Classes of divisibility commutative semigroups

Divisibility commutative semigroups are precisely the semigroups that have their L and R relations coincide. That is, they are the semigroups which act like commutative-semigroups with respect to divisibility, and they include all semigroups which are factorisation partially ordered. The consideration of these semigroups recently, as well as their specializations led to us to consider a number of subclasses of the class of divisibility commutative semigroups. An ontology of them is displayed below.

Clifford semigroups are particularly interesting because they are the most natural generalization of groups within the class of semigroups. In the theory of symmetric inverse semigroups, the Clifford semigroups are constructed entirely from charts that consist of only a permutation part and no nilpotent part. Clifford semigroups include both groups and semilattices as described above. We will now transition from our consideration of divisibility commutative semigroups to other generalisations of commutativity.