∗-homomorphisms. The resulting C ∗-algebras are identified as Toeplitz graph algebras. Graph algebras are proved to have inductive limit decompositions over any family of subgraphs with union equal to the whole graph. The construction is used to prove various structural properties of graph algebras. Introduction. Since the paper of Cuntz and Krieger in 1976, much work has gone into elucidating the brief remarks made there regarding the case of infinite 0 − 1 matrices. While perhaps the most far-reaching solution put forward has been a direct generalization to infinite 0 − 1 matrices ([9]), most of the papers on the subject generalize to a class of infinite directed graphs. (In fact, this is the direction indicated in [7].) In this paper

Dedicated to Hvedri Inassaridze for his 70th birthday The category of crossed complexes gives an algebraic model of CW-complexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the problem of calculating non-abelian extensions. We show how the strong properties of this category allow for the computation of free crossed resolutions for amalgamated sums and HNN-extensions of groups, and so obtain computations of higher homotopical syzygies in these cases. 1

We use the isomorphism between the categories of inverse semigroups and inductive groupoids to construct HNN extensions of inverse semigroups, where the associated inverse subsemigroups are order ideals of the base. Properties of groupoids then ensure that the base inverse semigroup always embeds in an HNN extension constructed in this way. Other properties of HNN extensions are determined, including a description of the maximal subgroups and of the maximal group image, and the structure of the inverse subsemigroup generated by the stable letters. Finally, some examples are given.

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