Different types of orders: Partial Order, Weak order, Strict partial Order, Weak partial order, Strict weak order, and Equivalence orders

Question

As I understand it, transitivity is compulsory to all types of orders.

In addition when the relation is also irreflexive (and consequently anti-symmetric), then it is called weak partial order. On the other hand, if the relation is reflexive, there can be two situations.

1. Either it's symmetric, in which case it is preorder (or equivalence relation).
2. Or it's anti-symmetric, in which case it is strong partial order.

I have three questions:

a. Is my above understanding correct?

b. Are partial order, weak order other terms for weak partial order?

c. Are strict partial order, strict weak order other terms for strong partial order?

EDIT: Citing different sources from where confusion arises.

• Partee, B. H., Meulen, A. G. B. ter, & Wall, R. E. (1990). Mathematical methods in linguistics. Kluwer Academic.

  • Pg. 47, "An order is a binary relation which is transitive and in addition either (i) reflexive and antisymmetric or else (ii) irreflexive and asymmetric. The former are weak orders; the latter are strict (or strong)".

  • Pg. 208-209, authors state that a relation is a weak partial order iff it's transitive, reflexive and antisymmetric; it's strict partial iff transitive, irreflexive and asymmetric.

• Narens, L. (2015). Probabilistic Lattices: With Applications to Psychology (Vol. 5). World Scientific.

  • On Pg. 29 author states conditions for different orders.
    • partial ordering: reflexive, transitive, and antisymmetric
    • weak ordering: transitive and connected
    • total ordering: weak ordered and antisymmetric.
  • He also states that weak orderings are also partial orderings, so each concept in the list is a generalization of the next.

• Lehman, Eric, Leighton, F Thomson, & Meyer, Albert R (2018). Mathematics for Computer Science.

  • Pg. 448, "R is a strict partial order iff R is transitive and irreflexive iff R is transitive and asymmetric." "R is a weak partial order iff R is transitive and anti-symmetric and reflexive."