New PhD thesis to be found at: http://xxx.lanl.gov/abs/math.CT/9812097 Summary of details: Title: Applications of Rewriting Systems and Groebner Bases to Computing Kan Extensions and Identities Among Relations. Authors: Anne Heyworth (University of Wales, Bangor). Comments: PhD thesis, 104 pages, LaTeX2e. Report-no: University of Wales, Bangor preprint number 98-23. Subj-class: Category Theory; Combinatorics. MSC-class: 18-04 (Primary) 05-02; 20F05; 68Q42; 68Q40; 16S15 (Secondary). \\ This thesis concentrates on the development and application of Groebner bases methods to a range of combinatorial problems (involving groups, semigroups, categories, category actions, algebras and K-categories). Chapter Two contains the generalisation of rewriting and Knuth-Bendix procedures to Kan extensions. Chapter Three shows that the standard Knuth-Bendix algorithm is step-for-step a special case of the Buchberger's algorithm for noncommutative Groebner bases. The one-sided cases and higher dimensions are considered, and the relations between these are made precise. Chapter Four relates rewrite systems, Groebner bases and automata. Reduction machines for rewrite systems are identified with standard output automata and the reduction machines devised for algebras are expressed as Petri-nets. Chapter Five introduces logged rewriting for group presentations. The completion of a logged rewriting system for a group determines a partial contracting homotopy which enables the computation of a set of generators for the module of identities among relations using the covering groupoid methods devised by Brown and Razak Sallah. Reducing the resulting set of submodule generators is identified as a Groebner basis problem. -- Anne Heyworth.