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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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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 ..."
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Cited by 9 (7 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.
The origins of combinatorics on words
, 2007
"... We investigate the historical roots of the field of combinatorics on words. They comprise applications and interpretations in algebra, geometry and combinatorial enumeration. These considerations gave rise to early results such as those of Axel Thue at the beginning of the 20th century. Other early ..."
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We investigate the historical roots of the field of combinatorics on words. They comprise applications and interpretations in algebra, geometry and combinatorial enumeration. These considerations gave rise to early results such as those of Axel Thue at the beginning of the 20th century. Other early results were obtained as a byproduct of investigations on various combinatorial objects. For example, paths in graphs are encoded by words in a natural way, and conversely, the Cayley graph of a group or a semigroup encodes words by paths. We give in this text an account of this twosided interaction.
The adjunction problem and a theorem of Serre
, 2005
"... Abstract In this note we prove injectivity and relative asphericity for “layered ” systems of equations over torsionfree groups, when the exponent matrix is invertible over Z. We also give elementary geometric proofs of results due to Bogley–Pride and Serre that are used in the proof of the main th ..."
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Abstract In this note we prove injectivity and relative asphericity for “layered ” systems of equations over torsionfree groups, when the exponent matrix is invertible over Z. We also give elementary geometric proofs of results due to Bogley–Pride and Serre that are used in the proof of the main theorem.
Geometry & Topology Monographs Volume 1: The Epstein birthday schrift Pages 159–166 Characterisation of a class of equations with solutions over torsionfree groups
"... Abstract We study equations over torsionfree groups in terms of their “t–shape ” (the occurences of the variable t in the equation). A t–shape is good if any equation with that shape has a solution. It is an outstanding conjecture [5] that all t–shapes are good. In [2] we proved the conjecture for ..."
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Abstract We study equations over torsionfree groups in terms of their “t–shape ” (the occurences of the variable t in the equation). A t–shape is good if any equation with that shape has a solution. It is an outstanding conjecture [5] that all t–shapes are good. In [2] we proved the conjecture for a large class of t–shapes called amenable. In [1] Clifford and Goldstein characterised a class of good t–shapes using a transformation on t–shapes 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 t–shapes is strictly larger than the class considered by Clifford and Goldstein. AMS Classification 20E34, 20E22; 20E06, 20F05
Solvability of Some Equations Over Groups
"... This project attempted to set bounds on the solvability of the equation 11122 −− − tbtattba = e, an equation that has not yet been fully classified. Attempts to explore the solvability of the equation were made by researching the orders of a and b in groups of the form G(m,n) = babababa nm 122,,, ..."
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This project attempted to set bounds on the solvability of the equation 11122 −− − tbtattba = e, an equation that has not yet been fully classified. Attempts to explore the solvability of the equation were made by researching the orders of a and b in groups of the form G(m,n) = babababa nm 122,,, −. Where solvability of the equation could not be proven, we attempted to find conditions on the order of the group, G(m,n), that would be necessary and sufficient to show the solvability of the equation.
Some odd flnitely presented groups
"... Once upon a time we set out to construct some truly exotic flnitely presented groups. As it turned out we didn’t succeed in flnding the groups we wanted, but we nevertheless managed to concoct some groups with rather interesting properties. The purpose of this note is to record these examples and co ..."
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Once upon a time we set out to construct some truly exotic flnitely presented groups. As it turned out we didn’t succeed in flnding the groups we wanted, but we nevertheless managed to concoct some groups with rather interesting properties. The purpose of this note is to record these examples and constructions.