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The word problem for an algebra is then to determine, given two expressions (words) involving the generators and operations, whether they represent the same element of the algebra modulo the identities. The word problems for groups and semigroups can be phrased as word problems for algebras. [1]
Then the word problem in is solvable: given two words , in the generators of , write them as words in and compare them using the solution to the word problem in . It is easy to think that this demonstrates a uniform solution of the word problem for the class K {\displaystyle K} (say) of finitely generated groups that can be embedded in G ...
This operation is known as reduction, and it does not change the group element represented by the word. Reductions can be thought of as relations (defined below) that follow from the group axioms. A reduced word is a word that contains no redundant pairs. Any word can be simplified to a reduced word by performing a sequence of reductions:
Word problem from the Līlāvatī (12th century), with its English translation and solution. In science education, a word problem is a mathematical exercise (such as in a textbook, worksheet, or exam) where significant background information on the problem is presented in ordinary language rather than in mathematical notation.
G is a word-hyperbolic group. [22] G has decidable conjugacy problem. [19] G is coherent, that is every finitely generated subgroup of G is finitely presentable. [23] The isomorphism problem is decidable for finitely generated one-relator groups with torsion, by virtue of their hyperbolicity. [24] G is residually finite. [25]
Word problem may refer to: Word problem (mathematics education) , a type of textbook exercise or exam question to have students apply abstract mathematical concepts to real-world situations Word problem (mathematics) , a decision problem for algebraic identities in mathematics and computer science
A particularly simple case of the word problem for groups and the isomorphism problem for groups asks if a finitely presented group is the trivial group. This is known to be intractable in general, even though there is a finite sequence of elementary Tietze transformations taking the presentation to the trivial presentation if and only if the ...
Given a freely reduced word w on X ±1, construct a sequence of freely reduced words w = w 0, w 1, w 2,..., as follows. Suppose w j is already constructed. If it is the empty word, terminate the algorithm. Otherwise check if w j contains a subword v such that v is also a subword of some defining relator r = vu ∈ R such that |v| > |r|/2.
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