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.

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) < ∞.

A r-maximal vector space not contained in any maximal vector space

1978 ◽  
Vol 43 (3) ◽  
pp. 430-441 ◽  
Author(s):  
J. Remmel
Keyword(s):  
Vector Space ◽  
Quotient Space ◽  
Independent Set ◽  
Scalar Multiplication ◽  
Effective Procedure ◽  
Natural Numbers ◽  
Infinite Dimensional ◽  
Vector Addition ◽  
Zero Vector

In [4], Metakides and Nerode define a recursively presented vector space V∞. over a (finite or infinite) recursive field F 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 any given n-tuple v0, …, vn−1 from V∞ is linearly dependent. Given a subspace W of V∞, we write dim(W) for the dimension of W. Given subspaces V and W of V∞, V + W will denote the weak sum of V and W and if V ∩ W = {0) (where 0 is the zero vector of V∞), we write V ⊕ W instead of V + W. If W ⊇ V, we write W mod V for the quotient space. An independent set A ⊆ V∞ is extendible if there is a r.e. independent set I ⊇ A such that I − A is infinite and A is nonextendible if it is not the case that A is extendible.

Automorphisms of supermaximal subspaces

1985 ◽  
Vol 50 (1) ◽  
pp. 1-9 ◽  
Author(s):  
R. G. Downey ◽  
G. R. Hird
Keyword(s):  
Vector Space ◽  
Recursive Relation ◽  
Effective Procedure ◽  
Dimensional Vector ◽  
Algebraic Systems ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Vector Addition ◽  
State Construction ◽  
Recursive Set

An infinite-dimensional vector space V∞ over a recursive field F is called fully effective if V∞ is a recursive set identified with ω upon which the operations of vector addition and scalar multiplication are recursive functions, identity is a recursive relation, and V∞ has a dependence algorithm, that is a uniformly effective procedure which when applied to x, a1,…,an, ∈ V∞ determines whether or not x is an element of {a1,…,an}* (the subspace generated by {a1,…,an}). The study of V∞, and of its lattice of r.e. subspaces L(V∞), was introduced in Metakides and Nerode [15]. Since then both V∞ and L(V∞) (and many other effective algebraic systems) have been studied quite intensively. The reader is directed to [5] and [17] for a good bibliography in this area, and to [15] for any unexplained notation and terminology.In [15] Metakides and Nerode observed that a study of L(V∞) may in some ways be modelled upon a study of L(ω), the lattice of r.e. sets. For example, they showed how an e-state construction could be modified to produce an r.e. maximal subspace, where M ∈ L(V∞) is maximal if dim(V∞/M) = ∞ and, for all W ∈ L(V∞), if W ⊃ M then either dim(W/M) < ∞ or dim(V∞/W) < ∞.However, some of the most interesting features of L(V∞) are those which do not have analogues in L(ω). Our concern here, which is probably one of the most striking characteristics of L(V∞), falls into this category. We say M ∈ L(V∞) is supermaximal if dim(V∞/M) = ∞ and for all W ∈ L(V∞), if W ⊃ M then dim(W/M) < ∞ or W = V∞. These subspaces were discovered by Kalantari and Retzlaff [13].

Maximal vector spaces under automorphisms of the lattice of recursively enumerable vector spaces

1977 ◽  
Vol 42 (4) ◽  
pp. 481-491 ◽  
Author(s):  
Iraj Kalantari ◽  
Allen Retzlaff
Keyword(s):  
Quotient Space ◽  
Vector Spaces ◽  
Infinite Field ◽  
Infinite Dimensional ◽  
Area Of Interest ◽  
Finite Dimensional ◽  
Recursively Enumerable ◽  
Simple Lattice ◽  
Vector Addition ◽  
Countably Infinite

The area of interest of this paper is recursively enumerable vector spaces; its origins lie in the works of Rabin [16], Dekker [4], [5], Crossley and Nerode [3], and Metakides and Nerode [14]. We concern ourselves here with questions about maximal vector spaces, a notion introduced by Metakides and Nerode in [14]. The domain of discourse is V∞ a fully effective, countably infinite dimensional vector space over a recursive infinite field F.By fully effective we mean that V∞, under a fixed Gödel numbering, has the following properties:(i) The operations of vector addition and scalar multiplication on V∞ are represented by recursive functions.(ii) There is a uniform effective procedure which, given n vectors, determines whether or not they are linearly dependent (the procedure is called a dependence algorithm).We denote the Gödel number of x by ⌈x⌉ By taking {εn ∣ n > 0} to be a fixed recursive basis for V∞, we may effectively represent elements of V∞ in terms of this basis. Each element of V∞ may be identified uniquely by a finitely-nonzero sequence from F Under this identification, εn corresponds to the sequence whose n th entry is 1 and all other entries are 0. A recursively enumerable (r.e.) space is a subspace of V∞ which is an r.e. set of integers, ℒ(V∞) denotes the lattice of all r.e. spaces under the operations of intersection and weak sum. For V, W ∈ ℒ(V∞), let V mod W denote the quotient space. Metakides and Nerode define an r.e. space M to be maximal if V∞ mod M is infinite dimensional and for all V ∈ ℒ(V∞), if V ⊇ M then either V mod M or V∞ mod V is finite dimensional. That is, M has a very simple lattice of r.e. superspaces.

The universal complementation property

1984 ◽  
Vol 49 (4) ◽  
pp. 1125-1136 ◽  
Author(s):  
R.G. Downey ◽  
J.B. Remmel
Keyword(s):  
Vector Space ◽  
Scalar Multiplication ◽  
Effective Procedure ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Recursively Enumerable ◽  
Finite Set ◽  
The Universe ◽  
Recursive Set

Let V∞ be a fully effective infinite dimensional vector space over a recursive field F. That is, we assume that the universe of V∞ is a recursive set, the operations of addition and scalar multiplication are recursive, and there is a uniform effective procedure to decide whether any finite set {υ0, …, υn} of vectors from V∞ is independent. The lattice of recursively enumerable subspaces has been extensively studied since its introduction by Metakides and Nerode [MN1] (see for example, [Do2], [Gu], [KR], [Re1], [Re2], and [Sh]). For those unfamiliar with the literature on , we shall give a list of basic definitions required for this paper in §0.It is well known that complements in V∞ are not unique. For example, in [Re2] Remmel constructed r.e. spaces M1 and M2 and co-r.e. spaces Q1 and Q2 such that for all i, j ∈ {1, 2}, Mi ⊕ Qj = V∞ and M1 is supermaximal, M2 is not maximal, Q1 has a fully extendible basis, and Q2 has no extendible basis. We say a subspace Q of V∞ is fully co-r.e. if Q is generated by a co-r.e. subset of some recursive basis of V∞. Downey [Do2] has shown that every r.e. subspace of V∞ has a complement which is a fully co-r.e. subspace. Moreover suppose Q is any fully co-r.e. subspace, say Q = (C)* where C is a co-r.e. subset of a recursive basis B of V∞; if C is nonrecursive, then it is shown in [Do2] that Q has a decidable complement as well as a nondecidable nowhere simple complement.

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.

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.

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.

Simple and hyperhypersimple vector spaces

1978 ◽  
Vol 43 (2) ◽  
pp. 260-269 ◽  
Author(s):  
Allen Retzlaff
Keyword(s):  
Vector Space ◽  
Modular Lattice ◽  
Final Section ◽  
Early Research ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Recursively Enumerable ◽  
Algebraic Properties

AbstractLet V∞ be a fixed, fully effective, infinite dimensional vector space. Let be the lattice consisting of the recursively enumerable (r.e.) subspaces of V∞, under the operations of intersection and weak sum (see §1 for precise definitions). In this article we examine the algebraic properties of .Early research on recursively enumerable algebraic structures was done by Rabin [14], Frölich and Shepherdson [5], Dekker [3], Hamilton [7], and Guhl [6]. Our results are based upon the more recent work concerning vector spaces of Metakides and Nerode [12], Crossley and Nerode [2], Remmel [15], [16], and Kalantari [8].In the main theorem below, we extend a result of Lachlan from the lattice of r.e. sets to . We define hyperhypersimple vector spaces, discuss some of their properties and show if A, B ∈ , and A is a hyperhypersimple subspace of B then there is a recursive space C such that A + C = B. It will be proven that if V ∈ and the lattice of superspaces of V is a complemented modular lattice then V is hyperhypersimple. The final section contains a summary of related results concerning maximality and simplicity.

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