The pseudo-hoops were originally introduced by Bosbach under the name of complementary semigroups. It was proved that a pseudo-hoop has the pseudo-divisibility condition and it is a meet-semilattice, so a bounded Rℓ-monoid can be viewed as a bounded pseudohoop together with the join-semilattice property. In other words, a bounded pseudohoop is a meet-semilattice ordered residuated, integral and divisible monoid.
In this chapter we present the main notions and results regarding the pseudo-hoops, we prove new properties of these structures, we introduce the notions of join-center and cancellative-center of pseudo-hoops and we define and study algebras on subintervals of pseudo-hoops.
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