Three theorems on linear ordered periodic semigroups

This paper is devoted to some classes of linearly ordered periodic semigroups ($op$-semigroups). Theorem 1 yields an abstract characteristic of $op$-semigroups with different contractibility and reversibility. Theorem 2 describes completely pre-ordered nilsemigroups. The Dubreil result that an $op$-semigroup with commutative subsemigroup of idempotents $E$ is residual of their convex nilsubsemigroups is extended in the last theorem, to the case of some non-commutative $E$.