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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.
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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Cited by 1 (0 self)
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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.