Controlling the dependence degree of a recursively enumerable vector space

1978 ◽  
Vol 43 (1) ◽  
pp. 13-22 ◽  
Author(s):  
Richard A. Shore
Keyword(s):  
Vector Space ◽  
Topological Structure ◽  
Independent Set ◽  
Original Data ◽  
Recursion Theory ◽  
Vector Spaces ◽  
Additional Structure ◽  
Recursively Enumerable ◽  
Added Benefit ◽  
And Algebra

Early work combining recursion theory and algebra had (at least) two different sets of motivations. First the precise setting of recursion theory offered a chance to make formal classical concerns as to the effective or algorithmic nature of algebraic constructions. As an added benefit the formalization gives one the opportunity of proving that certain constructions cannot be done effectively even when the original data is presented in a recursive way. One important example of this sort of approach is the work of Frohlich and Shepardson [1955] in field theory. Another motivation for the introduction of recursion theory to algebra is given by Rabin [1960]. One hopes to mathematically enrich algebra by the additional structure provided by the notion of computability much as topological structure enriches group theory. Another example of this sort is provided in Dekker [1969] and [1971] where the added structure is that of recursive equivalence types. (This particular structural view culminates in the monograph of Crossley and Nerode [1974].)More recently there is the work of Metakides and Nerode [1975], [1977] which combines both approaches. Thus, for example, working with vector spaces they show in a very strong way that one cannot always effectively extend a given (even recursive) independent set to a basis for a (recursive) vector space.

G. Metakides and A. Nerode. Recursion theory and algebra. Algebra and logic, Papers from the 1974 Summer Research Institute of the Australian Mathematical Society, Monash University, Australia, edited by J. N. Crossley, Lecture notes in mathematics, vol. 450, Springer-Verlag, Berlin, Heidelberg, and New York, 1975, pp. 209–219. - Iraj Kalantari and Allen Retzlaff. Maximal vector spaces under automorphisms of the lattice of recursively enumerable vector spaces. The journal of symbolic logic, vol. 42 no. 4 (for 1977, pub. 1978), pp. 481–491. - Iraj Kalantari. Major subspaces of recursively enumerable vector spaces. The journal of symbolic logic, vol. 43 (1978), pp. 293–303. - J. Remmel. A r-maximal vector space not contained in any maximal vector space. The journal of symbolic logic, vol. 43 (1978), pp. 430–441. - Allen Retzlaff. Simple and hyperhypersimple vector spaces. The journal of symbolic logic, vol. 43 (1978), pp. 260–269. - J. B. Remmel. Maximal and cohesive vector spaces. The journal of symbolic logic, vol. 42 no. 3 (for 1977, pub. 1978), pp. 400–418. - J. Remmel. On r.e. and co-r.e. vector spaces with nonextendible bases. The journal of symbolic logic, vol. 45 (1980), pp. 20–34. - M. Lerman and J. B. Remmel. The universal splitting property: I. Logic Colloquim '80, Papers intended for the European summer meeting of the Association for Symbolic Logic, edited by D. van Dalen, D. Lascar, and T. J. Smiley, Studies in logic and the foundations of mathematics, vol. 108, North-Holland Publishing Company, Amsterdam, New York, and Oxford, 1982, pp. 181–207. - J. B. Remmel. Recursively enumerable Boolean algebras. Annals of mathematical logic, vol. 15 (1978), pp. 75–107. - J. B. Remmel. r-Maximal Boolean algebras. The journal of symbolic logic, vol. 44 (1979), pp. 533–548. - J. B. Remmel. Recursion theory on algebraic structures with independent sets. Annals of mathematical logic, vol. 18 (1980), pp. 153–191. - G. Metakides and J. B. Remmel. Recursion theory on orderings. I. A model theoretic setting. The journal of symbolic logic, vol. 44 (1979), pp. 383–402. - J. B. Remmel. Recursion theory on orderings. II. The journal of symbolic logic, vol. 45 (1980), pp. 317–333.

1986 ◽  
Vol 51 (1) ◽  
pp. 229-232
Author(s):  
Henry A. Kierstead
Keyword(s):  
New York ◽  
Mathematical Logic ◽  
Vector Space ◽  
Recursion Theory ◽  
Boolean Algebras ◽  
Vector Spaces ◽  
Symbolic Logic ◽  
Recursively Enumerable ◽  
North Holland Publishing ◽  
And Algebra

scholarly journals Isomorphisms on countable vector spaces with recursive operations

1974 ◽  
Vol 18 (2) ◽  
pp. 230-235 ◽  
Author(s):  
Robert I. Soare
Keyword(s):  
Vector Space ◽  
Independent Set ◽  
Vector Spaces ◽  
Recursive Structure ◽  
Recursively Enumerable ◽  
Linearly Independent ◽  
Vector Addition

Terminology and notation may be found in Dekker [1] and [2]. Briefly, we fix a recursively enumerable (r.e.) field F with recursive structure, and let Ū be the vector space over F consisting of ultimately vanishing countable sequences of elements of F with the usual definitions of vector addition and multiplication by a scalar. A subspace V of Ū is called an α-space if V has a basis B which is contained in some r.e. linearly independent set S.

On r.e. and co-r.e. vector spaces with nonextendible bases

1980 ◽  
Vol 45 (1) ◽  
pp. 20-34 ◽  
Author(s):  
J. Remmel
Keyword(s):  
Vector Space ◽  
Quotient Space ◽  
Independent Set ◽  
Effective Procedure ◽  
Vector Spaces ◽  
Natural Numbers ◽  
Infinite Dimensional ◽  
Recursively Enumerable ◽  
Vector Addition ◽  
Zero Vector

The concern of this paper is with recursively enumerable and co-recursively enumerable subspaces of a recursively presented vector spaceV∞ over a (finite or infinite) recursive field F which is defined in [6] to consist of a recursive subset U of the natural numbers N and operations of vector addition and scalar multiplication which are partial recursive and under which V∞ becomes a vector space. Throughout this paper, we will identify V∞ with N, say via some fixed Gödel numbering, and assume V∞ is infinite dimensional and has a dependence algorithm, i.e., there is a uniform effective procedure which determines whether or not any given n-tuple v0, …, vn−1 from V∞ is linearly dependent. Various properties of V∞ and its sub-spaces have been studied by Dekker [1], Guhl [3], Metakides and Nerode [6], Kalantari and Retzlaff [4], and the author [7].Given a subspace W of V∞, we say W is r.e. (co-r.e.) if W(V∞ − W) is an r.e. subset of N and write dim(V) for the dimension of V. Given subspaces V, W of V∞, V + W will denote the weak sum of V and W and if V ⋂ M = {0} (where 0 is the zero vector of V∞), we write V ⊕ Winstead of V + W. If W ⊇ V, we write Wmod V for the quotient space. An independent set A ⊆ V∞ is extendible if there is an r.e. independent set I ⊇ A such that I − A is infinite and A is nonextendible if it is not the case An is extendible. A r.e. subspace M ⊇ V∞ is maximal if dim(V∞ mod M) = ∞ and for any r.e. subspace W ⊇ Meither dim(W mod M) < ∞ or dim(V∞ mod W) < ∞.

Simplicity in effective topology

1982 ◽  
Vol 47 (1) ◽  
pp. 169-183 ◽  
Author(s):  
Iraj Kalantari ◽  
Anne Leggett
Keyword(s):  
Topological Space ◽  
Recursion Theory ◽  
Activity Analysis ◽  
Vector Spaces ◽  
Topological Spaces ◽  
Mathematical Structures ◽  
Recursively Enumerable ◽  
Countable Basis ◽  
Recursive Analysis ◽  
And Algebra

The recursion-theoretic study of mathematical structures other thanωis now a field of much activity. Analysis and algebra are two such structures which have been studied for their effective contents. Studies in analysis began with the work of Specker on nonconstructive proofs in analysis [16] and in Lacombe's inspiring notes on relevant notions of recursive analysis [8]. Studies in algebra originated in the work of Frolich and Shepherdson on effective extensions of explicit fields [1] and in Rabin's study of computable fields [15]. Equipped with the richness of modern techniques in recursion theory, Metakides and Nerode [11]–[13] began investigating the effective content of vector spaces and fields; these studies have been extended by Kalantari, Remmel, Retzlaff, Shore and others.Kalantari and Retzlaff [5] began a foundational inquiry into effectiveness in topological spaces. They consider a topological spaceXwith a countable basis ⊿ for the topology. The space isfully effective, that is, the basis elements are coded intoωand the operation of intersection of basis elements and the relation of inclusion among them are both computable. Similar to, the lattice of recursively enumerable (r.e.) subsets ofω, the collection of r.e. open subsets ofXforms a latticeℒ(X)under the usual operations of union and intersection.

Recursion theory on orderings. I. A model theoretic setting

1979 ◽  
Vol 44 (3) ◽  
pp. 383-402 ◽  
Author(s):  
G. Metakides ◽  
J.B. Remmel
Keyword(s):  
Algebraic Closure ◽  
Recursion Theory ◽  
Boolean Algebras ◽  
Vector Spaces ◽  
Formal Definition ◽  
Recursively Enumerable ◽  
Closed Fields ◽  
Partial Orderings ◽  
Definition Of ◽  
Algebraically Closed Fields

In [6], Metakides and Nerode introduced the study of the lattice of recursively enumerable substructures of a recursively presented model as a means to understand the recursive content of certain algebraic constructions. For example, the lattice of recursively enumerable subspaces,, of a recursively presented vector spaceV∞has been studied by Kalantari, Metakides and Nerode, Retzlaff, Remmel and Shore. Similar studies have been done by Remmel [12], [13] for Boolean algebras and by Metakides and Nerode [9] for algebraically closed fields. In all of these models, the algebraic closure of a set is nontrivial. (The formal definition of the algebraic closure of a setS, denoted cl(S), is given in §1, however in vector spaces, cl(S) is just the subspace generated byS, in Boolean algebras, cl(S) is just the subalgebra generated byS, and in algebraically closed fields, cl(S) is just the algebraically closed subfield generated byS.)In this paper, we give a general model theoretic setting (whose precise definition will be given in §1) in which we are able to give constructions which generalize many of the constructions of classical recursion theory. One of the main features of the modelswhich we study is that the algebraic closure of setis just itself, i.e., cl(S) = S. Examples of such models include the natural numbers under equality 〈N, = 〉, the rational numbers under the usual ordering 〈Q, ≤〉, and a large class ofn-dimensional partial orderings.

scholarly journals Recursive equivalence types of vector spaces

1974 ◽  
Vol 18 (3) ◽  
pp. 376-384 ◽  
Author(s):  
Alan G. Hamilton
Keyword(s):  
Vector Space ◽  
Vector Spaces ◽  
Natural Numbers ◽  
Infinite Dimensional ◽  
Recursively Enumerable ◽  
Countably Infinite

We consider subspaces of a vector space UF, which is countably infinite dimensional over a recursively enumerable field F with recursive operations, where the operations in UF are also recursive, and where, of course, F and UF are sets of natural numbers. It is the object of this paper to investigate recursive equivalence types of such vector spaces and the ways in which their properties are analogous to and depend on properties of recursive equivalence types of sets.

Maximal and Cohesive vector spaces

1977 ◽  
Vol 42 (3) ◽  
pp. 400-418 ◽  
Author(s):  
J. B. Remmel
Keyword(s):  
Vector Space ◽  
Quotient Space ◽  
Effective Procedure ◽  
Vector Spaces ◽  
Recursively Enumerable Set ◽  
Infinite Dimensional ◽  
Recursively Enumerable ◽  
Vector Addition ◽  
Zero Vector ◽  
Recursive Set

Let N denote the natural numbers. If A ⊆ N, we write Ā for the complement of A in N. A set A ⊆ N is cohesive if (i) A is infinite and (ii) for any recursively enumerable set W either W ∩ A or ∩ A is finite. A r.e. set M ⊆ N is maximal if is cohesive.A recursively presented vector space (r.p.v.s.) U over a recursive field F consists of a recursive set U ⊆ N and operations of vector addition and scalar multiplication which are partial recursive and under which U becomes a vector space. A r.p.v.s. U has a dependence algorithm if there is a uniform effective procedure which applied to any n-tuple ν0, ν1, …, νn−1 of elements of U determines whether or not ν0, ν1 …, νn−1 are linearly dependent. Throughout this paper we assume that if U is a r.p.v.s. over a recursive field F then U is infinite dimensional and U = N. If W ⊆ U, then we say W is recursive (r.e., etc.) iff W is a recursive (r.e., etc.) subset of N. If S ⊆ U, we write (S)* for the subspace generated by S. If V1 and V2 are subspaces of U such that V1 ∩ V2 ={} (where is the zero vector of U), then we write V1 ⊕ V2 for (V1 ∪ V2)*. If V1 ⊆ V2⊆U are subspaces, we write V2/V1 for the quotient space.

Major subspaces of recursively enumerable vector spaces

1978 ◽  
Vol 43 (2) ◽  
pp. 293-303 ◽  
Author(s):  
Iraj Kalantari
Keyword(s):  
Vector Space ◽  
Space Structure ◽  
Vector Spaces ◽  
Recent Theory ◽  
Essential Difference ◽  
Infinite Dimensional ◽  
Recursively Enumerable ◽  
Partial Recursive Functions ◽  
Initial Work ◽  
Object Of Study

The main point of this paper is a further development of some aspects of the recent theory of recursively enumerable (r.e.) algebraic structures. Initial work in this area is due to Frölich and Shepherdson [4] and Rabin [10]. Here we are only concerned with vector space structure. The previous work on r.e. vector spaces is due to Dekker [2], [3], Metakides and Nerode [8], Remmel [11], Retzlaff [13], and the author [5].Our object of study is V∞ a countably infinite dimensional fully effective vector space over a countable recursive field . By fully effective we mean that V∞. under a fixed Godel numbering has the following properties:(i) Operations of vector addition and scalar multiplication on V∞ are presented by partial recursive functions on the Gödel numbers of elements of V∞.(ii) V∞ has a dependence algorithm, i.e., there is a uniform effective procedure which applied to any n vectors of V∞ determines whether or not they are linearly independent.We also study , the lattice of r.e. subspaces of V∞ (under the operations of intersection, ⋂ and (weak) sum, +). We note that if is not distributive and is merely modular (see [1]). This fact indicates the essential difference between the lattice of r.e. sets and .

Bases and α-dimensions of countable vector spaces with recursive operations

1970 ◽  
Vol 35 (1) ◽  
pp. 85-96
Author(s):  
Alan G. Hamilton
Keyword(s):  
Vector Space ◽  
Vector Spaces ◽  
Natural Numbers ◽  
Infinite Dimensional ◽  
Recursively Enumerable ◽  
Countably Infinite

This paper is based on the notions originally described by Dekker [2], [3], and the reader is referred to these for explanation of notation etc. Briefly, we are concerned with a countably infinite dimensional countable vector space Ū with recursive operations, regarded as being coded as a set of natural numbers. Necessarily, then, Ū must be a vector space over a field which itself is in some sense recursively enumerable and has recursive operations.

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