## Saturday, May 31, 2014

### Monogenic semigroups

The monogenic semigroups are semigroups that are generated by a single element. All semigroups that are generated by at least two elements must contain distinct maximal proper subsemigroups corresponding to those elements. This means that all semigroups that contain only a single maximal proper subsemigroup are necessarily monogenic. Nonetheless, not all monogenic semigroups contain only one maximal proper subsemigroup.

The monogenic semigroups that contain only one maximal proper subsemigroup are called primary monogenic semigroups. Non-primary monogenic semigroups include cyclic groups that are of non-prime power orders. The prime power cyclic groups are called primary cyclic groups and hence they are primary monogenic semigroups. The primary monogenic semigroups can be used to generate the elements of any semigroup.

## Friday, May 30, 2014

### Perfect graphs

Amongst the set of all graphs the perfect graphs are particularly interesting. Perfect graphs subsume both comparability graphs and co-comparability graphs so they can describe both the comparability and the incomparability conditions of partial ordering relations. As the the perfect graphs exclude odd holes and odd anti-holes they are far easier to describe using partial orders themselves.

The trivially perfect graphs and the co trivially perfect graphs are perfect graphs that are of particular interest to us. They can both be described entirely using preorders produced by the immediate reachability conditions on the graphs. Equivalence graphs are trivially perfect and their complements the complete partite graphs are co trivially perfect.

The threshold graphs are both trivially perfect and co trivially perfect. The threshold equivalence graphs form a Moore family of graphs with their own closure operation defined by split completion. Both empty graphs and complete graphs belong to the class of threshold equivalence graphs.