Bass-Serre theory for groupoids and the structure of full regular semigroup amalgams
While it has been known since 1975 that all amalgams in the category of inverse semigroups are strongly embeddable, detailed knowledge of the structure of the associated amalgamated free products has been scarce. The main purpose of \S 3 of this paper is an investigation of the maximal subgroups of the amalgamated free product \(S_1*_US_2\) of inverse semigroups \(S_1\), \(S_2\), over a common full inverse subsemigroup \(U\) (``full'' meaning that \(U\) contains all the idempotents of both \(S_1\) and \(S_2\)). In fact the general methods employed are applicable within the category of regular semigroups. That the amalgamated free product over a full common regular subsemigroup exists was shown by \textit{T. E. Hall} [Trans. Am. Math. Soc. 246, 395-406 (1978; Zbl 0401.20053)]. However, the methods in the paper under review are based on \textit{K. S. S. Nambooripad}'s association of an ``inductive groupoid'' with each regular semigroup. He and \textit{F. Pastijn} [Houston J. Math. 15, No. 2, 249-254 (1989; Zbl 0697.20048)] showed that when \(U\) is full in \(S_1\) and \(S_2\), the groupoid associated with the amalgamated free product is just the amalgamated free product of the groupoids associated with \(S_1\) and \(S_2\), over that associated with \(U\). The maximal subgroups of \(S_1*_US_2\) are the vertex groups of this groupoid. Now amalgamated free products of groupoids have been shown by E. T. Ordman to behave in a manner akin to those of groups. The major innovation of this paper is the recognition that the vertex groups in the groupoid product are the fundamental groups of a certain graph of groups, arising quite simply from the sets of \({\mathcal D}\)-classes of \(S_1\), \(S_2\) and \(U\). This involves the Bass-Serre theory of groups acting on graphs. Various applications of the general theory are given. For instance, it is shown that if both \(S_1\) and \(S_2\) are combinatorial (that is, they have trivial subgroups), then the subgroups of any amalgamated free product of \(S_1\) and \(S_2\) are free groups. A more detailed analysis is given for the case when both \(S_1\) and \(S_2\) are bicyclic; in this case all full inverse subsemigroups \(U\) are known.
scientific article; zbMATH DE number 923671