Topologies arising from metrics valued in abelian ℓ-groups

Every semifield of non-zero characteristic is either a field of prime characteristic or a semifield of characteristic 1. Semifields of characteristic 1 are equivalent to abelian lattice-ordered groups. It is proved that such a semifield A is algebraically closed if and only if the pure equations xn = a and certain quadratic equations are solvable in A. Using a sheaf representation for z-projectable abelian l-groups on the co-Zariski space of minimal primes, a sheaf-theoretic characterization of algebraic closedness in characteristic 1 is obtained. Concerning the solvability of quadratic equations, the criterion consists in a topological condition for the base space. The results are built upon an analysis of rational functions in characteristic 1. While polynomials satisfy the “fundamental theorem of algebra”, the multiplicative structure of rational functions is determined by means of “divisors”, extracted from the additive structure of A modulo parallelogram identities.

Complemented MacNeille completions and algebras of fractions

A commutative BCK-algebra with the relative cancellation property is a commutative BCK-algebra (X;*,0) which satisfies the condition: if a ≤ x, a ≤ y and x * a = y * a, then x = y. Such BCK-algebras form a variety, and the category of these BCK-algebras is categorically equivalent to the category of Abelian l-groups whose objects are pairs (G, G 0), where G is an Abelian l-group, G 0 is a subset of the positive cone generating G + such that if u, v ∈ G 0, then 0 ∨ (u - v) ∈ G 0, and morphisms are l-group homomorphisms h: (G, G 0) → (G′,G′0) with f(G 0) ⫅ G′0. Our methods in particular cases give known categorical equivalences of Cornish for conical BCK-algebras and of Mundici for bounded commutative BCK-algebras (= MV-algebras).

In a partly ordered space the orthogonality relation is defined by incomparability. We define integrally open and integrally semi-open ordered real vector spaces. We prove: if an ordered real vector space is integrally semi-open, then a complete lattice of double orthoclosed sets is orthomodular. An integrally open concept is closely related to an open set in the Euclidean topology in a finite dimensional ordered vector space. We prove: if V is an ordered Euclidean space, then V is integrally open and directed (and is also Archimedean) if and only if its positive cone, without vertex 0, is an open set in the Euclidean topology (and also the family of all order segments \( \{ z \in V:a < z < b\} \), a < b, is a base for the Euclidean topology).

Introduced by Chang in the late fifties, MV-algebras stand to Łukasiewicz’s infinite-valued propositional logic as boolean algebras stand to the classical propositional calculus. As stated by Chang in his original paper, “MV is supposed to suggest many-valued logics … for want of a better name”. The name has stuck. After some decades of relative quiescency, MV-algebras are today intensely investigated. On the one hand, these algebras find applications in such diverse fields as error-correcting feedback codes and logic-based control theory. On the other hand, MV-algebras are interesting mathematical objects in their own right. The main aim of this paper is to show that the interaction between MV-algebras and lattice-ordered abelian groups — including the timehonored theory of magnitudes — has much to offer, not only to specialists in these two fields, but also to people interested in the fan-theoretic description of toric varieties, and in the K0-theory of AF C*-algebras.

This paper introduces the notion of a functorial torsion class (FTC): in a concrete category $\mathfrak{C}$ which has image factorization, one considers monocoreflective subcategories which are closed under formation of subobjects. Here the interest is in FTCs in the category of abelian lattice-ordered groups with designated strong order unit. The FTCs $\mathfrak{T}$ consisting of archimedean latticeordered groups are characterized: for each subgroup A of the rationals with the identity 1, either $\mathfrak{T} = \mathfrak{S}\left( A \right)$, the class of all lattice-ordered groups of functions on a set X which have finite range in A, or $$\mathfrak{T} = \mathbb{T}\left( A \right)$$, the class of all subgroups of A with 1. As for FTCs possessing non-archimedean groups, it is shown that if $\mathfrak{T}$ is an FTC containing a subgroup A of the reals with 1, of rank two or greater, then $\mathfrak{T}$ contains all ℓ-groups of the form $A\vec \times G$, for all abelian lattice-ordered groups G. Finally, the least FTC that contains a non-archimedean group is the class of all $\mathbb{Z}\vec \times G$, for all abelian lattice-ordered groups G.

The representation of a ring as the ring of all the automorphisms of a primary abelian operator group expresses significant inner properties of this ring, since there exists essentially at most one such representation of a given ring. Thus it is the object of this investigation to expose invariant qualities of these automorphism rings. They are found to be pecularities of their ideal theory (Chapter III); and from these we select a complete set of postulates for the class of all the automorphism rings of primary abelian operator groups (Sections 12 and 13); thus proving incidentally that the structure of these rings is completely determined by their ideal theory (Section 14). The properties of the ideals in these rings reflect so completely the structure of the underlying operator group that we are able to prove-provided the rank of this group is at least three-the essential identity of the group of a-utomorphisms of the ring and of the group of projectivities (= biunivoque and monotone increasing transformations) of the system of admissible subgroups of the underlying operator group (Chapter II). Two [extreme] special cases may serve as an-i illustration for these theorems. Every projective geometry of finite dimension not less than three may be represented as the set of admissible subgroups of a suitable primary abelian operator group; and our characterization of the automorphisms of the automophism ring specializes in this case to the (well known) theorem that each automorphism of the automorphism ring (= ring of square matrices with coefficients from a suitable (not necessarily commutative) field) is induced by a so-called seini-linear transformation. If secondly G is an ordinary primary abelian group with the property that the least common multiple of the orders of the elements in G is a prime power p' and that G contains at least three independent elements of maximum order, then our theorem states that every automorphism of the automorphism ring is an inner auomorphism.

Massive integrable soliton theories

Residual properties of HNN-extensions with base group an abelian group

The space Lp(G), 1 > p < ∞, on a locally compact group G is known to be closed under convolution only if G is compact. However, the weighted spaces Lp(G, w) are Banach algebras with respect to convolution and natural norm under certain conditions on the weight. In the present paper, sufficient conditions for a weight defining a convolution algebra are stated in general form. These conditions are well known in some special cases. The spectrum (the maximal ideal space) of the algebra Lp(G,w) on an Abelian group G is described. It is shown that all algebras of this type are semisimple.

Algebraic structure of codes over Fq, closed under arbitrary abelian group G of permutations with exponent relatively prime to q, called G-invariant codes, is investigated using a transform domain approach. In particular, this general approach unveils algebraic structure of quasi-cyclic codes, abelian codes, cyclic codes, and quasi-abelian codes with restriction on G to appropriate special cases. Dual codes of G-invariant codes and self-dual G-invariant codes are characterized. The number of G-invariant self-dual codes for any abelian group G is found. In particular, this gives the number of self-dual l-quasi-cyclic codes of length ml over Fq when (m,q)=1. We extend Tanner's approach for getting a bound on the minimum distance from a set of parity check equations over an extension field and outline how it can be used to get a minimum distance bound for a G-invariant code. Karlin's decoding algorithm for a systematic quasi-cyclic code with a single row of circulants in the generator matrix is extended to the case of systematic quasi-abelian codes. In particular, this can be used to decode systematic quasi-cyclic codes with columns of parity circulants in the generator matrix.

Over the past few years, a family of interesting new inequalities for the entropies of sums and differences of random variables has been developed by Ruzsa, Tao and others, motivated by analogous results in additive combinatorics. The present work extends these earlier results to the case of random variables taking values in $\mathbb{R}^n$ or, more generally, in arbitrary locally compact and Polish abelian groups. We isolate and study a key quantity, the Ruzsa divergence between two probability distributions, and we show that its properties can be used to extend the earlier inequalities to the present general setting. The new results established include several variations on the theme that the entropies of the sum and the difference of two independent random variables severely constrain each other. Although the setting is quite general, the result are already of interest (and new) for random vectors in $\mathbb{R}^n$. In that special case, quantitative bounds are provided for the stability of the equality conditions in the entropy power inequality; a reverse entropy power inequality for log-concave random vectors is proved; an information-theoretic analog of the Rogers-Shephard inequality for convex bodies is established; and it is observed that some of these results lead to new inequalities for the determinants of positive-definite matrices. Moreover, by considering the multiplicative subgroups of the complex plane, one obtains new inequalities for the differential entropies of products and ratios of nonzero, complex-valued random variables.

Amenability is a notion of facial exposedness for convex cones that is stronger than being facially dual complete (or ‘nice’) which is, in turn, stronger than merely being facially exposed. Hyperbolicity cones are a family of algebraically structured closed convex cones that contain all spectrahedral cones (linear sections of positive semidefinite cones) as special cases. It is known that all spectrahedral cones are amenable. We establish that all hyperbolicity cones are amenable. As part of the argument, we show that any face of a hyperbolicity cone is a hyperbolicity cone. As a corollary, we show that the intersection of two hyperbolicity cones, not necessarily sharing a common relative interior point, is a hyperbolicity cone.

A class of suboptimal Wiener filters is considered, and their computational and statistical performances (and the trade-off between the two) are studied and compared with those for known classes of suboptimal Wiener filters. A general model of a suboptimal Wiener filter over a group is defined, which includes, as special cases, the known filters based on the discrete Fourier transform (DFT) in the case of a cyclic group and the Walsh-Hadamard transform (WHT) in the case of a dyadic group. Statistical and computational performances of various group filters are investigated. The cyclic and the dyadic group filters are known to be computationally the best ones among all the group filters. However, they are not always the best ones statistically and other (not necessarily Abelian) group filters are studied. Results are compared with those for the cyclic group filters (DFT), and the general problem of selecting the best group filter is posed. That problem is solved numerically for small-size signals <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(\leq 64)</tex> for the first-order Markov process and random sine wave corrupted by white noise. For the first-order Markov process with the covariance matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B^{(S,l)} = \rho^{|s-l|}</tex> as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\rho</tex> increases, the use of various non-Abelian groups results in improved statistical performance of the filter as compared to the DFT. Similarly, for the random sine wave with covariance matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B^{(s,l)} = \cos \lambda (s - l)</tex> as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\lambda</tex> decreases, non-Abelian groups result in a better statistical performance of the filter than the DFT does. However, that is compensated for by the increased number of computations to perform the filtering.

Projection method, cranking condition and collective energy