Algebra Relational sql GROUP BY SORT BY ORDER BY

Neither is possible in relational algebra but people have been creating some “extensions” for these operations (Note: in the original text, part of the text is written as subscript).

GROUP BY, According to the book Fundamentals of Database Systems (Elmasri, Navathe 2011 6th ed):

Another type of request that cannot be expressed in the basic relational algebra is to specify mathematical aggregate functions on collections of values from the database.
…
We can define an AGGREGATE FUNCTION operation, using the symbol ℑ (pronounced script F)7, to specify these types of requests as follows:

<grouping attributes> ℑ <function list> (R)

where <grouping attributes> is a list of attributes of the relation specified in R, and <function list> is a list of (<function> <attribute>) pairs. In each such pair, <function> is one of the allowed functions—such as SUM, AVERAGE, MAXIMUM, MINIMUM,COUNT—and <attribute> is an attribute of the relation specified by R. The resulting relation has the grouping attributes plus one attribute for each element in the function list.

ORDER BY (SORT BY), another source:

Since a relation is a set (or a bag), there is no ordering defined for a relation. That is, two relations are the same if they contain the same tuples, irrespective of ordering. However, a user frequently wants the output of a query to be listed in some particular order. We can define an additional operator τ which sorts a relation if we are willing to allow an operator whose output is not a relation, but an ordered list of tuples.

For example, the expression

τLastName,FirstName(Student)

generates a list of all the Student tuples, ordered by LastName (as the primary sort key) then FirstName (as a secondary sort key). (The secondary sort key is used only if two tuples agree on the primary sort key. A sorting operation can list any number of sort keys, from most significant to least significant.)

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