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Completeness criteria and invariants for operation and transformation algebras. (English) Zbl 1078.08001
In the paper operation algebras are investigated, i.e. algebras whose elements are operations (of fixed arity \(k\)) on a set \(A\) and whose fundamental operations are induced by an algebra (from some fixed quasivariety \(\mathcal K\)) on the base set \(A\) and also include composition; for unary mappings (\(k=1\)) such algebras are called transformation algebras. One of the motivations to study operation algebras comes from the observation that such algebras are concrete cases of the so-called composition algebras introduced by H. Lausch and W. Nöbauer [Algebra of polynomials. Amsterdam: North-Holland Publishing Company (1973; Zbl 0283.12101)] – the most general approach known to the authors generalizing the classical Cayley theorem for groups. Operation algebras are described and characterized via invariant relations; they are the Galois-closed elements with respect to a suitable Galois connection. Moreover, the completeness problem in operation algebras is considered and solved for concrete cases (e.g. for transformation \((\max ,\circ )\)-semirings).
MSC:
08A40 Operations and polynomials in algebraic structures, primal algebras
16Y60 Semirings
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