Cellularity of diagram algebras as twisted semigroup algebras

The Temperley–Lieb and Brauer algebras and their cyclotomic analogues, as well as the partition algebra, are all examples of twisted semigroup algebras. We prove a general theorem about the cellularity of twisted semigroup algebras of regular semigroups, which allows us to easily reproduce the cellularity of these algebras. This theorem generalizes a result of East about the cellularity of semigroup algebras of inverse semigroups.