2 problems

Let $S$ be the finite semigroup under consideration, let $u,u'\in S$, let $u^{{+}_r}$ and ${u'}^{{+}_l}$ denote their associated right and left idempotents, let $\boldsymbol{*}$ de…

Let $\mathcal{PODI}_n$ be the monoid of partial order-preserving or order-reversing permutations, let $\mathcal{POI}_n$ be the monoid of partial order-preserving permutations, and…