Universal Algebra Research Articles

CONDITIONAL FLATNESS, FIBERWISE LOCALIZATIONS, AND ADMISSIBLE REFLECTIONS

Abstract We extend the group-theoretic notion of conditional flatness for a localization functor to any pointed category, and investigate it in the context of homological categories and of semi-abelian categories. In the presence of functorial fiberwise localization, analogous results to those obtained in the category of groups hold, and we provide existence theorems for certain localization functors in specific semi-abelian categories. We prove that a Birkhoff subcategory of an ideal determined category yields a conditionally flat localization, and explain how conditional flatness corresponds to the property of admissibility of an adjunction from the point of view of categorical Galois theory. Under the assumption of fiberwise localization, we give a simple criterion to determine when a (normal epi)-reflection is a torsion-free reflection. This is shown to apply, in particular, to nullification functors in any semi-abelian variety of universal algebras. We also relate semi-left-exactness for a localization functor L with what is called right properness for the L-local model structure.

Канторовость квазиунитарных полигонов над вполне (0-)простыми полугруппами

An universal algebra $A$ is called cantorian if for any algebra $B$ of the same signature, the existence of injective homomorphisms $A\to B$ and $B \to A$ implies an isomorphism of algebras $A$ and $B$. A right act $X$ over a semigroup $S$ is called quasiunitary if $X=XS$. We prove that every quasiunitary act over a completely simple semigroup and also every quasiunitary act with zero over a completely 0-simple semigroup are cantorian.

Fibered universal algebra for first-order logics

We extend Lawvere-Pitts prop-categories (aka. hyperdoctrines) to develop a general framework for providing fibered algebraic semantics for general first-order logics. This framework includes a natural notion of substitution, which allows first-order logics to be considered as structural closure operators just as propositional logics are in abstract algebraic logic. We then establish an extension of the homomorphism theorem from universal algebra for generalized prop-categories and characterize two natural closure operators on the prop-categorical semantics. The first closes a class of structures (which are interpreted as morphisms of prop-categories) under the satisfaction of their common first-order theory and the second closes a class of prop-categories under their associated first-order consequence. It turns out that these closure operators have characterizations that closely mirror Birkhoff's characterization of the closure of a class of algebras under the satisfaction of their common equational theory and Blok and Jónsson's characterization of closure under equational consequence, respectively. These algebraic characterizations of the first-order closure operators are unique to the prop-categorical semantics. They do not have analogues, for example, in the Tarskian semantics for classical first-order logic. The prop-categories we consider are much more general than traditional intuitionistic prop-categories or triposes (i.e., topos representing indexed partially ordered sets). Nonetheless, to the best of our knowledge, our results are new, even when restricted to these special classes of prop-categories.

Barr-exact categories and soft sheaf representations

It has long been known that a key ingredient for a sheaf representation of a universal algebra A consists in a distributive lattice of commuting congruences on A. The sheaf representations of universal algebras (over stably compact spaces) that arise in this manner have been recently characterised by Gehrke and van Gool (J. Pure Appl. Algebra, 2018), who identified the central role of the notion of softness.In this paper, we extend the scope of this theory by replacing varieties of algebras with Barr-exact categories, thus encompassing a number of “non-algebraic” examples. Our approach is based on the notion of K-sheaf: intuitively, whereas sheaves are defined on open subsets, K-sheaves are defined on compact ones. Throughout, we consider sheaves on complete lattices rather than spaces; this allows us to obtain point-free versions of sheaf representations whereby spaces are replaced with frames.These results are used to construct sheaf representations for the dual of the category of compact ordered spaces, and to recover Banaschewski and Vermeulen's point-free sheaf representation of commutative Gelfand rings (Quaest. Math., 2011).

S3-permutation orbifolds of Virasoro vertex algebras

In this paper, a continuation of [24], we investigate the S3-orbifold subalgebra of (Vc)⊗3, that is, we consider the S3-fixed point vertex subalgebra of the tensor product of three copies of the universal Virasoro vertex operator algebras Vc. Our main result is construction of a minimal, strong set of generators of this subalgebra for any generic values of c. More precisely, we show that this vertex algebra is of type (2,4,62,82,9,102,11,123).We also investigate two prominent examples of simple S3-orbifold algebras corresponding to central charges c=12 (Ising model) and c=−225 (i.e. (2,5)-minimal model). We prove that the former is a new unitary W-algebra of type (2,4,6,8) and the latter is isomorphic to the affine simple W-algebra of type G2 at non-admissible level −196. We also provide another version of this isomorphism using the affine W-algebra of type G2 coming from a subregular nilpotent element.

Cognitive technologies of the subject area based on ontology

An important problem of the automated construction of ontologies as a knowledge system is intellectualization, as well as the related data integration and qualitative information search. In this connection, the task of automated formation of ontologies is quite urgent. The most common quality criterion of the ontology is based on the assessment of the convenience and efficiency of working with it, and the presence of a connection between cognitive and semantic processes, which contributes to the process of its construction and actualization. The main difference of the conducted analysis, the results of which are presented in this work, are cognitive information technologies of the subject area, cognitive-semantic analysis based on the theory of categories, mathematical logic and universal algebra, construction of an ontological dictionary and ontological constructions. The informational interaction of the intellectual environment of the object with the subject is defined. For the formation of the concept in the intellectual environment due to the attached information space, the possibility is determined and the application of the concept and its features within the framework of the problem is substantiated. The construction of ontologies is based on the connection of cognitive processes with semantic ones and actualization – description and identification of sources of cognitive information in intellectual environments of subject areas. Linguistic-semantic analyzer is the main method of analyzing the emergence of new concepts, identification of semantically compatible concepts from various sources of knowledge, formation of textual stereotypes, prototypes and invariants in the form of text written with the words of this dictionary and terms contained in the thesaurus – by searching in text arrays of varying degrees of structuring of minimal semantic units and the use of object-oriented multivariate analysis in the construction of a system of templates with fuzzy calculations for class fields.

Coalgebras for Bisimulation of Weighted Automata over Semirings

Weighted automata are a generalization of nondeterministic automata that associate a weight drawn from a semiring $K$ with every transition and every state. Their behaviours can be formalized either as weighted language equivalence or weighted bisimulation. In this paper we explore the properties of weighted automata in the framework of coalgebras over (i) the category $\mathsf{SMod}$ of semimodules over a semiring $K$ and $K$-linear maps, and (ii) the category $\mathsf{Set}$ of sets and maps. We show that the behavioural equivalences defined by the corresponding final coalgebras in these two cases characterize weighted language equivalence and weighted bisimulation, respectively. These results extend earlier work by Bonchi et al. using the category $\mathsf{Vect}$ of vector spaces and linear maps as the underlying model for weighted automata with weights drawn from a field $K$. The key step in our work is generalizing the notions of linear relation and linear bisimulation of Boreale from vector spaces to semimodules using the concept of the kernel of a $K$-linear map in the sense of universal algebra. We also provide an abstract procedure for forward partition refinement for computing weighted language equivalence. Since for weighted automata defined over semirings the problem is undecidable in general, it is guaranteed to halt only in special cases. We provide sufficient conditions for the termination of our procedure. Although the results are similar to those of Bonchi et al., many of our proofs are new, especially those about the coalgebra in $\mathsf{SMod}$ characterizing weighted language equivalence.

An $$\mathfrak {sl}_2$$-type tensor category for the Virasoro algebra at central charge 25 and applications

Let $\mathcal{O}_{25}$ be the vertex algebraic braided tensor category of finite-length modules for the Virasoro Lie algebra at central charge $25$ whose composition factors are the irreducible quotients of reducible Verma modules. We show that $\mathcal{O}_{25}$ is rigid and that its simple objects generate a semisimple tensor subcategory that is braided tensor equivalent to an abelian $3$-cocycle twist of the category of finite-dimensional $\mathfrak{sl}_2$-modules. We also show that this $\mathfrak{sl}_2$-type subcategory is braid-reversed tensor equivalent to a similar category for the Virasoro algebra at central charge $1$. As an application, we construct a simple conformal vertex algebra which contains the Virasoro vertex operator algebra of central charge $25$ as a $PSL_2(\mathbb{C})$-orbifold. We also use our results to study Arakawa's chiral universal centralizer algebra of $SL_2$ at level $-1$, showing that it has a symmetric tensor category of representations equivalent to $\mathrm{Rep}\,PSL_2(\mathbb{C})$. This algebra is an extension of the tensor product of Virasoro vertex operator algebras of central charges $1$ and $25$, analogous to the modified regular representations of the Virasoro algebra constructed earlier for generic central charges by I. Frenkel-Styrkas and I. Frenkel-M. Zhu.

On Hopfianity and Co-Hopfianity of Acts Over Groups

A universal algebra is called Hopfian if any of its surjective endomorphisms is an automorphism, and co-Hopfian if injective endomorphisms are automorphisms. In this paper, necessary and sufficient conditions are found for Hopfianity and co-Hopfianity of unitary acts over groups. It is proved that a coproduct of finitely many acts (not necessarily unitary) over a group is Hopfian if and only if every factor is Hopfian.

Universal corner symmetry and the orbit method for gravity

A universal symmetry algebra organizing the gravitational phase space has been recently found. It corresponds to the subset of diffeomorphisms that become physical at corners – codimension-2 surfaces supporting Noether charges. It applies to both finite distance and asymptotic corners. In this paper, we study this algebra and its representations, via the coadjoint orbit method. We show that generic orbits of the universal algebra split into sub-orbits spanned by finite distance and asymptotic corner symmetries, such that the full universal symmetry algebra gives rise to a unified treatment of corners in a manifold. We then identify the geometric structure that captures these algebraic properties on corners, which is the Atiyah Lie algebroid associated to a principal GL(2,R)⋉R2-bundle. This structure is suggestive of the existence of a novel quantum gravitational theory which would unitarily glue such geometric structures, with spacetime geometries appearing as semi-classical configurations.

Extensions and tangent prolongations of differentiable loops

The tangent Akivis algebra and Sabinin algebra of degree 3 of a differentiable loop is the tangential object determined by the third-order Taylor polynomial of the multiplication function of the loop. It is endowed with a bilinear skew-symmetric and a trilinear operation defined by the infinitesimal commutator and associator of the loop. The aim of our work is to study tangent algebras of degree 3 of abelian extensions of differentiable loops, which are affine extensions of the tangent algebras of the loop by abelian algebras. This class of loop extensions has previously been studied in terms of computational complexity and in terms of universal algebra. We apply the obtained results to the determination of tangent algebras of degree 3 of tangent prolongation of differentiable loops.

On the number of universal algebraic geometries

The algebraic geometry of a universal algebra {textbf{A}} is defined as the collection of solution sets of systems of term equations. Two algebras {textbf{A}}_1 and {textbf{A}}_2 are called algebraically equivalent if they have the same algebraic geometry. We prove that on a finite set A with |A| there are countably many algebraically inequivalent Mal’cev algebras and that on a finite set A with |A| there are continuously many algebraically inequivalent algebras.

BL-global representations

A subdirect product [Formula: see text] is global if there is a topology on [Formula: see text] which makes of [Formula: see text] the algebra of all global sections of a sheaf whose stalks are the algebras [Formula: see text] with [Formula: see text]. A quasivariety [Formula: see text] has BL-global representations if there is a class [Formula: see text] containing but close to the class of all relatively subdirectly irreducible members of [Formula: see text], satisfying that every member of [Formula: see text] is isomorphic to a global subdirect product whose factors are in [Formula: see text]. The adjective “BL” refers to “Birkhoff-like” since this type of representations are analogous to the classical Birkhoff subdirect representation by relatively subdirectly irreducibles. This paper has two main contributions. The first one is a generalization of a theorem proved in [D. Vaggione, Sheaf representation and Chinese remainder theorems, Algebra Universalis 29 (1992) 232–272] which characterizes the existence of a global representation of an algebra in terms of the solvability of certain congruence systems. The second one is a theorem assuring the existence of BL-global representations for quasivarieties with a near unanimity term.

{U}_{mathfrak{q}}{mathfrak{sl}}_2 -invariant non-compact boundary conditions for the XXZ spin chain

We introduce new {U}_{mathfrak{q}}{mathfrak{sl}}_2 -invariant boundary conditions for the open XXZ spin chain. For generic values of mathfrak{q} we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for mathfrak{q}={e}^{frac{ipi}{p}} a 2p-th root of unity — to its p-dimensional analogue. Both cases are parametrised by a continuous “spin” α ∈ ℂ.To motivate our construction, we first specialise to mathfrak{q} = i, where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions. In the continuum, this model is identified with the (η, ξ) ghost CFT on the upper-half plane with a continuum of conformally invariant boundary conditions on the real axis. The different sectors of the Hamiltonian are identified with irreducible Virasoro representations.Going back to generic q we investigate the algebraic properties of the underlying lattice algebras. We show that if mathfrak{q} α ∉± mathfrak{q} ℤ, the new boundary coupling provides a faithful representation of the blob algebra which is Schur-Weyl dual to {U}_{mathfrak{q}}{mathfrak{sl}}_2 . Then, modifying the boundary conditions on both the left and the right, we obtain a representation of the universal two-boundary Temperley-Lieb algebra. The generators and parameters of these representations are computed explicitly in terms of mathfrak{q} and α. Finally, we conjecture the general form of the Schur-Weyl duality in this case.This paper is the first in a series where we will study, at all values of the parameters, the spectrum and its continuum limit, the representation content of the relevant lattice algebras and the fusion properties of these new spin chains.

Dynamical symmetries of homogeneous minisuperspace models

We investigate the phase space symmetries and conserved charges of homogeneous gravitational minisuperspaces. These ($0+1$)-dimensional reductions of general relativity are defined by spacetime metrics in which the dynamical variables depend only on a time coordinate and are formulated as mechanical systems with a nontrivial field space metric (or supermetric) and effective potential. We show how to extract conserved charges for those minisuperspaces from the homothetic Killing vectors of the field space metric. In the case of two-dimensional field spaces, we exhibit a universal eight-dimensional symmetry algebra $\mathcal{A}=(\mathfrak{sl}(2,\mathbb{R})\ensuremath{\bigoplus}\mathbb{R})⨭{\mathfrak{h}}_{2}$, based on the two-dimensional Heisenberg algebra ${\mathfrak{h}}_{2}\ensuremath{\simeq}{\mathbb{R}}^{4}$. We apply this to the systematic study of the Bianchi models for homogeneous cosmology. This extends the previous results on the $\mathfrak{sl}(2,\mathbb{R})$ algebra for Friedmann-Lema\^{\i}tre-Robertson-Walker cosmology and the Poincar\'e symmetry for Kantowski-Sachs metrics describing the black hole interior. The presence of this rich symmetry structure already in minisuperspace models opens new doors toward quantization and the study of solution generating mechanisms.

Congruences of finite multi-base universal algebras isomorphic to $n$-ary groups derived from groups

Гомоморфизм многоосновной универсальной алгебры на многоосновную универсальную алгебру будем называть $mb$-гомоморфизмом, а соответствующие конгруэнции - $mb$-конгруэнциями. Получено описание $mb$-конгруэнций конечных многоосновных универсальных алгебр, $mb$-изоморфных $n$-арным группам, производных от групп, т. е. допускающих представление в виде последовательного применения операций некоторой группы. В частности, в данный класс входят универсальные алгебры, для $n$-арных операций которых справедлива теорема Глускина - Хоссу и ее аналоги.

AF-algebras with lattice-ordered K0: Logic and computation

Following Bratteli's original definition, an AF-algebra A is the norm closure of the union of an ascending sequence of finite-dimensional C⁎-algebras, all with the same unit. Elliott proved that the unital dimension (Grothendieck) group K0(A) uniquely determines A up to isomorphism. AF-algebras with lattice-ordered K0, (AFℓ-algebras) have a preeminent role in the AF-algebraic literature. The Elliott classifier E(A) of any AFℓ-algebra A—i.e., the unit interval of K0(A), or equivalently, the countable set of Murray-von Neumann equivalence classes of projections of A—has the structure of a countable MV-algebra. Every countable MV-algebra arises as E(A) for some AFℓ-algebra A. Since E(A) is the Lindenbaum algebra of some countable set of formulas in Łukasiewicz logic Ł∞, every Ł∞-formula ϕ naturally codes an equivalence class ϕA of projections in A. The deductive-algorithmic machinery of Ł∞ can be thus applied to decide, among many others, the following basic problems, with ⪯ the Murray-von Neumann order:Is ϕA⪯ψA?IsϕA={0}? Is ϕA central ? For any AFℓ-algebra A we construct mutual polytime reductions of the first two problems. The third problem is polytime reducible to both problems. If E(A) is finitely generated and A either has no quotient isomorphic to C or is simple, then the first two problems are polytime reducible to the third one. The complexity of these problems is polytime for many AFℓ-algebras in the literature, including the Behncke-Leptin algebras Am,n, the Farey AFℓ-algebra M1 and all its principal quotients, the CAR algebra and Glimm's universal UHF algebra, and every Effros-Shen algebra Fθ for θ∈[0,1]∖Q a real algebraic integer, or for θ=1/e. For any fixed n=1,2,… we consider the class Kn of AFℓ-algebras presented by generators X1,…,Xn and relations θ1=0,…,θm=0, for an arbitrary finite set Θ={θ1,…,θm} of Ł∞-formulas in the variables X1,…,Xn. We prove that all decision problems studied in this paper are polytime decidable for Kn—uniformly over the presentation Θ.

A theorem of Mumford and Ramanujam for universal algebras

A theorem of Mumford and Ramanujam for universal algebras

Bifold Algebras and Commutants for Enriched Algebraic Theories

Commuting pairs of algebraic structures on a set have been studied by several authors and may be described equivalently as algebras for the tensor product of Lawvere theories, or more basically as certain bifunctors that here we call bifold algebras. The much less studied notion of commutant for Lawvere theories was first introduced by Wraith and generalizes the notion of centralizer clone in universal algebra. Working in the general setting of enriched algebraic theories for a system of arities, we study the interaction of the concepts of bifold algebra and commutant. We show that the notion of commutant arises via a universal construction in a two-sided fibration of bifold algebras over various theories. On this basis, we study special classes of bifold algebras that are related to commutants, introducing the notions of commutant bifold algebra and balanced bifold algebra. We establish several adjunctions and equivalences among these categories of bifold algebras and related categories of algebras over various theories, including commutative, contracommutative, saturated, and balanced algebras. We also survey and develop examples of commutant bifold algebras, including examples that employ Pontryagin duality and a theorem of Ehrenfeucht and Łoś on reflexive abelian groups. Along the way, we develop a functorial treatment of fundamental aspects of bifold algebras and commutants, including tensor products of theories and the equivalence of bifold algebras and commuting pairs of algebras. Because we work relative to a (possibly large) system of arities in a closed category $${\mathscr {V}}$$ , our main results are applicable to arbitrary $${\mathscr {V}}$$ -monads on a finitely complete $${\mathscr {V}}$$ , the enriched theories of Borceux and Day, the enriched Lawvere theories of Power relative to a regular cardinal, and other notions of algebraic theory.

On the definitions of direct product in universal algebra

On the definitions of direct product in universal algebra