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Algebraic logic, varieties of algebras, and algebraic varieties
, 1995
"... Abstract. The aim of the paper is discussion of connections between the three kinds of objects named in the title. In a sense, it is a survey of such connections; however, some new directions are also considered. This relates, especially, to sections 3, 4 and 5, where we consider a field that could ..."
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Cited by 13 (5 self)
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Abstract. The aim of the paper is discussion of connections between the three kinds of objects named in the title. In a sense, it is a survey of such connections; however, some new directions are also considered. This relates, especially, to sections 3, 4 and 5, where we consider a field that could be understood as an universal algebraic geometry. This geometry is parallel to universal algebra. In the monograph [51] algebraic logic was used for building up a model of a database. Later on, the structures arising there turned out to be useful for solving several problems from algebra. This is the position which the present paper is written from.
Characterisation of a Class of Equations With Solutions Over TorsionFree Groups
"... We study equations over torsionfree groups in terms of their "tshape" (the occurences of the variable t in the equation). A tshape is good if any equation with that shape has a solution. It is an outstanding conjecture [5] that all tshapes are good. In [2] we proved the conjecture for a large cl ..."
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Cited by 8 (6 self)
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We study equations over torsionfree groups in terms of their "tshape" (the occurences of the variable t in the equation). A tshape is good if any equation with that shape has a solution. It is an outstanding conjecture [5] that all tshapes are good. In [2] we proved the conjecture for a large class of tshapes called amenable. In [1] Clifford and Goldstein characterised a class of good tshapes using a transformation on tshapes called the Magnus derivative. In this note we introduce an inverse transformation called blowing up. Amenability can be defined using blowing up; moreover the connection with differentiation gives a useful characterisation and implies that the class of amenable tshapes is strictly larger than the class considered by Clifford and Goldstein.
Diagrams and the second homotopy group
 Comm. Anal. Geom
"... Abstract We use Klyachko’s methods [2,3,4,6] to prove that, if a 1–cell and a 2–cell are added to a complex with torsionfree fundamental group, and with the 2–cell attached by an amenable t–shape, then π2 changes by extension of scalars. We also prove that the normal closure of the attaching word c ..."
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Cited by 8 (5 self)
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Abstract We use Klyachko’s methods [2,3,4,6] to prove that, if a 1–cell and a 2–cell are added to a complex with torsionfree fundamental group, and with the 2–cell attached by an amenable t–shape, then π2 changes by extension of scalars. We also prove that the normal closure of the attaching word contains no words of smaller complexity. AMS Classification 57M20, 57Q05; 20E22, 20F05
UDC 512.543.7 HOW TO GENERALIZE THE KNOWN RESULTS ON EQUATIONS OVER GROUP
, 2005
"... The known facts about solvability of equations over groups are considered from a more general point of view. A generalized version of the theorem about solvability of unimodular equations over torsionfree groups is proved. In a special case, this generalized version become a multivariable variant o ..."
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The known facts about solvability of equations over groups are considered from a more general point of view. A generalized version of the theorem about solvability of unimodular equations over torsionfree groups is proved. In a special case, this generalized version become a multivariable variant of this theorem. For unimodular (ordinary and generalized) equations over torsion free groups, we prove an analogue of Magnus’s Freiheitssatz, which asserts that there exists a solution with good behavior with respect to free factors of the initial group. Key words: equations over groups, Kervaire–Laudenbach conjecture, Freiheitssatz. 1.
FREE SUBGROUPS OF ONERELATOR RELATIVE PRESENTATIONS
, 2005
"... ,..., x±1 n}. It is proved that for n � 2 the group ˜ G = 〈G, x1, x2,..., xn  w = 1 〉 always contains a nonabelian free subgroup. For n = 1 the question Suppose that G is a nontrivial torsionfree group and w is a word over the alphabet G ∪ {x ±1 1 about the existence of nonabelian free subgroups i ..."
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,..., x±1 n}. It is proved that for n � 2 the group ˜ G = 〈G, x1, x2,..., xn  w = 1 〉 always contains a nonabelian free subgroup. For n = 1 the question Suppose that G is a nontrivial torsionfree group and w is a word over the alphabet G ∪ {x ±1 1 about the existence of nonabelian free subgroups in ˜ G is answered completely in the unimodular case (i.e., when the exponent sum of x1 in w is one). Some generalisations of these results are discussed. Key words: relative presentations, onerelator groups, free subgroups. MSC: 20F05, 20E06, 20E07. The following theorem was stated in [Ma32]; as far as we know, the proof appeared for the first time in [Mo69]. Free subgroup theorem for onerelator groups. A onerelator group 〈x1, x2,..., xn  w = 1〉 contains no nonabelian free subgroups if and only if it is either cyclic or isomorphic to the Baumslag–Solitar group
HOW TO GENERALIZE THE KNOWN RESULTS ON EQUATIONS OVER GROUP
, 2005
"... The known facts about solvability of equations over groups are considered from a more general point of view. A generalized version of the theorem about solvability of unimodular equations over torsionfree groups is proved. In a special case, this generalized version become a multivariable variant o ..."
Abstract
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The known facts about solvability of equations over groups are considered from a more general point of view. A generalized version of the theorem about solvability of unimodular equations over torsionfree groups is proved. In a special case, this generalized version become a multivariable variant of this theorem. For unimodular (ordinary and generalized) equations over torsion free groups, we prove an analogue of Magnus’s Freiheitssatz, which asserts that there exists a solution with good behavior with respect to free factors of the initial group.
ON ASPHERICAL PRESENTATIONS OF GROUPS
"... (Communicated by Efim Zelmanov) Abstract. The Whitehead asphericity conjecture claims that if 〈A‖R 〉 is an aspherical group presentation, then for every S⊂Rthe subpresentation 〈A‖S 〉 is also aspherical. This conjecture is generalized for presentations of groups with periodic elements by introduction ..."
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(Communicated by Efim Zelmanov) Abstract. The Whitehead asphericity conjecture claims that if 〈A‖R 〉 is an aspherical group presentation, then for every S⊂Rthe subpresentation 〈A‖S 〉 is also aspherical. This conjecture is generalized for presentations of groups with periodic elements by introduction of almost aspherical presentations. It is proven that the generalized Whitehead asphericity conjecture (which claims that every subpresentation of an almost aspherical presentation is also almost aspherical) is equivalent to the original Whitehead conjecture and holds for standard presentations of free Burnside groups of large odd exponent, Tarski monsters and some others. Next, it is proven that if the Whitehead conjecture is false, then there is an aspherical presentation E = 〈A‖R∪z〉 of the trivial group E, where the alphabet A is finite or countably infinite and z ∈ A, such that its subpresentation 〈A‖R 〉 is not aspherical. It is also proven that if the Whitehead conjecture fails for finite presentations (i.e., with finite A and R), then there is a finite aspherical presentation 〈A‖R〉, R = {R1,R2,...,Rn}, such that for every S⊆Rthe subpresentation 〈A‖S〉 is aspherical and the subpresentation 〈A‖R1R2,R3,...,Rn 〉 of aspherical 〈A‖R1R2,R2,R3,...,Rn 〉 is not aspherical. Now suppose a group presentation H = 〈A‖R 〉 is aspherical, x ∈ A, W (A ∪x) is a word in the alphabet (A ∪x) ±1 with nonzero sum of exponents on x, and the group H naturally embeds in G = 〈A∪x‖R∪W(A∪x)〉. It is conjectured that the presentation G = 〈A∪x‖R∪W(A∪x)〉is aspherical if and only if G is torsion free. It is proven that if this conjecture is false and G = 〈A∪x‖R∪W(A∪x)〉is a counterexample, then the integral group ring Z(G) of the torsion free group G will contain zero divisors. Some special cases where this conjecture holds are also indicated.
Notation
, 2008
"... c ○ P. J. Davidson 2008I should have written about pigeons. Everyone is interested in pigeons. ..."
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c ○ P. J. Davidson 2008I should have written about pigeons. Everyone is interested in pigeons.
THE APPLICATION OF PICTURES TO DECISION PROBLEMS AND RELATIVE PRESENTATIONS
, 1995
"... problems and relative presentations. ..."