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In this paper it is shown how the graphical methods developed by Stephen for analyzing inverse semigroup presentations may be used to study varieties of inverse semigroups. In particular, these methods may be used to solve the word problem for the free objects in the variety of inverse semigroups generated by the five-element combinatorial Brandt semigroup and in the variety of inverse semigroups determined by laws of the form xn = xn + 1. Covering space methods are used to study the free objects in a variety of the form ∨ where is a variety of inverse semigroups and is the variety of groups.

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Let 5 and T be inverse semigroups. Their free product S inv T is their coproduct in the category of inverse semigroups, defined by the usual commutative diagram. Previous descriptions of free products have been based, like that for the free product of groups, on quotients of the free semigroup product 5 sgp T. In that framework, a set of canonical forms for S inv T consists of a transversal of the classes of the congruence associated with the quotient. The general result [4] The approach here is to make use of the relationships among the presentations of 5, T and 5 inv T. Recall that if X is a nonempty set and X~l is a set of formal inverses of X then the free inverse semigroup ¥\S{X) over X is (" is the free semigroup on XUX' 1 and p is the Vagner congruence (see [10]). If P is a relation on X then the inverse semigroup with presentation (X \ P), Inv(A r \ P), is the quotientwhere x is the congruence generated by p U P . (Clearly, Inv(A" | P) is also isomorphic to a quotient of the free inverse semigroup on X, by a suitable congruence.)Now if the given inverse semigroups 5 and T are presented as 5 = Inv (A" | P) and T = Inv(y | Q), where X and Y are disjoint, then it is an exercise in universal algebra to verify that S inv T = Inv (A' U Y | P U Q). The graph-theoretical techniques developed by Stephen to study presentations of inverse semigroups may then be used and this is the point of view that will be taken in this paper. These techniques originated with Munn's use of trees to study free inverse semigroups [9] and the current paper is therefore in a sense also a sequel to his.Stephen's techniques are reviewed rather summarily in Section 1. For many further details the reader is referred to the paper [12] and the thesis [13] by Stephen (see also [8]). Section 2 treats in the abstract the particular types of graphs that appear in the construction of the Schiitzenberger automata for free products, the construction itself appearing in Section 3 (Theorem 3.4). Exactly which automata can be the Schiitzenberger automaton of an element of a free product is established in Section 4 (Theorem 4.1), thus providing a set of canonical forms for the product. From these forms, the canonical forms found by Jones in [4] may be quite easily found and proven unique (Theorem 4.5 and its corollary). Yet another set of canonical forms is also produced (Theorem 4.7 and its corollary). These are again graphical and are very similar to those previously given by Jones [3] for the free product of £-unitary inverse semigroups and by Margolis and Meakin [7] for the analogous product of inverse monoids, in the category of inverse monoids.

We study amalgamated free products in the category of inverse semigroups. Our approach is combinatorial. Graphical techniques are used to relate the structures of the inverse semigroups in a pushout square, and we then examine amalgamated free products. We show that an amalgam of inverse semigroups strongly embeds in the amalgamated free product, thus providing an alternative proof of the strong amalgamation property for inverse semigroups, a result due to T. E. Hall. We provide sufficient conditions for an amalgamated free product to be E-unitary, and we give necessary and sufficient conditions for the amalgamated free product of a special amalgam to be E-unitary. ᮊ 1998 Academic PressThe first steps toward a combinatorial theory of inverse semigroups are w x found in the results of Munn 18 . The Munn representation associates an element of the free inverse semigroup with a rooted finite connected subgraph of the Cayley graph of the free group. The graphical methods w x developed by Stephen 22, 23 then provided a general setting for the combinatorial study of inverse semigroups, via presentations. These results w x have been utilized in the study of word problems 25, 23 , formal languages w x w x 3, 21, 15, 16 , and universal objects 10, 25 . The graphical representation w of an inverse semigroup is reminiscent of the Cayley graph of a group 4, x wx 11, 13, 27 , and the results of 23 reduce to a discussion of the Cayley graph and its construction, if one considers an inverse semigroup which is also a group. This work extends the basic set of tools for the combinatorial study of inverse semigroups by providing a detailed combinatorial study of amalgamated free products of inverse semigroups.We first investigate the general case of pushouts in the category of inverse semigroups, and then examine the more specific case of amalga- J. B. STEPHEN 400 mated free products. Our investigations will establish that the category of inverse semigroups has the strong amalgamation property. T. E. Hall provided the original proof of the strong amalgamation property for inverse semigroups and investigated embeddings of amalgams and amalgaw x mated free products in a series of papers 5᎐7 . Our investigation is constructive and provides an insight into amalgamated free products that is reminiscent of combinatorial group theory. The representation is transparent to those familiar with graphical methods, and can be used with facility to investigate the structure of an amalgamated free product.The paper is composed of six sections.1. Terminology and notation are introduced and discussed.2. Our results derive from a procedure to construct the graphical representations of the inverse semigroups considered. Properties of the categories in which these constructions take place are discussed.3. Technical particulars of the graphical representations are considered.4. The relationships between graphical representations of inverse semigroups in a pushout diagram are discussed. Pushouts in the category of inverse semigroups are discussed. The g...