Classification of variants of partial Brauer monoids

A variant of a semigroup \(S\) with respect to an element \(a\in S\) is the semigroup \(S^{a}=(S,\star_{a})\), where \(x\star_{a}y=xay\) for any \(x,y\in S\). Here, \(a\) is the sandwich element of \(S^{a}\). In this article, we study variants of the partial Brauer monoid \(\mathcal{PB}_{n}\) for \(n\in\mathbb{N}\). We give the classification of these variants in the case when the rank of the sandwich element is nonzero.