In this article, we provide a description of the Auslander–Reiten quiver for certain posets endowed with an involution, which we denote as types \(\mathfrak {U}_n\) and \(\mathfrak {U}_\infty \) . These posets appear in Zavadskij’s Differentiation III (Can Math Soc Conf Proc 11:299–323, 1991). Our approach follows the classical Auslander–Reiten theory developed by Auslander, Reiten, and Smalø (Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics, UK, 1992). To this end, we establish a natural exact structure for the category of representations of a partially ordered set with an involution, and subsequently delineate the projective and injective representations.
