ABSTRACT
The chapter contains a study of irreducible and weakly irreducible FR-fragments where the filtration FR is filtered over the integers and is right bounded. We explain that irreducibility is preserved under taking quotients and why we can restrict to fragments of essential length not exceeding the length of the filtration. For algebra filtrations, i.e. when all R are rings or algebras, we discuss various decomposition results for two types of direct sums, the fragment direct sum and the strong fragment direct sum. The chapter also contains a non-exhaustive list of examples of fragments and glider representations appearing in various branches of mathematics.
