A method is presented for proving primality and functional completeness theorems, which makes use of the operation-relation duality. By the result of Sierpiński, we have to investigate relations generated by the two-element subsets of $A^{k}$ only. We show how the method applies for proving Słupecki's classical theorem by generating diagonal relations from each pair of k-tuples.
A diagrammatic statement is developed for the generalized semidistributive law in case of single algebras assuming that their congruences are permutable. Without permutable congruences, a diagrammatic statement is developed for the ∧-semidistributive law.
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