Commutative Structures Research Articles

TBicomR: Event Prediction in Temporal Knowledge Graphs with Bicomplex Rotation

Temporal knowledge graphs (TKGs) capture relationships and entities evolving over time, making event prediction a challenging task due to the complex temporal and relational dynamics. In this work, we propose BiCoTime, a novel model using bicomplex embeddings to represent entities, relations, and time. While quaternions capture asymmetric relations through non-commutativity, bicomplex numbers provide a commutative algebraic structure, ideal for modeling both symmetric and asymmetric relations. Unlike quaternions, bicomplex embeddings maintain interpretability in symmetric relations while preserving key algebraic properties like distributivity. Temporal rotations further enhance BiCoTime's ability to model the interaction between relations and time, capturing how entities and relationships evolve. This combination of bicomplex embeddings and temporal rotations ensures a more interpretable and accurate modeling of TKGs. Our experiments show that TBiComR achieved a 21% improvement in Mean Reciprocal Rank (MRR) on the ICEWS14 dataset, which emphasizes time points, and a 15% improvement on the YAGO11k dataset, which focuses on time spans. The choice of bicomplex numbers balances computational complexity and expressive power, offering efficient training and better predictive performance compared to models using quaternions or octonions.

Biderivations, commuting linear maps, post-Lie algebra structure on solvable Lie algebras of maximal rank

In this paper, all biderivations of solvable Lie algebras of maximal rank g = Q ⊕ N are characterized. Namely, we consider a solvable Lie algebra of the form g = Q ⊕ N , where Q is the maximal torus subalgebra of g , N is the nilradical of g and dim Q = dim N / N 2 .In the case dim Q = N we characterize the form of skew-symmetric and symmetric biderivations of g . As applications, the forms of linear commuting (skew-commuting) maps and the commutative post-Lie algebra structures on g are given.

Derivations and biderivations of the non-square linear Lie algebra gl(m×n, 𝔽)

Let m, n be integers with m > n . Denote M m , n by the set of all m × n matrices over the field F of characteristic zero. Let I be an n × m matrix with ( i , i ) -position 1 for any 1 ≤ i ≤ n , and 0 in other position. Define a bracket [ A , B ] = AIB − BIA , where A , B ∈ M m , n . Then M m , n with this bracket is a Lie algebra, called non-square linear Lie algebra, denoted by gl ( m × n , F ) . In this paper, all derivations and biderivations of gl ( m × n , F ) are determined. As applications of biderivations, the linear commuting maps and commutative post-Lie algebra structures on gl ( m × n , F ) are given.

Strolling through common meadows

This paper establishes a connection between rings, lattices and common meadows. Meadows are commutative and associative algebraic structures with two operations (addition and multiplication) with additive and multiplicative identities and for which inverses are total. Common meadows are meadows that introduce, as the inverse of zero, an error term a which is absorbent for addition. We show that common meadows are unions of rings which are ordered by a partial order that defines a lattice. These results allow us to extend some classical algebraic constructions to the setting of common meadows. We also briefly consider common meadows from a categorical perspective.

Divisor graphs and symmetries of finite abelian groups

The algebraic composition of an element [Formula: see text] of a commutative binary structure [Formula: see text] is modeled by its divisor (pseudo)graph [Formula: see text] (which can be regarded as a simple graph if [Formula: see text] is an abelian group), whose vertices are the elements of [Formula: see text] such that two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. With respect to the action of a group of symmetries on the set of divisor pseudographs of [Formula: see text], the numbers of orbits are considered, and these numbers are shown to yield groupoid-isotopy invariants when [Formula: see text] is a finite abelian group. The minimum number of orbits is computed for every abelian group of order at most 11, and it is also determined for all finite cyclic groups. Moreover, systems of pseudographs that can be realized as those of finite abelian groups are completely characterized. In fact, recurrence relations are given for constructing systems of divisor pseudographs of finite cyclic groups, and all commutative binary structures that are isotopic to a finite abelian group are established.

Canonical Supermultiplets and Their Koszul Duals

The pure spinor superfield formalism reveals that, in any dimension and with any amount of supersymmetry, one particular supermultiplet is distinguished from all others. This “canonical supermultiplet” is equipped with an additional structure that is not apparent in any component-field formalism: a (homotopy) commutative algebra structure on the space of fields. The structure is physically relevant in several ways; it is responsible for the interactions in ten-dimensional super Yang–Mills theory, as well as crucial to any first-quantised interpretation. We study the L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_\\infty $$\\end{document} algebra structure that is Koszul dual to this commutative algebra, both in general and in numerous examples, and prove that it is equivalent to the subalgebra of the Koszul dual to functions on the space of generalised pure spinors in internal degree greater than or equal to three. In many examples, the latter is the positive part of a Borcherds–Kac–Moody superalgebra. Using this result, we can interpret the canonical multiplet as the homotopy fiber of the map from generalised pure spinor space to its derived replacement. This generalises and extends work of Movshev–Schwarz and Gálvez–Gorbounov–Shaikh–Tonks in the same spirit. We also comment on some issues with physical interpretations of the canonical multiplet, which are illustrated by an example related to the complex Cayley plane, and on possible extensions of our construction, which appear relevant in an example with symmetry type G2×A1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_2 \ imes A_1$$\\end{document}.

ON TOPOLOGICAL COMPLEXITY OF GORENSTEIN SPACES

In this paper, using Sullivan's approach to rational homotopy theory of simply-connected finite type CW complexes, we endow the $\mathbb{Q}$-vector space $\mathcal{E}xt_{C^{\ast}(X;\mathbb{Q})}(\mathbb{Q}, C^{\ast}(X;\mathbb{Q}))$ with a graded commutative algebra structure. This leads us to introduce the $\mathcal{E}xt$-version of higher (resp. module, homology) topological complexity of $X_0$, the rationalization of $X$ (resp. of $X$ over $\mathbb{Q}$). We then compare these invariants and their respective ordinary ones for Gorenstein spaces.
 We also highlight, in this context, the benefit of Adams-Hilton models over a field of odd characteristics especially through two cases, the first one when the space is a $2$-cell CW-complex and the second one when it is a suspension.

Addressing the algebraic structure of a linear mixed model with balanced design towards model extension

Linear mixed models provide a general and versatile approach for analyzing data collected in experiments, suitable for modeling repeated, longitudinal, or clustered observations. Important results for estimation can be obtained when subclasses of these mixed models are considered, based on some characteristics of their algebraic structure. The class of the models with commutative orthogonal block structure, for which least squares estimators are the best linear unbiased estimators, is of great interest. In an approach based on the algebraic structure of the models, and availing ourselves of U matrices, we study the possibility of extending a balanced mixed model, which could lead to a model with commutative orthogonal block structure.

IDENTITIES WITH MULTIPLICATIVE GENERALIZED (α,α)-DERIVATIONS OF SEMIPRIME RINGS

Let R be a semiprime ring and α be an automorphism of R. A mapping F : R → R (not necessarily additive) is called multiplicative generalized (α,α)-derivation if there exists a unique (α,α)-derivation d of R such that F(xy) = F(x)α(y) + α(x)d(y) for all x,y ∈ R. In the present paper, we intend to study several algebraic identities involving multiplicative generalized (α,α)-derivations on appropriate subsets of semiprime rings and collect the information about the commutative structure of these rings.

Designs for half-diallel experiments with commutative orthogonal block structure

In some experiments, the experimental units are all pairs of individuals who have to undertake a given task together. The set of such pairs forms a triangular association scheme. Appropriate randomization then gives two non-trivial strata. The design is said to have commutative orthogonal block structure (COBS) if the best linear unbiased estimators of treatment contrasts do not depend on the stratum variances. There are precisely three ways in which such a design can have COBS. We give a complete description of designs for which all treatment contrasts are in the same stratum. Then we give a very general construction for designs with COBS which have some treatment contrasts in each stratum.

Joining Iso-Structured Models with Commutative Orthogonal Block Structure

In this work, we focus on a special class of mixed models, named models with commutative orthogonal block structure (COBS), whose covariance matrix is a linear combination of known pairwise orthogonal projection matrices that add to the identity matrix, and for which the orthogonal projection matrix on the space spanned by the mean vector commutes with the covariance matrix. The COBS have least squares estimators giving the best linear unbiased estimators for estimable vectors. Our approach to COBS relies on their algebraic structure, based on commutative Jordan algebras of symmetric matrices, which proves to be advantageous as it leads to important results in the estimation. Specifically, we are interested in iso-structured COBS, applying to them the operation of models joining. We show that joining iso-structured COBS gives COBS and that the estimators for the joint model may be obtained from those for the individual models.

Symmetry in the Asymmetric Universe: Remarks on Kayne (2022)

Abstract Kayne (2022) has proposed that the asymmetry of syntax be built into the fundamental operation of Merge itself. This squib reviews some of his proposals and supporting evidence. Departing from Kayne, this squib hypothesized that the asymmetric patterns mainly lie in the functional domain of syntax, and the lexical domain may remain symmetric either within a language or cross-linguistically. The Functional Asymmetry Hypothesis (FAH) is supported by the global symmetry of the VO/OV word order, the commutative conjunction structures in the lexical domain, and the free ordering of event-internal adverbs. If the observation is on the right track, it suggests that the asymmetry of syntax, while empirically robust, cannot be entirely reduced to the operation Merge.

Biderivations of a Lie algebra of Block type and applications

Let be a Lie algebra of Block type with basis and relations . In the present paper, the biderivations of are explicitly described. In particular, it is shown that any biderivation of is inner. As applications, the linear commuting maps and commutative post-Lie algebra structures on this algebra are given.

The classification of non-commutative torsion-free rings of rank two

The paper contains the classification of non-commutative torsion-free associative rings of rank two. Furthermore, torsion-free abelian groups of rank two supporting an associative, but not commutative ring structure are classified.

On the Kantor product, II

We describe the Kantor square (and Kantor product) of multiplications, extending the classification proposed in [J. Algebra Appl. 2017, 16 (9), 1750167]. Besides, we explicitly describe the Kantor square of some low dimensional algebras and give constructive methods for obtaining new transposed Poisson algebras and Poisson-Novikov algebras; and for classifying Poisson structures and commutative post-Lie structures on a given algebra.

Derivations and biderivations of the n-th Schrödinger algebra

The n-th Schrödinger algebra is the semi-direct of the Lie algebra with the n-th Heisenberg Lie algebra . In this paper, all derivations and biderivations of the n-th Schrödinger algebra are determined. As applications, all linear commuting maps and commutative post-Lie algebra structures on are obtained.

Algebraic Constructions for Novikov–Poisson Algebras

A Novikov–Poisson algebra (A,∘,·) is a vector space with a Novikov algebra structure (A,∘) and a commutative associative algebra structure (A,·) satisfying some compatibility conditions. Give a Novikov–Poisson algebra (A,∘,·) and a vector space V. A natural problem is how to construct and classify all Novikov–Poisson algebra structures on the vector space E=A⊕V such that (A,∘,·) is a subalgebra of E up to isomorphism whose restriction on A is the identity map. This problem is called extending structures problem. In this paper, we introduce the definition of a unified product for Novikov–Poisson algebras, and then construct an object GH2(V,A) to answer the extending structures problem. Note that unified product includes many interesting products such as bicrossed product, crossed product and so on. Moreover, the special case when dim(V)=1 is investigated in detail.

On a family of vertex operator superalgebras

This paper is to study vertex operator superalgebras which are strongly generated by their weight-2 and weight-32 homogeneous subspaces. Among the main results, it is proved that if such a vertex operator superalgebra V is simple, then V(2) has a canonical commutative associative algebra structure equipped with a non-degenerate symmetric associative bilinear form and V(32) is naturally a V(2)-module equipped with a V(2)-valued symmetric bilinear form and a non-degenerate (C-valued) symmetric bilinear form, satisfying a set of conditions. On the other hand, assume that A is any commutative associative algebra equipped with a non-degenerate symmetric associative bilinear form and assume that U is an A-module equipped with a symmetric A-valued bilinear form and a non-degenerate (C-valued) symmetric bilinear form, satisfying the corresponding conditions. Then we construct a Lie superalgebra L(A,U) and a simple vertex operator superalgebra LL(A,U)(ℓ,0) for every nonzero number ℓ such that LL(A,U)(ℓ,0)(2)=A and LL(A,U)(ℓ,0)(32)=U.

Twisted forms of commutative monoid structures on affine spaces

In this paper we study commutative algebraic monoid structures on affine spaces over an arbitrary field of characteristic zero. We obtain full classification of such structures on AK2 and AK3 and describe some generalizations on AKn for any dimension n.

Mixed Witt rings of algebras with involution

AbstractAlthough there is no natural internal product for hermitian forms over an algebra with involution of the first kind, we describe how to multiply two $\varepsilon $ -hermitian forms to obtain a quadratic form over the base field. This allows to define a commutative graded ring structure by taking together bilinear forms and $\varepsilon $ -hermitian forms, which we call the mixed Witt ring of an algebra with involution. We also describe a less powerful version of this construction for unitary involutions, which still defines a ring, but with a grading over $\mathbb {Z}$ instead of the Klein group.We first describe a general framework for defining graded rings out of monoidal functors from monoidal categories with strong symmetry properties to categories of modules. We then give a description of such a strongly symmetric category $\mathbf {Br}_h(K,\iota )$ which encodes the usual hermitian Morita theory of algebras with involutions over a field K.We can therefore apply the general framework to $\mathbf {Br}_h(K,\iota )$ and the Witt group functors to define our mixed Witt rings, and derive their basic properties, including explicit formulas for products of diagonal forms in terms of involution trace forms, explicit computations for the case of quaternion algebras, and reciprocity formulas relative to scalar extensions.We intend to describe in future articles further properties of those rings, such as a $\lambda $ -ring structure, and relations with the Milnor conjecture and the theory of signatures of hermitian forms.