We report on the Equational Theories Project (ETP), an online collaborative pilot project to explore new ways to collaborate in mathematics with machine assistance. The project successfully determined all 22 028 942 edges of the implication graph between the 4694 simplest equational laws on magmas, by a combination of human-generated and automated proofs, all validated by the formal proof assistant language Lean. As a result of this project, several new constructions of magmas satisfying specific laws were discovered, and several auxiliary questions were also addressed, such as the effect of restricting attention to finite magmas.

To each meet-semilattice $E$ is associated an inverse semigroup $T_{E}$ called the Munn semigroup of $E$. We generalise this construction by replacing the meet-semilattice $E$ by a presheaf of sets $X$ over a meet-semilattice. The inverse semigroup $T_{X}$ that results is called the generalised Munn semigroup. Our construction can be viewed as a generalisation of one due to Zhitomirskiy as well as a restriction of one due to Reilly. We prove that idempotent-separating representations in to the generalised Munn semigroup characterise étale actions of inverse semigroups.

We extend the existing skew polynomial representations of matrix algebras which are direct sum of matrix spaces over division rings. In this representation, the sum-rank distance between two tuples of matrices is captured by a weight function on their associated skew polynomials, defined through degrees and greatest common right divisors with the polynomial that defines the representation. We exploit this representation to construct new families of maximum sum-rank distance (MSRD) codes over finite and infinite fields, and over division rings. These constructions generalize many of the known existing constructions of MSRD codes as well as of optimal codes in the rank and in the Hamming metric. As a byproduct, in the case of finite fields we obtain new families of MDS codes which are linear over a subfield and whose length is close to the field size.

We consider the action of the projective group $PGL(3,\mathbb{R})$ on the $n$-fold first-order jet space of point configurations on the plane. Using the method of moving frames, we construct an explicit complete generating set for the field of absolute first-order joint projective differential invariants $\mathcal{I}_{n,0}$ for any $n \ge 3$. This approach provides a unified construction for all $n$, immediately ensuring functional independence of the fundamental invariants and yielding formulas suitable for both symbolic and numerical implementation. Next, we study the field of relative first-order invariants $\mathcal{I}_n$ with Jacobian multiplier. It is shown that the invariantization of the Jacobian under the projective action yields a primitive element of the field extension $\mathcal{I}_n / \mathcal{I}_{n,0}$. Finally, we introduce a multiplicative cochain complex $C^\bullet$ associated with the action of $PGL(3,\mathbb{R})$ on the jet space, and show that the invariantization operator induced by the moving frame generates an explicit contracting homotopy. This provides a constructive proof of the vanishing of higher cohomology and an interpretation of the "defect" of invariantization as an exact cocycle in $C^\bullet$.

Our constructions provide a systematic way to study cohomology pre-algebraic structures via classical cohomology, simplifying computations and enabling the use of established techniques.

Unlike the Hochschild chain complex of an algebra, the Hochschild cochain complex of an algebra is not functorial. Nonetheless, we show that the Hochschild cochain complex of an algebra even a dg category is of lax functoriality, i.e., there exists a lax functor from bicategory of dg categories to bicategory of $B_\infty$-algebras which sends every dg category to its Hochschild cochain complex. This result is a homotopy version of the lax functoriality of center of an algebra obtained by Davydov, Kong, Runkel, Grady, Oren, et al, in the more general context of dg categories, and extends the restricted functoriality of Hochschild cochain complex of a dg category obtained by Keller to global lax functoriality.

This paper aims to continue the studies initiated by Botha in [Linear Algebra Appl. 273 (1998), 65-82; Linear Algebra Appl. 286 (1999), 37-44; Linear Algebra Appl. 315 (2000), 1-23] by extending them to matrices over noncommutative division rings. In particular, we show that every such matrix can be written as either a sum or a product of two diagonalizable matrices. The number $2$ is not valid under mild conditions on the center, similar to those in Botha's work on fields. By applying this result and other results obtained so far, we latter establish some Waring-type results for matrices.

We investigate the relation between the notion of $e$-exactness, recently introduced by Akray and Zebary, and some functors naturally related to it, such as the functor $P\colon\operatorname{Mod} R\to \operatorname{Spec}(\operatorname{Mod} R)$, where $\operatorname{Spec}(\operatorname{Mod} R)$ denotes the spectral category of $\operatorname{Mod} R$, and the localization functor with respect to the singular torsion theory.

We study the analogue of the Hopkins-Levitzky Theorem for dg-algebras $(A,d)$. We first consider the Hopkins approach. Here we show that for acyclic dg-algebras with graded-Artinian algebras of cycles $\ker(d)$, we also have that $(A,d)$ is left dg-Noetherian, and we show that acyclic dg-Artinian dg-algebras are dg-Noetherian. Then, studying the Levitzki approach, we consider a definition of a dg-semiprimary algebra. For dg-semiprimary dg-artinian dg-algebras $(A,d)$, we show that all dg-simple dg-modules are acyclic, and so are all dg-modules with finite dg-composition length. We finally show that dg-Artinian dg-semiprimary dg-algebras with nilpotent dg-radical $dgrad_2(A,d)$ are dg-Noetherian and acyclic.

We show a certain existence of a lifting of modules under the self-$\mathrm{Ext}^2$-vanishing condition over the "derived quotient" by using the notion of higher algebra. This refines a work of Auslander-Ding-Solberg's solution of the Auslander-Reiten conjecture for complete interesctions. Together with Auslander's zero-divisor theorem, we show that the existence of such $\mathrm{Ext}$-vanishing module over derived quotients is equivalent to being local complete intersections.

As a generalization of the classical killing-contractible-complexes lemma, we present algebraic Morse theory via homological perturbation lemma, in a form more general than existing presentations in the literature. Two-sided Anick resolutions due to E.~Sköldberg are generalised to algebras given by quivers with relations and a minimality criterion is provided as well. Two applications of algebraic Morse theory are presented. It is shown that the Chinese algebra of rank $n\geq 1$ is homologically smooth and of global dimension $\frac{n(n+1)}{2}$, and the minimal two-sided projective resolution of a Koszul algebra is constructed.

We first construct an action of the extended double affine braid group $\mathcal{\ddot{B}}$ on the quantum toroidal algebra $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ in untwisted and twisted types. As a crucial step in the proof, we obtain a finite Drinfeld new style presentation for a broad class of quantum affinizations. In the simply laced cases, using our action and certain involutions of $\mathcal{\ddot{B}}$ we produce automorphisms and anti-involutions of $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ which exchange the horizontal and vertical subalgebras. Moreover, they switch the central elements $C$ and $k_{0}^{a_{0}}\dots k_{n}^{a_{n}}$ up to inverse. This can be viewed as the analogue, for these quantum toroidal algebras, of the duality for double affine braid groups used by Cherednik to realise the difference Fourier transform in his celebrated proof of the Macdonald evaluation conjectures. Our work generalises existing results in type $A$ due to Miki which have been instrumental in the study of the structure and representation theory of $U_{q}(\mathfrak{sl}_{n+1,\mathrm{tor}})$.

Let $\mathcal{L}$ be the noncrossing partition lattice associated to a finite Coxeter group $W$. In this paper we construct explicit bases for the top homology groups of intervals and rank-selected subposets of $\mathcal{L}$. We define a multiplicative structure on the Whitney homology of $\mathcal{L}$ in terms of the basis, and the resulting algebra has similarities to the Orlik-Solomon algebra. As an application, we obtain four chain complexes which compute the integral homology of the Milnor fibre of the reflection arrangement of $W$, the Milnor fibre of the discriminant of $W$, the hyperplane complement of $W$ and the Artin group of type $W$, respectively. We also tabulate some computational results on the integral homology of the Milnor fibres.

We introduce invariant algebras and representation$^{(c_1,..., c_8)}$ of algebras, and give many ways of constructing Lie algebras, Jordan algebras, Leibniz algebras, pre-Lie algebras and left-symmetric algebras in an invariant algebras.

Order and symmetry are main structural principles in mathematics. We give five examples where on the face of it order is not apparent, but deeper investigations reveal that they are governed by order structures. These examples are finite topologies, associative algebras, subgroups of matrix groups, ideals in polynomial rings, and classes of bipartite graphs.

We study automorphisms, isomorphisms, and derivations of the incidence algebra $I(X,R)$, where $X$ is preordered set and $R$ is an algebra over some commutative ring $T$.

Let $\Bbbk$ be an algebraically closed field of characteristic 0 and $H$ a finite-dimensional Hopf algebra over $\Bbbk$ with the dual Chevalley property. In this paper, we show that $\operatorname{gr}^c(H)$ is of tame corepresentation type if and only if $\operatorname{gr}^c(H)\cong (\Bbbk\langle x,y\rangle/I)^* \times H^\prime$ for some finite-dimensional semisimple Hopf algebra $H^\prime$ and some special ideals $I$. Then, by the method of link quiver and bosonization, we discuss which of the above ideals will occur when $(\Bbbk\langle x,y\rangle/I)^* \times H_0$ is a Hopf algebra of tame corepresentation type under some assumptions.

In 1971, Kac and Weisfeiler made two influential conjectures describing the dimensions of simple modules of a restricted Lie algebra $\mathfrak{g}$. The first predicts the maximal dimension of simple $\mathfrak{g}$-modules and in this paper we apply the Lefschetz principle and classical techniques from Lie theory to prove this conjecture for all restricted Lie subalgebras of $\mathfrak{gl}_n(k)$ whenever $k$ is an algebraically closed field of characteristic $p \gg n$. As a consequence we deduce that the conjecture holds for the the Lie algebra of a group scheme when specialised to an algebraically closed field of almost any characteristic. In the appendix to this paper, written by Akaki Tikaradze, a short proof of the first Kac--Weisfeiler conjecture is given for the Lie algebra of group scheme over a finitely generated ring $R \subseteq \mathbb{C}$, after base change to a field of large positive characteristic.

In 1971, Kac and Weisfeiler made two influential conjectures describing the dimensions of simple modules of a restricted Lie algebra $\mathfrak{g}$. The first predicts the maximal dimension of simple $\mathfrak{g}$-modules and in this paper we apply the Lefschetz principle and classical techniques from Lie theory to prove this conjecture for all restricted Lie subalgebras of $\mathfrak{gl}_n(k)$ whenever $k$ is an algebraically closed field of characteristic $p \gg n$. As a consequence we deduce that the conjecture holds for the the Lie algebra of a group scheme when specialised to an algebraically closed field of almost any characteristic. In the appendix to this paper, written by Akaki Tikaradze, a short proof of the first Kac--Weisfeiler conjecture is given for the Lie algebra of group scheme over a finitely generated ring $R \subseteq \mathbb{C}$, after base change to a field of large positive characteristic.