Stebletsova, V.; Venema, Y. \(Q\)-algebras. (English) Zbl 0933.03084 Algebra Univers. 40, No. 1, 19-49 (1998). Let \(R\) be a square matrix of type \(n\times n\) whose elements \(R_{ij}\) are binary relations. Then \(Q^{kl}_n(R)\) is the binary relation defined by \(\langle s,t\rangle \in Q^{kl}_n(R)\) iff there are \(u_0,\ldots ,u_{n-1}\) such that \(s=u_k\), \(t=u_1\) and \(\langle u_i,u_j\rangle \in R_{ij}\). Then for a given set \(U\), the full relation set \(Q\)-algebra over \(U\) is defined as the relation algebra of all binary relations on \(U\) equipped with all \(Q^{kl}_n\). The class of all these algebras is denoted by \(FQ\). \(Q\)-type algebras are algebras \(A=(A;\cdot ,+,-,0,1,',Q^{kl}_n)\). A representation of \(A\) is an embedding of \(A\) into a product of algebras from \(FQ\). The class of all representable \(Q\)-type algebras is denoted by \(RQ\). A \(Q\)-algebra is a \(Q\)-type algebra satisfying 10 axiom schemes introduced in the paper. Theorem 1. \(RQ\) is a discriminator variety of Boolean algebras with operators. Theorem 2. A \(Q\)-type algebra is representable iff it is a \(Q\)-algebra. Theorem 3. For \(n>3\), an algebra \(A\) is a relation algebra of dimension \(n\) iff it can be embedded in the Tarski reduct of a \(Q_n\)-algebra. Reviewer: I.Chajda (Olomouc) MSC: 03G15 Cylindric and polyadic algebras; relation algebras 08A02 Relational systems, laws of composition 08B99 Varieties 06E25 Boolean algebras with additional operations (diagonalizable algebras, etc.) Keywords:relation algebra; representable relation algebra; Boolean algebra with operators; \(Q\)-type algebras; discriminator variety; Tarski reduct PDFBibTeX XMLCite \textit{V. Stebletsova} and \textit{Y. Venema}, Algebra Univers. 40, No. 1, 19--49 (1998; Zbl 0933.03084) Full Text: DOI