30,159 research outputs found
On the periodicity of irreducible elements in arithmetical congruence monoids
Arithmetical congruence monoids, which arise in non-unique factorization
theory, are multiplicative monoids consisting of all positive
integers satsfying . In this paper, we examine the
asymptotic behavior of the set of irreducible elements of , and
characterize in terms of and when this set forms an eventually periodic
sequence
Factorization Properties of Leamer Monoids
The Huneke-Wiegand conjecture has prompted much recent research in
Commutative Algebra. In studying this conjecture for certain classes of rings,
Garc\'ia-S\'anchez and Leamer construct a monoid S_\Gamma^s whose elements
correspond to arithmetic sequences in a numerical monoid \Gamma of step size s.
These monoids, which we call Leamer monoids, possess a very interesting
factorization theory that is significantly different from the numerical monoids
from which they are derived. In this paper, we offer much of the foundational
theory of Leamer monoids, including an analysis of their atomic structure, and
investigate certain factorization invariants. Furthermore, when S_\Gamma^s is
an arithmetical Leamer monoid, we give an exact description of its atoms and
use this to provide explicit formulae for its Delta set and catenary degree
Some new results on decidability for elementary algebra and geometry
We carry out a systematic study of decidability for theories of (a) real
vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces,
Banach spaces and metric spaces, all formalised using a 2-sorted first-order
language. The theories for list (a) turn out to be decidable while the theories
for list (b) are not even arithmetical: the theory of 2-dimensional Banach
spaces, for example, has the same many-one degree as the set of truths of
second-order arithmetic.
We find that the purely universal and purely existential fragments of the
theory of normed spaces are decidable, as is the AE fragment of the theory of
metric spaces. These results are sharp of their type: reductions of Hilbert's
10th problem show that the EA fragments for metric and normed spaces and the AE
fragment for normed spaces are all undecidable.Comment: 79 pages, 9 figures. v2: Numerous minor improvements; neater proofs
of Theorems 8 and 29; v3: fixed subscripts in proof of Lemma 3
Defining Recursive Predicates in Graph Orders
We study the first order theory of structures over graphs i.e. structures of
the form () where is the set of all
(isomorphism types of) finite undirected graphs and some vocabulary. We
define the notion of a recursive predicate over graphs using Turing Machine
recognizable string encodings of graphs. We also define the notion of an
arithmetical relation over graphs using a total order on the set
such that () is isomorphic to
().
We introduce the notion of a \textit{capable} structure over graphs, which is
one satisfying the conditions : (1) definability of arithmetic, (2)
definability of cardinality of a graph, and (3) definability of two particular
graph predicates related to vertex labellings of graphs. We then show any
capable structure can define every arithmetical predicate over graphs. As a
corollary, any capable structure also defines every recursive graph relation.
We identify capable structures which are expansions of graph orders, which are
structures of the form () where is a partial order. We
show that the subgraph order i.e. (), induced subgraph
order with one constant i.e. () and an expansion
of the minor order for counting edges i.e. ()
are capable structures. In the course of the proof, we show the definability of
several natural graph theoretic predicates in the subgraph order which may be
of independent interest. We discuss the implications of our results and
connections to Descriptive Complexity
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