Repetitions and permutations of columns in the semijoin algebra

Codd defined the relational algebra [E.F. Codd, Communications of the ACM 13 (1970) 377–387; E.F. Codd, Relational completeness of data base sublanguages, in Data Base Systems, R. Rustin, Ed., Prentice-Hall (1972) 65–98] as the algebra with operations projection, join, restriction, union and difference. His projection operator can drop, permute and repeat columns of a relation. This permuting and repeating of columns does not really add expressive power to the relational algebra. Indeed, using the join operation, one can rewrite any relational algebra expression into an equivalent expression where no projection operator permutes or repeats columns. The fragment of the relational algebra known as the semijoin algebra, however, lacks a full join operation. Nevertheless, we show that any semijoin algebra expression can still be simulated in a natural way by a set of expressions where no projection operator permutes or repeats columns