BCK-monoids

We generalize the notions of a pseudo BCK-algebra and a residuated lattice by introducing, respectively, extended BCK-algebras and BCK-monoids, and prove a decomposition theorem for BCK-monoids, generalising a decomposition theorem of Galatos and Tsinakis for GBL-algebras. Also, we specialise this theorem to hoop monoids and Wajsberg monoids, which generalize, respectively, GBL-algebras and GMV-algebras. Finally, we include a discussion on Iséki monoids, which extend the concept of left pseudo Iséki algebras of Iorgulescu.