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This is how I would propose to lay out material on a subclass of semirings, in this case complete star semirings.  Firstly, define complete semirings and star semirings, each with a set of examples  Finally define complete star semirings, again with a set of examples.   In this case continuous semirings are closely related to complete semirings, so it makes sense to include them: an additional motivation here is that this emphasises the relationship between partial orders defined on the semirings and the star structure: for example, Conway uses the order approach to prove Kleene's theorem in his 1971 book.  Of course there are choices here.  For example, should the examples of complete star semirings be introduced early on, or only after the last definition?  There's no right answer to that.  I would like to see examples of a concept which are not examples of the richer structure, folloewd by examples of the richer structure.  Examples of continuous semirings would have been helpful, but as some editors have said, there is no deadline.  In my view, a definition without an example is hugely preferable to no definition at all: the reader in search of information will still be able to see what sort of thing it is and find references to further resources on the subject.  That, it seems to me, is what an encyclopaedia is for.
The current state of the material seems to me to be intended to do one thing only, which is to introduce complete star semirings in a way which makes the star operation as close as possible to the traditional Kleene star.  This is not the only approach.  A reason for the axiomatisation and separation out of the various properties is the entirely reasonable mathematical activity of finding out which of the properties of the Kleene star derive from what assumptions.  On the way, various structures of interest emerge, and various theorems are proved about the relationships between them.  That may not suit the point of view that star semirings are ""really"" or ""completely unsurprisingly"" the original formal languages: but that's only a point of view; it isn't correct, or at least, isn't the whole truth; and it isn't helpful to the reader of the encyclopaedia.      "