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Perhaps things will be clearer if you begin by thinking of the full data pipeline involved, and recognize when an expression is just an abbreviated form.

In constructing an unabbreviated pipeline, the basic principles regarding "dots" are quite simple:

• . refers to the input
• .foo is for accessing the value of the key "foo"
• .[] is for expanding an array or object

There are many allowed abbreviations. The two that seem to have confused you the most are:

• .foo.bar for .foo | .bar
• {foo} for {"foo": .foo}

Another important abbreviation is:

• E[] for E | .[] where E is a suitably compact expression

With these principles and examples in mind, you should be able to master the details as explained in the jq documentation.