An algebra structure on the linear span of signed permutations

The study of algebra structures on the linear span of all signed permutations has attracted considerable attention over the past two decades. In this work, we introduce a new algebra structure on this underlying space and show that the map which sends each signed permutation to its corresponding standard permutation induces a surjective algebra homomorphism from the newly defined algebra to the Malvenuto-Reutenauer Hopf algebra.