Algebraic Characterization Research Articles

Finite algebras with Hom-sets of polynomial size

We provide an internal characterization of those finite algebras (i.e., algebraic structures) A \mathbf {A} such that the number of homomorphisms from any finite algebra X \mathbf {X} to A \mathbf {A} is bounded from above by a polynomial in the size of X \mathbf {X} . Namely, an algebra A \mathbf {A} has this property if, and only if, no subalgebra of A \mathbf {A} has a nontrivial strongly abelian congruence. We also show that the property can be decided in polynomial time for algebras in finite signatures. Moreover, if A \mathbf {A} is such an algebra, the set of all homomorphisms from X \mathbf {X} to A \mathbf {A} can be computed in polynomial time given X \mathbf {X} as input. As an application of our results to the field of computational complexity, we characterize inherently tractable constraint satisfaction problems over fixed finite structures, i.e., those that are tractable and remain tractable after expanding the fixed structure by arbitrary relations or functions.

Auslander‐type conditions and weakly Gorenstein algebras

AbstractLet be an Artin algebra. Under certain Auslander‐type conditions, we give some equivalent characterizations of (weakly) Gorenstein algebras in terms of the properties of Gorenstein projective modules and modules satisfying Auslander‐type conditions. As applications, we provide some support for several homological conjectures. In particular, we prove that if is left quasi‐Auslander, then is Gorenstein if and only if it is (left and) right weakly Gorenstein; and that if satisfies the Auslander condition, then is Gorenstein if and only if it is left or right weakly Gorenstein. This is a reduction of an Auslander–Reiten's conjecture, which states that is Gorenstein if satisfies the Auslander condition.

Characterising large‐type Artin groups

AbstractWe show that the class of large‐type Artin groups is invariant under isomorphism, in stark contrast with the corresponding situation for Coxeter groups. We obtain this result by providing a purely algebraic characterisation of large‐type Artin groups (i.e. independent of the presentation graph). As a corollary, we completely describe the Artin groups isomorphic to a given large‐type Artin group, and characterise those large‐type Artin groups that are rigid.

Algebraic and geometric characterizations related to the quantization problem of the $$C_{2,8}$$ channel

Algebraic and geometric characterizations related to the quantization problem of the $$C_{2,8}$$ channel

MPCC: strong stability of weakly nondegenerate S-stationary points

ABSTRACT In this paper, we consider the class of mathematical programmes with complementarity constraints (MPCC). Specifically, we focus on strong stability of M- and S-stationary points for MPCC. Kojima introduced this concept for standard nonlinear optimization problems. It refers to several well-posedness properties of the underlying problem. Besides its topological definition, the challenge is to state an algebraic characterization of strong stability. We obtain such a description for S-stationary points whose components of Lagrange vectors corresponding to bi-active constraints do not mutually vanish. We call these points weakly nondegenerate. Moreover, we show that a particular constraint qualification is necessary for strong stability.

Algebraic characterizations of homeomorphisms between algebraic varieties

We address the question of finding algebraic properties that are respectively equivalent, for a morphism between algebraic varieties over an algebraically closed field of characteristic zero, to be a homeomorphism for the Zariski topology and for a strong topology that we introduce. Our answers involve a study of seminormalization and saturation for morphisms between algebraic varieties, together with an interpretation in terms of continuous rational functions on the closed points of an algebraic variety. The continuity refers to the strong topology which is the usual Euclidean topology in the complex case and which comes from the theory of real closed fields otherwise.

Free Constructions in Hoops via $$\ell $$-Groups

AbstractLattice-ordered abelian groups, or abelian$$\ell $$ ℓ -groups in what follows, are categorically equivalent to two classes of 0-bounded hoops that are relevant in the realm of the equivalent algebraic semantics of many-valued logics: liftings of cancellative hoops and perfect MV-algebras. The former generate the variety of product algebras, and the latter the subvariety of MV-algebras generated by perfect MV-algebras, that we shall call $$\textsf{DLMV}$$ DLMV . In this work we focus on these two varieties and their relation to the structures obtained by forgetting the falsum constant 0, i.e., product hoops and DLW-hoops. As main results, we first show a characterization of the free algebras in these two varieties as particular weak Boolean products; then, we show a construction that freely generates a product algebra from a product hoop and a DLMV-algebra from a DLW-hoop. In other words, we exhibit the free functor from the two algebraic categories of hoops to the corresponding categories of 0-bounded algebras. Finally, we use the results obtained to study projective algebras and unification problems in the two varieties (and the corresponding logics); both varieties are shown to have (strong) unitary unification type, and as a consequence they are structurally and universally complete.

Normed Amenability and Bounded Cohomology over Non-Archimedean Fields

We study continuous bounded cohomology of totally disconnected locally compact groups with coefficients in a non-Archimedean valued field K \mathbb {K} . To capture the features of classical amenability that induce the vanishing of bounded cohomology with real coefficients, we start by introducing the notion of normed K \mathbb {K} -amenability, of which we prove an algebraic characterization. It implies that normed K \mathbb {K} -amenable groups are locally elliptic, and it relates an invariant, the norm of a K \mathbb {K} -amenable group, to the order of its discrete finite p p -subquotients, where p p is the characteristic of the residue field of K \mathbb {K} . Moreover, we prove a characterization of discrete normed K \mathbb {K} -amenable groups in terms of vanishing of bounded cohomology with coefficients in K \mathbb {K} . The algebraic characterization shows that normed K \mathbb {K} -amenability is a very restrictive condition, so the bounded cohomological one suggests that there should be plenty of groups with rich bounded cohomology with trivial K \mathbb {K} coefficients. We explore this intuition by studying the injectivity and surjectivity of the comparison map, for which surprisingly general statements are available. Among these, we show that if either K \mathbb {K} has positive characteristic or its residue field has characteristic 0 0 , then the comparison map is injective in all degrees. If K \mathbb {K} is a finite extension of Q p \mathbb {Q}_p , we classify unbounded and non-trivial quasimorphisms of a group and relate them to its subgroup structure. For discrete groups, we show that suitable finiteness conditions imply that the comparison map is an isomorphism; this applies in particular to finitely presented groups in degree 2 2 . A motivation as to why the comparison map is often an isomorphism, in stark contrast with the real case, is given by moving to topological spaces. We show that over a non-Archimedean field, bounded cohomology is a cohomology theory in the sense of Eilenberg–Steenrod, except for a weaker version of the additivity axiom which is however equivalent for finite disjoint unions. In particular there exists a Mayer–Vietoris sequence, the main missing piece for computing real bounded cohomology.

Transcendental properties of entropy-constrained sets II

In this work, we address the question of the impossibility of certain single-letter formulas by exploiting the semi-algebraic nature of various entropy-constrained sets. The focus lies on studying the properties of the level sets of relative entropy, mutual information, and Rényi entropies. We analyze the transcendental structure of the set of states in which one of the aforementioned entropy quantities is fixed. Our results rule out (semi)algebraic single-shot characterizations of these entropy measures with bounded ancilla for both the classical and quantum cases.

From Entanglement to Universality: A Multiparticle Spacetime Algebra Approach to Quantum Computational Gates Revisited

Alternative mathematical explorations in quantum computing can be of great scientific interest, especially if they come with penetrating physical insights. In this paper, we present a critical revisitation of our application of geometric (Clifford) algebras (GAs) in quantum computing as originally presented in [C. Cafaro and S. Mancini, Adv. Appl. Clifford Algebras 21, 493 (2011)]. Our focus is on testing the usefulness of geometric algebras (GAs) techniques in two quantum computing applications. First, making use of the geometric algebra of a relativistic configuration space (namely multiparticle spacetime algebra or MSTA), we offer an explicit algebraic characterization of one- and two-qubit quantum states together with a MSTA description of one- and two-qubit quantum computational gates. In this first application, we devote special attention to the concept of entanglement, focusing on entangled quantum states and two-qubit entangling quantum gates. Second, exploiting the previously mentioned MSTA characterization together with the GA depiction of the Lie algebras SO3;R and SU2;C depending on the rotor group Spin+3,0 formalism, we focus our attention to the concept of universality in quantum computing by reevaluating Boykin’s proof on the identification of a suitable set of universal quantum gates. At the end of our mathematical exploration, we arrive at two main conclusions. Firstly, the MSTA perspective leads to a powerful conceptual unification between quantum states and quantum operators. More specifically, the complex qubit space and the complex space of unitary operators acting on them merge in a single multivectorial real space. Secondly, the GA viewpoint on rotations based on the rotor group Spin+3,0 carries both conceptual and computational advantages compared to conventional vectorial and matricial methods.

Operator theory on the pentablock

The pentablock, denoted as P, is defined as follows:P={(a21,tr(A),det(A)):A=[aij]2×2 with ‖A‖<1}. It originated from the work of Agler–Lykova–Young in connection with a particular case of the μ-synthesis problem. It is a non-convex, polynomially convex, C-convex, star-like about the origin, and inhomogeneous domain.This paper deals with operator theory on the pentablock. We study pentablock unitaries and isometries, providing an algebraic characterization of pentablock isometries. En route, we provide the Wold-type decomposition for pentablock isometries, which consists of three parts: the unitary part, the pure part, and a new component. We define this novel component as the quasi-pentablock unitary and provide a functional model for it. Additionally, a model for a class of pure pentablock isometries has been found, along with some examples. Furthermore, a representation resembling the Beurling-Lax-Halmos paradigm has been presented for the invariant subspaces of pentablock pure isometries.

An algebraic characterization of binary CSS-T codes and cyclic CSS-T codes for quantum fault tolerance

CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes (C1,C2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(C_1, C_2)$$\\end{document} such that C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_1$$\\end{document} contains C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_2$$\\end{document}, C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_2$$\\end{document} is even, and the shortening of the dual of C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_1$$\\end{document} with respect to the support of each codeword of C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_2$$\\end{document} is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes (C1,C2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(C_1, C_2)$$\\end{document} is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed–Muller, cyclic and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature.

On a algebraic characterization of the generalized Fredholm operators

On a algebraic characterization of the generalized Fredholm operators

Residue maps, Azumaya algebras, and buildings

The goal of this note is to give an explicit formula for the residues of twists of the matrix algebra in terms of the twisting cocycle. Combined with the Fixed Point Theorem for actions of finite groups on affine buildings, this leads to a quick proof of the well-known characterization of unramified algebras in terms of Azumaya algebras.

Cohomology operations for moment-angle complexes and resolutions of Stanley–Reisner rings

A fundamental result in toric topology identifies the cohomology ring of the moment-angle complex Z K \mathcal {Z}_K associated to a simplicial complex K K with the Koszul homology of the Stanley–Reisner ring of K K . By studying cohomology operations induced by the standard torus action on the moment-angle complex, we extend this to a topological interpretation of the minimal free resolution of the Stanley–Reisner ring. The exterior algebra module structure in cohomology induced by the torus action recovers the linear part of the minimal free resolution, and we show that higher cohomology operations induced by the action (in the sense of Goresky–Kottwitz–MacPherson [Invent. Math. 131 (1998), pp. 25–83]) can be assembled into an explicit differential on the resolution. Describing these operations in terms of Hochster’s formula, we recover and extend a result due to Katthän [Mathematics 7 (2019), no. 7, p. 605]. We then apply all of this to study the equivariant formality of torus actions on moment-angle complexes. For these spaces, we obtain complete algebraic and combinatorial characterisations of which subtori of the naturally acting torus act equivariantly formally.

Non-commutative effect algebras, L-algebras, and local duality

Abstract GPE-algebras were introduced by Dvurečenskij and Vetterlein as unbounded pseudo-effect algebras. Recently, they have been characterized as partial L-algebras with local duality. In the present paper, GPE-algebras with an everywhere defined L-algebra operation are investigated. For example, linearly ordered GPE-algebra are of that type. They are characterized by their self-similar closures which are represented as negative cones of totally ordered groups. More generally, GPE-algebras with an everywhere defined multiplication are identified as negative cones of directed groups. If their partial L-algebra structure is globally defined, the enveloping group is lattice-ordered. For any self-similar L-algebra A, exponent maps are introduced, generalizing conjugation in the structure group. It is proved that the exponent maps are L-algebra automorphisms of A if and only if A is a GPE-algebra. As an application, a new characterization of cone algebras is obtained. Lattice GPE-algebras are shown to be equivalent to ∧-closed L-algebras with local duality.

Characterization of Ordered Semigroups Generating Well Quasi-Orders of Words

The notion of a quasi-order generated by a homomorphism from the semigroup of all words onto a finite ordered semigroup was introduced by Bucher et al. (Theor. Comput. Sci. 40, 131–148 1985). It naturally occurred in their studies of derivation relations associated with a given set of context-free rules, and they asked a crucial question, whether the resulting relation is a well quasi-order. We answer this question in the case of the quasi-order generated by a semigroup homomorphism. We show that the answer does not depend on the homomorphism, but it is a property of its image. Moreover, we give an algebraic characterization of those finite semigroups for which we get well quasi-orders. This characterization completes the structural characterization given by Kunc (Theor. Comput. Sci. 348, 277–293 2005) in the case of semigroups ordered by equality. Compared with Kunc’s characterization, the new one has no structural meaning, and we explain why that is so. In addition, we prove that the new condition is testable in polynomial time.

New Characterizations of Semisimple Alternative Algebras

New Characterizations of Semisimple Alternative Algebras

When is C(X) a p.p. local, a p.f. local or an r-local ring?

An element [Formula: see text] of a ring [Formula: see text] is called a p.p. (respectively, p.f.) element if the principal ideal [Formula: see text] is projective (respectively, flat) and a ring is called a p.p. (respectively, p.f.) ring if its every element is a p.p. (respectively, p.f.) element. As a natural generalization of these notions, a ring [Formula: see text] is called a p.p. (respectively, p.f.) local ring if for each [Formula: see text] either [Formula: see text] or [Formula: see text] is a p.p. (respectively, p.f.) element. We have shown that [Formula: see text] is a p.p. (respectively, p.f.) local ring if and only if at most one point of the space [Formula: see text] fails to be a basically disconnected point (respectively, [Formula: see text]-point) or equivalently, [Formula: see text] contains a maximal ideal [Formula: see text] such that each element of [Formula: see text], is a p.p. (respectively, p.f.) element. A ring with a unique regular maximal ideal is called an [Formula: see text]-local ring. To continue this approach, we introduce a space in which at most one point fails to be an almost [Formula: see text]-point, namely, essential almost[Formula: see text]-space, and we observe that non-almost [Formula: see text]-spaces which are essential almost [Formula: see text]-spaces are exactly the spaces which guarantee that [Formula: see text] to be an [Formula: see text]-local ring. Algebraic characterizations of these spaces are given and it is shown that [Formula: see text] is an essential almost [Formula: see text]-space if and only if every regular [Formula: see text]-ideal of [Formula: see text] is prime, if and only if the smallest [Formula: see text]-ideal containing each regular ideal of [Formula: see text] is a maximal ideal. Finally, we give a diagram summarizing the relations between the spaces we studied in the paper.

A generalization of immanants based on partition algebra characters

A generalization of immanants based on partition algebra characters