We investigate the algebra and category theory arising from the Geometry of Interaction series of papers, with the aim of abstracting the essential ideas behind these models of fragments of linear logic. The main tools used in this investigation are inverse semigroups (in particular the polycyclic monoids, and an inverse monoid of partial bijections on a term language that we call
the clause semigrou p) and the theory of symmetric monoidal categories, (in particular traced and com pact closed categories).
Applications of the above program are given to the following
(i) Ring theory - the conditions for a ( corner of a) ring R to be isomorphic to all matrix rings over R.
(ii) The construction of a composition and tensor preserving map from a category to a monoid, giving almost monoidal structures on monoids satisfying certain algebraic or categorical conditions,
and a partial dual to this construction given by a restriction of the Karoubi envelope.
(iii) The Geometry of Interaction I system - the identification of the Resolution formula as a categorical trace and the cut-elimination procedure as compact closure.
(iv) Two-way automata - the identification of the composition of global transition relations as the composition in an endomorphism monoid of a compact closed category, and an explicit description of global transition relations of singleton words in terms of Girard 's Resolution formula.
The thesis is not intended to be a study of the logical models used in the Geometry of Interaction, rather, it aims to identify the underlying algebra and category theory, and give applications.