Investigation of the structure of clones and related algebras of finitary functions, selected problems in the theory of varieties.
An excellent survey of principal results obtained by the present author and by other researchers in the theory of functional algebras and in the theory of equational classes of finitary algebras.NEWLINENEWLINEThe chapter titles are as follows: 1. Basic concepts, 2. Important results of other authors, 3. \(\underleftarrow\sigma\)-completeness in \(P_ k\) and properties of hypo-closedness operators, 4. \(\underrightarrow\sigma\)-completeness in \(P_ k\) and \(\sigma\)-envelopes of clones, 5. Three variants of \(\sigma\)-deformation stability for simple and semisimple bases of \(P_ 2\), 6. On \(\sigma\)-deformation stability for simple and semisimple bases in \(P_ k\;(k\geq3)\), comparison with the \(P_ 2\) case; sufficient conditions of \(\underarrow\sigma\)-stable simplicity for simple bases of the clones \(F_ k\), 7. \(\underrightarrow{\tilde G}\)-stable and \(\underrightarrow G\)-stable simple bases in \(P_ 2\), \(\underarrow L\)-stable simple bases in \(P_ k\) \((k=2,3)\), lower and upper base \(k\)-types, 8. Some special independent sets of finitary functions, 9. Congruences on clones and iterative algebras, 10. The problem of Salomaa on fundamental subgroups of symmetric groups, 11. Other results, 12. Simple criteria of functional completeness in finite algebras in varieties, 13. Arity ranks of independent sets of Rosenberg relations, 14. On independent bases of varieties having modular filter of over-varieties, 15. On varieties of almost distributive lattices: towards the problems of Tarski, McKenzie and Jonsson, various ways of forming varieties, 16. Non-standard subdirect representations in axiomatic classes and reduct-multiplicative decompositions in classes of varieties, 17. Termal and polynomial ranks of varieties, their induced closedness and pseudo-closedness, 18. Miscellaneous. 442 references are listed.
scientific article; zbMATH DE number 1648162