Hereditary undecidability of some theories of finite structures

1994 ◽  
Vol 59 (4) ◽  
pp. 1254-1262 ◽  
Author(s):  
Ross Willard
Keyword(s):  
Finite Field ◽  
Word Problem ◽  
Vertex Degree ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Finite Graphs ◽  
Finite Dimensional ◽  
Finite Semigroups ◽  
Finite Structures ◽  
Hereditary Undecidability

AbstractUsing a result of Gurevich and Lewis on the word problem for finite semigroups, we give short proofs that the following theories are hereditarily undecidable: (1) finite graphs of vertex-degree at most 3; (2) finite nonvoid sets with two distinguished permutations; (3) finite-dimensional vector spaces over a finite field with two distinguished endomorphisms.

scholarly journals On nonzero component graph of vector spaces over finite fields

2017 ◽  
Vol 16 (01) ◽  
pp. 1750007 ◽  
Author(s):  
Angsuman Das
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Clique Number ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Nonzero Component ◽  
Component Graph

In this paper, we study nonzero component graph [Formula: see text] of a finite-dimensional vector space [Formula: see text] over a finite field [Formula: see text]. We show that the graph is Hamiltonian and not Eulerian. We also characterize the maximal cliques in [Formula: see text] and show that there exists two classes of maximal cliques in [Formula: see text]. We also find the exact clique number of [Formula: see text] for some particular cases. Moreover, we provide some results on size, edge-connectivity and chromatic number of [Formula: see text].

scholarly journals $q$-Exponential Families

10.37236/1789 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Kent E. Morrison
Keyword(s):  
Finite Field ◽  
Exponential Families ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Finite Sets ◽  
Finite Dimensional ◽  
Exponential Formula ◽  
Positive Integers

We develop an analog of the exponential families of Wilf in which the label sets are finite dimensional vector spaces over a finite field rather than finite sets of positive integers. The essential features of exponential families are preserved, including the exponential formula relating the deck enumerator and the hand enumerator.

scholarly journals Neutrosophic Quadruple Vector Spaces and Their Properties

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 758 ◽  
Author(s):  
Vasantha Kandasamy W.B. ◽  
Ilanthenral Kandasamy ◽  
Florentin Smarandache
Keyword(s):  
Finite Field ◽  
Final Section ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Dimensional Vector ◽  
Characteristic P ◽  
Finite Dimensional ◽  
Classical Property ◽  
First Time

In this paper authors for the first time introduce the concept of Neutrosophic Quadruple (NQ) vector spaces and Neutrosophic Quadruple linear algebras and study their properties. Most of the properties of vector spaces are true in case of Neutrosophic Quadruple vector spaces. Two vital observations are, all quadruple vector spaces are of dimension four, be it defined over the field of reals R or the field of complex numbers C or the finite field of characteristic p, Z p ; p a prime. Secondly all of them are distinct and none of them satisfy the classical property of finite dimensional vector spaces. So this problem is proposed as a conjecture in the final section.

scholarly journals An Introduction to $q$-Species

10.37236/1959 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Kent E. Morrison
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Vector Spaces ◽  
Combinatorial Theory ◽  
Dimensional Vector ◽  
Finite Sets ◽  
Enumerative Combinatorics ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Linear Maps

The combinatorial theory of species developed by Joyal provides a foundation for enumerative combinatorics of objects constructed from finite sets. In this paper we develop an analogous theory for the enumerative combinatorics of objects constructed from vector spaces over finite fields. Examples of these objects include subspaces, flags of subspaces, direct sum decompositions, and linear maps or matrices of various types. The unifying concept is that of a "$q$-species", defined to be a functor from the category of finite dimensional vector spaces over a finite field to the category of finite sets.

A closer look at the stable completion

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Inverse Limit ◽  
Unit Vector ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Compact Sets ◽  
Linear Topology ◽  
Topological Embedding ◽  
Definable Set ◽  
Finite Dimensional ◽  
The One

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

scholarly journals A note on extensions of multilinear maps defined on multilinear varieties

2021 ◽  
pp. 1-26
Author(s):  
W. T. Gowers ◽  
L. Milićević
Keyword(s):  
Finite Fields ◽  
Vector Spaces ◽  
Restriction Map ◽  
Zero Set ◽  
Dimensional Vector ◽  
Prime Field ◽  
Finite Dimensional ◽  
Multilinear Maps ◽  
Multilinear Map

Abstract Let $G_1, \ldots , G_k$ be finite-dimensional vector spaces over a prime field $\mathbb {F}_p$ . A multilinear variety of codimension at most $d$ is a subset of $G_1 \times \cdots \times G_k$ defined as the zero set of $d$ forms, each of which is multilinear on some subset of the coordinates. A map $\phi$ defined on a multilinear variety $B$ is multilinear if for each coordinate $c$ and all choices of $x_i \in G_i$ , $i\not =c$ , the restriction map $y \mapsto \phi (x_1, \ldots , x_{c-1}, y, x_{c+1}, \ldots , x_k)$ is linear where defined. In this note, we show that a multilinear map defined on a multilinear variety of codimension at most $d$ coincides on a multilinear variety of codimension $O_{k}(d^{O_{k}(1)})$ with a multilinear map defined on the whole of $G_1\times \cdots \times G_k$ . Additionally, in the case of general finite fields, we deduce similar (but slightly weaker) results.

The genus of graphs associated with vector spaces

2019 ◽  
Vol 19 (05) ◽  
pp. 2050086 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
K. Prabha Ananthi
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Undirected Graph ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Vertex Connectivity ◽  
Dimensional Vector Space ◽  
Nonzero Component ◽  
Vertex Set ◽  
Component Graph

Let [Formula: see text] be a k-dimensional vector space over a finite field [Formula: see text] with a basis [Formula: see text]. The nonzero component graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as nonzero vectors of [Formula: see text] such that there is an edge between two distinct vertices [Formula: see text] if and only if there exists at least one [Formula: see text] along which both [Formula: see text] and [Formula: see text] have nonzero scalars. In this paper, we find the vertex connectivity and girth of [Formula: see text]. We also characterize all vector spaces [Formula: see text] for which [Formula: see text] has genus either 0 or 1 or 2.

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