We present some results about chains of monoids S₀ → S₁ → S.₂. and their associated representation rings, with particular emphasis to behavior as the index n (viz. S_n) varies. A rich supply of such chains of monoids can be found via specializations of diagrammatic algebras or variations of diagrammatic algebras, where the inclusions involve the addition of loose strands. This thesis comprises of original results along three themes associated with the above: (1) Identifying a certain polynomial property featuring operators on representation rings, and a characterization of chains of groups G₀ →G₁ → G₂ .. which satisfy this polynomial property. (2) Understanding the induced action on homology from topological actions of the chain of Temperley-Lieb monoids TL₁ → TL₂ → TL₃ ... . Making the analogy to classical representation stability. (3) Identifying chains of diagrammatic monoids S₀ → S₁ → S₂ .. on which cryptographic protocols resist linear attacks. Explicitly computes lower bounds on the dimensions of all representations of various truncations of diagrammatic monoids.