## Definition. [Pl1] Groups G1, G2 ∈ Θ are called X-equivalent if T ′′ (2004)

### BibTeX

@MISC{04definition.[pl1],

author = {},

title = {Definition. [Pl1] Groups G1, G2 ∈ Θ are called X-equivalent if T ′′},

year = {2004}

}

### OpenURL

### Abstract

Assume that Θ is an arbitrary variety of groups. Let W(X) be a free group of the variety Θ over the finite set X and G is a group in this variety (G ∈ Θ). We can consider the ”affine space over the group G”: HomΘ(W(X), G). For every set T ⊂ W(X) we can consider the ”algebraic variety”

### Citations

16 |
Varieties of algebras and algebraic varieties. Categories of algebraic varieties
- Plotkin
- 1997
(Show Context)
Citation Context ...on ”∼” is an equivalence. It was proved that groups G1, G2 ∈ Θ are geometrically equivalent if and only if every finitely generated subgroup of G1 can be approximated by subgroup of G2 and vice versa =-=[Pl2]-=-. We denote G ≺ H if the group G can be approximated by the group H, i.e., there is {ϕi | i ∈ I} ⊂ Hom(G, H) such that ⋂ kerϕi = {1}. It’s clear that the relation ”≺” is the order. If we consider only... |

13 | Algebraic logic, varieties of algebras and algebraic varieties
- Plotkin
(Show Context)
Citation Context ... group G”: HomΘ(W(X), G). For every set T ⊂ W(X) we can consider the ”algebraic variety” and G-closure of T : A(T)G = T ′ G = {µ ∈ HomΘ(W(X), G) | T ⊂ kerµ} T ′′ ⋂ G = µ∈T′ kerµ ⊂ W(X). G Definition. =-=[Pl1]-=- Groups G1, G2 ∈ Θ are called X-equivalent if T ′′ G1 = T ′′ G2 holds for every set T ⊂ W(X). Groups G1, G2 ∈ Θ are called geometrically equivalent (denoted G1 ∼ G2 ) if they are X-equivalent for ever... |

9 |
Geometrical equivalence of algebras
- Berzins
(Show Context)
Citation Context ...ly equivalent groups in the variety of nilpotent groups of some fixed class s, or, in other words, to describe all quasivarietes generated by single group of this variety. It was proved by A. Berzins =-=[Be]-=-, that two Abelian groups are geometrically equivalent if and only if for every prime number p the exponents of their corresponding p-Sylow subgroups coincide, and if one of these group is not periodi... |

9 |
Tsurkov, Geometrical equivalence of groups
- Plotkin, Plotkin, et al.
- 1999
(Show Context)
Citation Context ...oups G and H are geometrically equivalent , iff G ≺ H and H ≺ G. Also, it was proved that if two groups are geometrically equivalent, they have the same identities [Pl1] and the same quasi-identities =-=[PPT]-=-. Definitions. A variety Θ of groups is called Noetherian if the Noether chain condition holds for the normal subgroups of every finitely generated free group W(X) of this variety. [Pl3] A group G in ... |

1 |
On the residual nilpotence of some varietal product
- Baumslag
- 1963
(Show Context)
Citation Context ...n ∈ √ ( G, such that x 1 ) n n = x. The √ G is the nilpotent group of the same class as G. The element x 1 n ∈ √ G is uniquely defined by x ∈ √ G and n ∈ N. Groups G and √ G have the same identities (=-=[Bau]-=-). In the [PPT] the question was asked when is the nilpotent torsion free group G geometrically equivalent to its Mal’tsev completion √ G and, so, groups G and √ G have the same quasiidentities. Theor... |

1 |
About geometrical equivalence of groups, Theses of report
- Bludov, Gusev
(Show Context)
Citation Context ...s and Mal’tsev coordinats. Corollary. A nilpotent class 2 torsion free group and its Mal’tsev completion have same the quasiidentities. This result was achieved independently also by Bludov and Gusev =-=[BG]-=-. In these theses there is the example of nilpotent class 3 torsion free group (with 4 generators) which is not geometrically equivalent to its Mal’tsev completion. So this group and its Mal’tsev comp... |

1 |
A lattice of quasivarieties of nilpotent groups
- Budkin
- 1994
(Show Context)
Citation Context ... they have the same quasiidentities. The topic of quasiidentities and quasivarieties of nilpotent groups (in most cases of nilpotent class 2 groups) was researched in many papers: [Is], [Fd1], [Fd2], =-=[Bu]-=-, [Sh]. i∈I 1Now we can resolve some questions in this topic by comparison of finitely generated groups by the relation ≺. In particular, there is one-to-one correspondence between classes of geometr... |

1 |
O podkvzimnogoobrazijah nil’potentnyh minimal’nyh neabelevyh mnogoobrazij grupp
- Fedorov
- 1980
(Show Context)
Citation Context ...if and only if they have the same quasiidentities. The topic of quasiidentities and quasivarieties of nilpotent groups (in most cases of nilpotent class 2 groups) was researched in many papers: [Is], =-=[Fd1]-=-, [Fd2], [Bu], [Sh]. i∈I 1Now we can resolve some questions in this topic by comparison of finitely generated groups by the relation ≺. In particular, there is one-to-one correspondence between class... |

1 |
Kvazitozhdestva svobodnoj 2-nilpotentnoj gruppy
- Fedorov
- 1986
(Show Context)
Citation Context ...if and only if they have the same quasiidentities. The topic of quasiidentities and quasivarieties of nilpotent groups (in most cases of nilpotent class 2 groups) was researched in many papers: [Is], =-=[Fd1]-=-, [Fd2], [Bu], [Sh]. i∈I 1Now we can resolve some questions in this topic by comparison of finitely generated groups by the relation ≺. In particular, there is one-to-one correspondence between class... |

1 | A note on finitely generated torsion free nilpotent groups of class 2 - Grunewald, Scharlau - 1979 |

1 |
O kvzimnogoobrazijah 2-stepenno nilpotentnyh grup, Izv. Akademii Nauk UzSSR
- Isakov
- 1976
(Show Context)
Citation Context ...alent if and only if they have the same quasiidentities. The topic of quasiidentities and quasivarieties of nilpotent groups (in most cases of nilpotent class 2 groups) was researched in many papers: =-=[Is]-=-, [Fd1], [Fd2], [Bu], [Sh]. i∈I 1Now we can resolve some questions in this topic by comparison of finitely generated groups by the relation ≺. In particular, there is one-to-one correspondence betwee... |

1 |
Merzljakov Ju.I. Fundamentals of the Theory of Groups
- Kargaplov
- 1979
(Show Context)
Citation Context ...n). To prove this Lemma we use the known theorem that every nilpotent group G can be generated by some subset M ⊂ G and the commutant γ2 (G) (G = 〈M, γ2 (G)〉) can be generated by set M (G = 〈M〉) (see =-=[KM]-=-, 16.2.5). Proof. Denote L the Lie Q-algebra, such that L◦ ∼ = √ G. We can identify the elements of L, L◦ and √ G. If a, b ∈ √ G = L, k ∈ N , then, by the CampbellHausdorff formula, a + b ≡ a · bmod [... |

1 | On the lattice of quasivarieties of nilpotent groups of class 2 - Shakhova - 1997 |