PhD Thesis Summary: The work in my PhD thesis revolves around some questions regarding string algebras linked by the common theme of generalizations of bridge quiver introduced by Schr ̈oer. String algebras are monomial algebras presented using quivers with relations. Finite-dimensional indecomposable modules over string algebras are associated with certain walks known as strings and bands on the underlying quiver. The finiteness of bands characterizes the class of domestic string algebras. Band-free connecting strings between bands are known as weak bridges, and bridges are irreducible with respect to a certain natural partial binary operation.

In the context of domestic string algebras along the lines of ‘Ringel’s quilt’, we aim at the exact computation of a version of hammocks that are linear orders up to isomorphism. We further extend the notion of the bridge quiver to the extended arch bridge quiver. Linearizing this quiver, we obtain a decorated tree, and we develop an algorithm using this tree as a tool to produce terms that label the paths on hammocks. Finally, we characterize the class of order types of these hammocks as the bounded discrete ones amongst the class of finitely presented linear orders--the smallest class of linear orders containing finite linear orders as well as ω, and that is closed under isomorphism, order reversal, finite order sums and lexicographic products. The rank of a domestic string algebra is an ordinal-valued function that takes value ∞ for non-domestic string algebras. In the latter context, the appropriate numerical invariant is the stable rank. Motivated by the factorization system introduced by Prest, we introduce the notion of recursive terms and use it to prove a dichotomy result for a subclass of non-domestic string algebras which states that if the rank of a morphism is not finite, then it is ∞ and show that the stable rank of algebras in this subclass lies in the set {ω, ω + 1, ω + 2}.