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It is shown that Tarski's set of ten axioms for the calculus of relations isindependent in the sense that no axiom can be derived from the remainingaxioms. It is also shown that by modifying one of Tarski's axioms slightly, andin fact by replacing the right-hand distributive law for relativemultiplication with its left-hand version, we arrive at an equivalent set ofaxioms which is redundant in the sense that one of the axioms, namely thesecond involution law, is derivable from the other axioms. The set of remainingaxioms is independent. Finally, it is shown that if both the left-hand andright-hand distributive laws for relative multiplication are included in theset of axioms, then two of Tarski's other axioms become redundant, namely thesecond involution law and the distributive law for converse. The set ofremaining axioms is independent and equivalent to Tarski's axiom system.

On Tarski's axiomatic foundations of the calculus of relations