  In this article we describe the Auslander-Reiten quiver for some posets with
an involution, that we call types $\mathfrak{U}_n$ and $\mathfrak{U}_\infty$.
These posets appear in the differentiation III of Zavadskij [12]. We follow the
approach to classical Auslander-Reiten theory due to Auslander, Reiten and
Smal\o{}[1]. For this purpose, we give a natural exact structure for the
category of representations of a partially ordered set with an involution. We
describe the projective and injective representations
