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By definition,
single means
(1) : consisting of or having only one part, feature, or portion(2) : consisting of one as opposed to or in contrast with many
(3) : consisting of only one in number
by Merriam-Webster
So why do we still say
"a single rose"
"hold to a single ideal"
Not
"single rose"
"hold to single ideal"
Repetition can be used for emphasis.
The indefinite article "a" does imply that there is only one [thing](otherwise you would normally use no article and a plural noun). So maybe you could say "single" is redundant in some sentences. But people do use it for emphasis when they want to make an explicit point that there is only one [thing].
Perhaps the word "single" could be left out in some situations, but the article "a" is a feature of the language, and you can't leave that word out (as you seem to suggest.) If you feel "a single" is repetitive, then leave out the "single". The "a" isn't optional.
Single serves the same role here that any other adjective would do, whether we are talking about a single rose, a solitary rose, a red rose or any other rose.
Introducing an adjective to qualify the rose does not change the need for the article a (or the)
The role of single is to emphasise that there are no other roses, not to replace the article.
It makes the difference between statements such as:
a pupil raised her hand
and
a single pupil raised her hand.
The adjective changes the nuance. The emphasis moves from the raising of the hand to the fact that only a single pupil does so. The same is true for saying a rose stood in the vase and a single rose stood in the vase.
The same rule applies regardless of whether you are talking about a rose, an ideal or any other noun.
We can say things like "six single roses" to mean something like six roses, but none of them grouped together. Since there's nothing redundant about "six single roses", there's nothing redundant about "one single rose" or "a single rose".
It's hard to describe the meaning of "single" without using the word "one". However, that doesn't mean that the adjective "single" is a good determiner. We use some other word to fill the determiner role when that role is appropriate: a single rose, the single rose, this single rose, whichever single rose, and so on.
Just because the meanings of two words overlap, that doesn't mean they are redundant. "single" is adjective that merely indicates singularity. "a" is an article that indicates singularity and indefiniteness, and is a determiner. So "single" provides only one out of three of the roles of "a". You can also say "the single rose", and that would mean something different from "a single rose", so "a" adds meaning.
