The torsion-free part of the Ziegler spectrum of orders over Dedekind domains

We study the R-torsion-free part of the Ziegler spectrum of an order Λ over a Dedekind domain R. We underline and comment on the role of lattices over Λ. We describe the torsion-free part of the spectrum when Λ is of finite lattice representation type

A factorisation theory for generalised power series and omnific integers

We prove that in every ring of generalised power series with non-positive real exponents and coefficients in a field of char- acteristic zero, every series admits a factorisation into finitely many irreducibles with infinite support, the number of which can be bounded in terms of the order type of the series, and a unique product, up to multiplication by a unit, of factors with finite support. We deduce analogous results for the ring of omnific integers within Conway’s surreal numbers, using a suitable notion of infinite product. In turn, we solve Gonshor’s conjecture that theomnificintegerω 2+ω+1isprime. We also exhibit new classes of irreducible and prime gener- alised power series and omnific integers, generalising previous work of Berarducci and Pitteloud

Factorisation theorems for generalised power series

Fields of generalised power series (or Hahn fields), with coefficients in a field and exponents in a divisible ordered abelian group, are a fundamental tool in the study of valued and ordered fields and asymptotic expansions. The subring of the series with non-positive exponents appear naturally when discussing exponentiation, as done in transseries, or integer parts. A notable example is the ring of omnific integers inside the field of Conway's surreal numbers. In general, the elements of such subrings do not have factorisations into irreducibles. In the context of omnific integers, Conway conjectured in 1976 that certain series are irreducible (proved by Berarducci in 2000), and that any two factorisations of a given series share a common refinement. Here we prove a factorisation theorem for the ring of series with non-positive real exponents: every series is shown to be a product of irreducible series with infinite support and a factor with finite support which is unique up to constants. From this, we shall deduce a general factorisation theorem for series with exponents in an arbitrary divisible ordered abelian group, including omnific integers as a special case. We also obtain new irreducibility and primality criteria. To obtain the result, we prove that a new ordinal-valued function, which we call degree, is a valuation on the ring of generalised power series with real exponents, and we formulate some structure results on the associated RV monoid

Quantum state transfer in spin chains with q-deformed interaction terms

We study the time evolution of a single spin excitation state in certain linear spin chains, as a model for quantum communication. Some years ago it was discovered that when the spin chain data (the nearest neighbour interaction strengths and the magnetic field strengths) are related to the Jacobi matrix entries of Krawtchouk polynomials or dual Hahn polynomials, so-called perfect state transfer takes place. The extension of these ideas to other types of discrete orthogonal polynomials did not lead to new models with perfect state transfer, but did allow more insight in the general computation of the correlation function. In the present paper, we extend the study to discrete orthogonal polynomials of q-hypergeometric type. A remarkable result is a new analytic model where perfect state transfer is achieved: this is when the spin chain data are related to the Jacobi matrix of q-Krawtchouk polynomials. The other cases studied here (affine q-Krawtchouk polynomials, quantum q-Krawtchouk polynomials, dual q-Krawtchouk polynomials, q-Hahn polynomials, dual q-Hahn polynomials and q-Racah polynomials) do not give rise to models with perfect state transfer. However, the computation of the correlation function itself is quite interesting, leading to advanced q-series manipulations

Teoria dei Modelli, Cultura (e Società?),

Questo articolo è il seguito di [4] dove è stata introdotta la Teoria dei Modelli. Lo scopo di queste note è quello di approfondire gli sviluppi piu' recenti del settore e sottolinearne le applicazioni con altre discipline scientifiche

The lattice coordinatized by a ring

A coordinatization functor L(−,1):Ring→Latt is defined from the category of rings to the category of modular lattices. The main features of this coordinatization functor are 1) that it extends the functor R↦L(R) of von Neumann that associates to a regular ring its lattice of principal right ideals; 2) it respects the respective (−)op endofunctors on Ring and Latt; and 3) it admits localization at a left R-module. The complemented elements of L(R,1) form a partially ordered set S(R) isomorphic to the space of direct summands of RR. The right nonsingular rings for which the embedding of S(R) into the localization L(R,1)Q at the right maximal ring of quotients QR is an isomorphism are characterized by a property that every finite matrix subgroup φ(RR) of the left R-module RR is essential in an element of S(R). In that case, the space S(R) obtains the structure of a complemented modular lattice coordinatized by the dominion, or equivalently, the ring of definable scalars, of the maximal ring of quotients. The class of rings with this property is elementary, in contrast to the class of rings whose space of right summands is coordinatized by the maximal ring of quotients

Weakly minimal modules over integral group rings and over some related classes of rings

A module is weakly minimal if and only if every pp-definable subgroup is either finite or of finite index. We study weakly minimal modules over several classes of rings, including valuation domains, Pr¨ufer domains and integral group rings

Minimalities and modules over Dedekind-like rings

We are interested in (right) modules M satisfying the following weak divisibility condition: If R is the underlying ring, then for every r ∈ R either Mr = 0 or Mr = M. Over a commutative ring, this is equivalent to say that M is connected with regular generics. Over arbitrary rings, modules which are “minimal” in several model theoretic senses satisfy this condition. In this article, we investigate modules with this weak divisibility property over Dedekind-like rings and over other related classes of rings

Cinquant'anni di Teoria dei Modelli

La teoria dei Modelli ha ufficialmente compiuto 50 anni di età. Lo scopo di queste note è quello di illustrare la Teoria dei modelli, le sue motivazioni, la sua evoluzione e soprattutto le sue applicazioni con altre discipline matematiche, come l'Analisi, la Geometria e l'Algebra