ord and Preston in their monograph cited above. Later on, in the 1970s, T.E. Hall formulated the Representation Extension Properties to study the amalgamation of semigroups.