Elementary classification and decidability of theories of derived structures.

*(English. Russian original)*Zbl 1105.03016
Russ. Math. Surv. 60, No. 3, 395-432 (2005); translation from Usp. Mat. Nauk 60, No. 3, 3-40 (2005).

By the derived structure of an algebraic system (a.s.) \(A\) the authors mean an a.s. \(D(A)\) uniquely determined by \(A\) (and containing some information about the structure of \(A\)). Automorphism groups Aut(\(A\)) and endomorphism monoids End(\(A\)) are examples of such structures.

The paper is a broad and well organized survey of results related to elementary theories of derived structures of universal algebras. These results are connected first of all with the following two questions:

1. Find necessary and sufficient conditions on a.s. \(A\) and \(B\) of a given class under which \(D(A)\equiv D(B)\) holds for their derived structures of type \(D\).

2. Find necessary and sufficient conditions on a.s. \(A\) under which the elementary theory of \(D(A)\) is decidable.

From the paper: “The…death of Yu. M. Vazhenin in 2003 interrupted this work for some time. The first author considers it his duty to complete the programme and decidate the resulting research to the memory of Yu. M.”

The paper is a broad and well organized survey of results related to elementary theories of derived structures of universal algebras. These results are connected first of all with the following two questions:

1. Find necessary and sufficient conditions on a.s. \(A\) and \(B\) of a given class under which \(D(A)\equiv D(B)\) holds for their derived structures of type \(D\).

2. Find necessary and sufficient conditions on a.s. \(A\) under which the elementary theory of \(D(A)\) is decidable.

From the paper: “The…death of Yu. M. Vazhenin in 2003 interrupted this work for some time. The first author considers it his duty to complete the programme and decidate the resulting research to the memory of Yu. M.”

Reviewer: S. R. Kogalovskij (Ivanovo)

##### MSC:

03B25 | Decidability of theories and sets of sentences |

03C05 | Equational classes, universal algebra in model theory |

03C85 | Second- and higher-order model theory |

03C07 | Basic properties of first-order languages and structures |

08A35 | Automorphisms and endomorphisms of algebraic structures |

08B20 | Free algebras |

20E15 | Chains and lattices of subgroups, subnormal subgroups |

06A12 | Semilattices |