completeness of the relational lattice

There seems to be some disagreement on what algebra we are studying. So let's discuss this first. It is important this get's fixed, because otherwise any time I spent on proving things might be lost. I think it's also important that we settle on notation, because I noticed I was getting confused. Since it's bascially Vadim's idea I'll try to stick to his notation as much as possible.

So the syntax of the algebra is as follows: (E is the non-terminal for algebra expressions)
- R : a relation name

• E /\ E : the natural join
• E \/ E : the inner union
• 00 : the empty relation with empty header
• 01 : the relation with the empty tuple and empty header
• 10 : the empty relation with the set of all attributes as header
• 11 : the relation with all tuples over all attributes
• [x] : the empty relation over header {x} with x a single attribute
• 'x=y' : the binary equality relation with header {x,y} and the restriction that x<>y

I don't like having 10 and 11 because it clearly steps outside the relational model and first order logic, but I see no direct reasons to remove them. There was some doubt on the presence of [x] but I don't see how we can otherwise define projection. Also I don't see how we can avoid mentioning attributes since we already have 'x=y' where the are already mentioned.

So can we agree on this algebra and notation?

• Jan Hidders