What is a pre-order that is not reflexive nor irreflexive? so just transitive and total. What is this relation called?
I am aware that total pre-order is transitive, total, and reflexive.
But I am wondering is there a variation of it that not reflexive nor irreflexive?
I would call your construction simply a total transitive order. Transitivity is the absolute minimum requirement for a binary relation to qualify as an order.
Alternatively, it could be called a thin, connected semicategory. The thin part accounts for the fact that your semicategory has only one morphism between objects, and the connected part accounts for the totality of your order.
Does this answer your question, or even make sense? If not, happy to edit it to improve.