What is the standardization if any when it comes to ordered sets?.
Specifically I'm always confused in the following cases:
1) When someone say "a partial ordered set": to me it can mean a strict partial ordered set (The relation is asymmetric and transitive) or a non strict partial ordered set (the relation is antisymmetric, reflexive and transitive), but many books refer to partial ordered sets like they were non strict and excluding the other case, why?.
2) When someone say "a total ordering": to me it can mean a strict total ordered set or it can mean a non strict total ordered set, but many books refer to total orders like they were non strict and excluding the other case.
3) When someone say "an ordered set" : to me it can mean any ordered set, partial, complete, strict, or non strict, but many books don't make this distintion and they refer to a "non strict partial orderd".
Am I missing something? Is it just me the one making these distintions?, please help me clarify this.