We investigate the generalized Kronecker algebra 𝒦r = kΓr with r ≥ 3 arrows. Given a regular component 𝒞 of the Auslander-Reiten quiver of 𝒦r, we show that the quasi-rank rk(𝒞) ∈ ℤ≤1 can be described almost exactly as the distance 𝒲(𝒞) ∈ ℕ0 between two non-intersecting cones in 𝒞, given by modules with the equal images and the equal kernels property; more precisley, we show that the two numbers are linked by the inequality
−W(C)≤rk(C)≤−W(C)+3.\documentclass[12pt]{minimal}
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\begin{document}$$-\mathcal{W}(\mathcal{C}) \leq \text{rk}(\mathcal{C}) \leq - \mathcal{W}(\mathcal{C}) + 3.$$\end{document}Utilizing covering theory, we construct for each n ∈ ℕ0 a bijection φn between the field k and {𝒞∣𝒞 regular component, 𝒲(𝒞) = n}. As a consequence, we get new results about the number of regular components of a fixed quasi-rank.
