Is it true that for all integer n > 1 and k ≤ n there exists a prime number in the interval [kn, (k + 1)n]? The case k = 1 is the Bertrand's postulate which was proved for the first time by P. L. Chebyshev in 1850, and simplified later by P. Erdős in 1932, see [2]. The present paper deals with the case k = 2. A positive answer to the problem for any k ≤ n implies a positive answer to the old problem whether there is always a prime in the interval [n 2 , n 2 + n], see [1, p. 11].
Square matrices over a relation algebra are relation algebras in a natural way. We show that for fixed n, these algebras can be characterized as reducts of some richer kind of algebra. Hence for fixed n, the class of n × n matrix relation algebras has a first-order characterization. As a consequence, homomorphic images and proper extensions of matrix relation algebras are isomorphic to matrix relation algebras.
We show that a relation algebra and its n-matrix relation algebra have the same degree for all positive integers n. An intermediate result relates the degree of a relation algebra to the degree of a relativization with respect to equivalence elements.
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