The word problem for free fields

1973 ◽  
Vol 38 (2) ◽  
pp. 309-314 ◽  
Author(s):  
P. M. Cohn
Keyword(s):  
Normal Form ◽  
Word Problem ◽  
Free Field ◽  
Free Associative Algebra ◽  
Commutative Field ◽  
Skew Field ◽  
Solvable Word Problem ◽  
Skew Fields ◽  
Free Fields ◽  
Solvable Word

It has long been known that every free associative algebra can be embedded in a skew field [11]; in fact there are many different embeddings, all obtainable by specialization from the ‘universal field of fractions’ of the free algebra (cf. [5, Chapter 7]). This makes it reasonable to call the latter the free field; see §2 for precise definitions. The existence of this free field was first established by Amitsur [1], but his proof is rather indirect and does not provide anything like a normal form for the elements of the field. Actually one cannot expect to find such a normal form, since it does not even exist in the field of fractions of a commutative integral domain, but at least one can raise the word problem for free fields: Does there exist an algorithm for deciding whether a given expression for an element of the free field represents zero?Now some recent work has revealed a more direct way of constructing free fields ([4], [5], [6]), and it is the object of this note to show how this method can be used to solve the word problem for free fields over infinite ground fields. In this connexion it is of interest to note that A. Macintyre [9] has shown that the word problem for skew fields is recursively unsolvable. Of course, every finitely generated commutative field has a solvable word problem (see e.g. [12]).The construction of universal fields of fractions in terms of full matrices is briefly recalled in §2, and it is shown quite generally for a ring R with a field of fractions inverting all full matrices, that if the set of full matrices over R is recursive, then the universal field has a solvable word problem. This holds more generally if the precise set of matrices over R inverted over the field is recursive, but it seems difficult to exploit this more general statement.

The word problem for free fields: a correction and an addendum

1975 ◽  
Vol 40 (1) ◽  
pp. 69-74 ◽  
Author(s):  
P. M. Cohn
Keyword(s):  
Word Problem ◽  
Free Field ◽  
Skew Field ◽  
Finite Dimensional ◽  
Finite Set ◽  
Free Fields

In [1] it was claimed that the word problem for free fields with infinite centre can be solved. In fact it was asserted that if K is a skew field with infinite central subfield C, then the word problem in the free field on a set X over K can be solved, relative to the word problem in K.As G. M. Bergman has pointed out (in a letter to the author), it is necessary to specify rather more precisely what type of problem we assume to be soluble for K: We must be able to decide whether or not a given finite set in K is linearly dependent over its centre. This makes it desirable to prove that the free field has a corresponding property (and not merely a soluble word problem). This is done in §2; interestingly enough it depends only on the solubility of the word problem in the free field (cf. Lemma 2 and Theorem 1′ below).Bergman also notes that the proof given in [1] does not apply when K is finite-dimensional over its centre; this oversight is rectified in §4, while §3 lifts the restriction on C (to be infinite). However, we have to assume C to be the precise centre of K, and not merely a central subfield, as claimed in [1].I am grateful to G. M. Bergman for pointing out the various inaccuracies as well as suggesting remedies.

scholarly journals Linearizing the word problem in (some) free fields

2018 ◽  
Vol 28 (07) ◽  
pp. 1209-1230 ◽  
Author(s):  
Konrad Schrempf
Keyword(s):  
Normal Form ◽  
Linear Algebra ◽  
Word Problem ◽  
Commutative Field ◽  
Linear Representations ◽  
Free Fields

We describe a solution of the word problem in free fields (coming from non-commutative polynomials over a commutative field) using elementary linear algebra, provided that the elements are given by minimal linear representations. It relies on the normal form of Cohn and Reutenauer and can be used more generally to (positively) test rational identities. Moreover, we provide a construction of minimal linear representations for the inverse of nonzero elements.

On Weighted Geodesics in Groups

1987 ◽  
Vol 30 (1) ◽  
pp. 86-91
Author(s):  
Seymour Lipschutz
Keyword(s):  
Weight Function ◽  
Free Product ◽  
Word Problem ◽  
Minimum Weight ◽  
Minimum Length ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Solvable Word Problem ◽  
Geodesic Problem ◽  
Solvable Word

AbstractA word W in a group G is a geodesic (weighted geodesic) if W has minimum length (minimum weight with respect to a generator weight function α) among all words equal to W. For finitely generated groups, the word problem is equivalent to the geodesic problem. We prove: (i) There exists a group G with solvable word problem, but unsolvable geodesic problem, (ii) There exists a group G with a solvable weighted geodesic problem with respect to one weight function α1, but unsolvable with respect to a second weight function α2. (iii) The (ordinary) geodesic problem and the free-product geodesic problem are independent.

scholarly journals Operator algebras with hyperarithmetic theory

2020 ◽  
Author(s):  
Isaac Goldbring ◽  
Bradd Hart
Keyword(s):  
Word Problem ◽  
Operator Algebras ◽  
Cuntz Algebra ◽  
Finitely Generated ◽  
C Algebra ◽  
Finitely Generated Group ◽  
C Algebras ◽  
Existentially Closed ◽  
Solvable Word Problem ◽  
Solvable Word

Abstract We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II$_1$ factor $\mathcal R$, $L(\varGamma )$ for $\varGamma $ a finitely generated group with solvable word problem, $C^*(\varGamma )$ for $\varGamma $ a finitely presented group, $C^*_\lambda (\varGamma )$ for $\varGamma $ a finitely generated group with solvable word problem, $C(2^\omega )$ and $C(\mathbb P)$ (where $\mathbb P$ is the pseudoarc). We also show that the Cuntz algebra $\mathcal O_2$ has a hyperarithmetic theory provided that the Kirchberg embedding problems have affirmative answers. Finally, we prove that if there is an existentially closed (e.c.) II$_1$ factor (resp. $\textrm{C}^*$-algebra) that does not have hyperarithmetic theory, then there are continuum many theories of e.c. II$_1$ factors (resp. e.c. $\textrm{C}^*$-algebras).

Separating Classes of Groups by First-Order Sentences

2003 ◽  
Vol 13 (03) ◽  
pp. 287-302 ◽  
Author(s):  
André Nies
Keyword(s):  
Nilpotent Group ◽  
Word Problem ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
First Order ◽  
Solvable Word Problem ◽  
Rank 2 ◽  
Finitely Presented ◽  
Solvable Word

For various proper inclusions of classes of groups [Formula: see text], we obtain a group [Formula: see text] and a first-order sentence φ such that H⊨φ but no G∈ C satisfies φ. The classes we consider include the finite, finitely presented, finitely generated with and without solvable word problem, and all countable groups. For one separation, we give an example of a f.g. group, namely ℤp ≀ ℤ for some prime p, which is the only f.g. group satisfying an appropriate first-order sentence. A further example of such a group, the free step-2 nilpotent group of rank 2, is used to show that true arithmetic Th(ℕ,+,×) can be interpreted in the theory of the class of finitely presented groups and other classes of f.g. groups.

Artin's problem for skew field extensions

1985 ◽  
Vol 97 (1) ◽  
pp. 1-6 ◽  
Author(s):  
A. H. Schofield
Keyword(s):  
Artinian Ring ◽  
Field Extension ◽  
Subsequent Paper ◽  
Commutative Field ◽  
Finite Representation ◽  
New Approach ◽  
Skew Field ◽  
Skew Fields ◽  
Left And Right ◽  
The Right

For a commutative field extension, L ⊃ K, it is clear that a left basis of L over K; is also a right basis of L over K; however, for an extension of skew fields, this may easily fail, though it is hard to determine whether the right and left dimension may be different. Cohn ([4], ch. 5), however, was able to find extensions of skew fields such that the left and right dimensions were an arbitrary pair of cardinals subject only to the restrictions that neither were 1 and at least one of them was infinite. In this paper, I shall present a new approach that allows us to construct extensions of skew fields such that the left and right dimensions are arbitrary integers not equal to 1. In a subsequent paper, [7], I shall present related results and consequences; in particular, there is a construction of a hereditary artinian ring of finite representation type corresponding to the Coxeter diagram I2(5) answering the question raised by Dowbor, Ringel and Simson[5].

scholarly journals 2-Stratifold groups have solvable word problem

2017 ◽  
Vol 112 (3) ◽  
pp. 803-810 ◽  
Author(s):  
J. C. Gómez-Larrañaga ◽  
F. González-Acuña ◽  
Wolfgang Heil
Keyword(s):  
Word Problem ◽  
Solvable Word Problem ◽  
Solvable Word
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