Words like 'the', 'a', and 'of' are often called syncategorematic words, words "that do not stand by themselves... (i.e. prepositions, logical connectives, etc.)" (here).
Examples of syncategorematic terms include:
- articles (for example, 'the' and 'a')
- connectives (for example, 'and' and 'or')
- prepositions (for exmaple, 'in' and 'at')
- quantifiers (for example, 'some' and 'all')
These contrast with categorematic words, "words that designate self-sufficient entities (i.e. nouns or adjectives)" (here).
Merriam-Webster defines categorematic as "capable of standing alone as the subject or predicate of a logical proposition : expressing a complete substantive meaning" (here).
Categorematic words include names (for example, 'John') and predicates (for example, 'tiger' and 'smokes').
You're correct in pointing out that syncategorematic words seem constant in some sense. That's why logicians and semanticists studying them call them logical constants. Their meanings do not vary from interpretation to interpretation (unlike names and predicates) (see here).
Bertrand Russell (1872-1970) was one of the first scholars to think about the meaning of the English definite article 'the'. He said that 'the' works like an existential quantifier, but with uniqueness. On his view, a sentence like 'the F is G' means:
- There is a unique F and it's G.
If you think about a sentence like 'The president of the US smokes', you can paraphrase it as 'There is a unique president of the US and he smokes'. Russell thought that this was the meaning of 'the': to express unique existence.
Similarly, he thought that the indefinite article 'a/an' had the same meaning as an existential quantifier. Thus, a sentence like 'A dog barked' means 'There is a dog and it barked'. The meaning of 'a/an' is to express existence.
Both of Russell's claims are quite controversial among logicians and semanticists concerned with natural language. They have spawned a century of literature. But that should get you thinking.
By the way, there is a semantic difference between your two "sentences":
- The play starts at 10:00pm at Rosedale
- Play starts 10:00pm Rosedale
As a matter of semantics, only well-formed or grammatical sentences express something. The second sentence is not well-formed and thus has no semantic content. Somebody who heard this "sentence" would be able to pragmatically interpret it, but this would likely involve their restoring the omitted 'the' and 'at', i.e. converting it to the first sentence. If you're wondering why English makes use of articles and prepositions when other languages get by without them, well, that's a different question entirely.