Symmetric algebras, skew category algebras and inverse semigroups

We use results from the theory of inverse semigroups to identify and define a class of symmetric algebras, which we call inverse-symmetric algebras. For a finite inverse category C we define the skew category algebra associated to a C-algebra, which satisfies some additional properties, and we prove that remains inverse-symmetric. The classical examples of symmetric algebras: matrix algebra, Iwahori–Hecke algebra and the (twisted) group algebra are inverse-symmetric.