Arithmetical invariants of local quaternion orders

Let $D$ be a DVR, let $K$ be its quotient field, and let $R$ be a $D$-order in a quaternion algebra $A$ over $K$. The elasticity of $R^\bullet$ is $ρ(R^\bullet) = \sup\{\, k/l : u_1\cdots u_k = v_1 \cdots v_l \text{ with $u_i$,$v_j$atoms of$R^\bullet$and$k$,$l \ge 1$} \,\}$ and is one of the basic arithmetical invariants that is studied in factorization theory. We characterize finiteness of $ρ(R^\bullet)$ and show that the set of distances $Δ(R^\bullet)$ and all catenary degrees $\mathsf c_\mathsf d(R^\bullet)$ are finite. In the setting of noncommutative orders in central simple algebras, such results have only been understood for hereditary orders and for a few individual examples.

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