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\(Q\)-algebras. (English) Zbl 0933.03084

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.)
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