1. This paper deals with orthodox semirings whose additive idempotents satisfy permutation identities.

2. Unlike the quaternion algebra, the split-quaternions contain nontrivial zero divisors, nilpotent elements, and idempotents.

3. The matrices that have just 1 over one element in the main diagonal and 0 otherwise are the orthogonal idempotents.

4. 21 A left seminormal orthodox semigroup is an orthodox semigroup whose idempotents form a left seminormal band. We deal with natures of a left seminormal orthodox semigroup with an inverse transversal.