A Generalization of Symmetric Inverse Semigroups

The set of difunctional binary relations D_X plays a special role in representing inverse semigroups by binary relations. However, D_X is not an inverse semigroup either with the standard operation ∘, or with an alternative operation introduced in 6]. We introduce a new binary operation ⋄ on the set B_X of binary relations. We demonstrate that (D_X, ⋄) is an inverse semigroup, and the symmetric inverse semigroup (I_X, ∘) is a subsemigroup of (D_X,⋄).