Considering only basic relational algebra operations, how can we implement the full outer-join ?
Full outer join returns null in the columns of rows from one table that don't match the other table -- that's a description of SQL.
Standard Relational Algebra doesn't have nulls. Here explains what would be the consequences.
Short answer: 'we' don't want to implement outer joins. If you think you do, think some more until you see why it leads to incoherence of queries.