Finite Basis Problem for Annular Monoids with Rotation

Let [Formula: see text] be the involution monoid of Annular monoid [Formula: see text] under the rotation involution [Formula: see text]. The involution monoids [Formula: see text] and [Formula: see text] are easily seen to be finitely based; Auinger et al. proved that [Formula: see text] is inherently non-finitely based if [Formula: see text]. In this paper, we show that [Formula: see text] is finitely based by providing a finite identity basis for [Formula: see text], which answers an open question posed by Auinger et al. Therefore, the involution monoid [Formula: see text] is finitely based if and only if [Formula: see text].