arXiv subjects
Publications and source records attributed to Alberto Facchini.
In the category of digroups there is a natural pretorsion theory in which the torsion-free digroups are all groups, and torsion digroups form a category isomorphic to the category of non-empty sets. It is also possible to extend the theory of heaps from groups to digroups. The corresponding notion is that of a diheap.
A right group is a semigroup $(S,\cdot)$ in which, for every $a,b\in S$, there is a unique $x\in S$ such that $a\cdot x=b$. In this article, we develop the theory of heaps starting not from groups, but from right groups. We thus get a natural definition of right heap. It is even possible to develop part of the theory starting from a left quasigroup, which is the non-associative analogue of a right group. Our motivation for this study is the investigation of left non-degenerate set-theoretic solutions of the Yang--Baxter equation. Thus, we are led to an analogue of the skew left trusses introduced by T.~Brzezi\'nski.
In this paper, we provide techniques to obtain left non-degenerate set-theoretic solutions of the Yang-Baxter equation, drawing on the class of right groups. To this end, we introduce the new algebraic structures of left $RG$-semibraces, which include left (cancellative) semibraces as a proper subclass.
Let $S$ be a right group. Then there exist two congruences $\sim$ and $\equiv$ on $S$ such that $S$ is the product of its quotient semigroups $S/{\sim}$ and $S/{\equiv}$, where $S/{\sim}$ is a group and $S/{\equiv}$ is a right zero semigroup. If $E$ is the set of all idempotents of $S$ and we fix an element $e_0\in E$, then the pointed right group $(S,e_0)$ is the coproduct of its pointed subsemigroups $(Se_0,e_0)$ and $(E,e_0)$ in the category of pointed right groups. In general, there is a pretorsion theory in the category of right groups in which the torsion objects are right zero semigroups and the torsion-free objects are groups.
In this paper we extend to left skew trusses $(T,+,\circ,\sigma)$ previous work on left skew rings. We had presented a left skew ring as a group $(N,+)$ with two binary operations $\circ$ and $\cdot$ with $\circ$ associative, $\cdot$ left distributive over the addition $+$ of the group, and such that the difference of the two operations $\circ$ and $\cdot$ is the binary operation $\pi_1\colon N\times N\to N$. Here we extend this idea to the left skew trusses introduced in 2019 by Brzezi\'nski, replacing the operation $\pi_1$ with the binary operation $\sigma\pi_1\colon T\times T\to T$. The case where the semigroup morphism $\lambda^T\colon T\to \End_\Gp(T,+)$ is constant turns out to be particular interesting. We get several canonical category isomorphisms. For instance, we get a category isomorphism between the category of all left skew trusses $(T,+,\circ,\sigma)$ with $\lambda^T\colon (T,\circ)\to \End_\Gp(T,+)$ a constant semigroup morphism and $\sigma,\lambda^T_0$ image-commuting idempotent endomorphisms and the category of all associative interchange near-rings. Interchange near-rings were introduced by Edmunds in 2016. When $\sigma$ is an idempotent group endomorphism of the group $(T,+)$ and $\lambda^T\colon (T,\circ)\to \End_\Gp(T,+)$ is a semigroup morphism constantly equal to a group endomorphism $\tau$, we also get a sort of duality exchanging the mappings $\sigma$ and $\tau$.
We introduce a new point of view to present classical notions related to set-theoretic solutions of the Yang-Baxter equation: left skew braces, dirings, left skew rings. The idea is to replace the single multiplication on a left near-ring by two operations, one associative and the other left distributive. Two algebraic structures naturally appear: left skew rings and left weak rings, whose categories turn out to be canonically isomorphic.
We study the category $\operatorname{Morph}(\operatorname{Mod} R)$ whose objects are all morphisms between two right $R$-modules. The behavior of objects of $\operatorname{Morph}(\operatorname{Mod} R)$ whose endomorphism ring in $\operatorname{Morph}(\operatorname{Mod} R)$ is semilocal is very similar to the behavior of modules with a semilocal endomorphism ring. For instance, direct-sum decompositions of a direct sum $\oplus_{i=1}^nM_i$, that is, block-diagonal decompositions, where each object $M_i$ of $\operatorname{Morph}(\operatorname{Mod} R)$ denotes a morphism $\mu_{M_i}\colon M_{0,i}\to M_{1,i}$ and where all the modules $M_{j,i}$ have a local endomorphism ring $\operatorname{End}(M_{j,i})$, depend on two invariants. This behavior is very similar to that of direct-sum decompositions of serial modules of finite Goldie dimension, which also depend on two invariants (monogeny class and epigeny class). When all the modules $M_{j,i}$ are uniserial modules, the direct-sum decompositions (block-diagonal decompositions) of a direct-sum $\oplus_{i=1}^nM_i$ depend on four invariants.
We describe the endomorphism ring of a short exact sequences $0 \to A_R \to B_R \to C_R \to 0$ with $A_R$ and $C_R$ uniserial modules and the behavior of these short exact sequences as far as their direct sums are concerned.
We study the structure of the commutative multiplicative monoid $\mathbb N_0[x]^*$ of all the non-zero polynomials in $\mathbb Z[x]$ with non-negative coefficients. We show that $\mathbb N_0[x]^*$ is not a half-factorial monoid and is not a Krull monoid, but has a structure very similar to that of Krull monoids, replacing valuations into $\mathbb N_0$ with derivations into $\mathbb N_0$. We study ideals, chain of ideals, prime ideals and prime elements of $\mathbb N_0[x]^*$. Our monoid $\mathbb N_0[x]^*$ is a submonoid of the multiplicative monoid of the ring $\mathbb Z[x]$, which is a left module over the Weyl algebra $A_1(\mathbb Z)$.
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.
First of all, we recall the well known notion of semidirect product both for classical algebraic structures (like groups and rings) and for more recent ones (digroups, left skew braces, heaps, trusses). Then we analyse the concept of semidirect product for an arbitrary algebra $A$ in a variety $\cal{V}$ of type~$\cal{F}$. An inner semidirect-product decomposition $A=B \ltimes\omega$ of $A$ consists of a subalgebra $B$ of $A$ and a congruence $\omega$ on $A$ such that $B$ is a set of representatives of the congruence classes of $A$ modulo $\omega$. An outer semidirect product is the restriction to $B$ of a functor from a suitable category $\cal{C}_B$ containing $B$, called the enveloping category of $B$, to the category Set$_*$ of pointed sets.
We study commutators of congruences, idempotent endomorphisms and semidirect-product decompositions of heaps and trusses.
We link the recent theory of $L$-algebras to previous notions of Universal Algebra and Categorical Algebra concerning subtractive varieties, commutators, multiplicative lattices, and their spectra. We show that the category of $L$-algebras is subtractive and normal in the sense of Zurab Janelidze, but neither the category of $L$-algebras nor that of pre-$L$-algebras are Mal'tsev categories, hence in particular they are not semi-abelian. Therefore $L$-algebras are a rather peculiar example of an algebraic structure.
In the study of pre-Lie algebras, the concept of pre-morphism arises naturally as a generalization of the standard notion of morphism. Pre-morphisms can be defined for arbitrary (not-necessarily associative) algebras over any commutative ring $k$ with identity, and can be dualized in various ways to generalized morphisms (related to pre-Jordan algebras) and anti-pre-morphisms (related to anti-pre-Lie algebras). We consider idempotent pre-endomorphisms (generalized endomorphisms, anti-pre-endomorphisms). Idempotent pre-endomorphisms are related to semidirect-product decompositions of the sub-adjacent anticommutative algebra.
We study the notions of action, semidirect product and commutator of ideals for digroups and skew braces.
We introduce the notions of pre-morphism and pre-derivation for arbitrary non-associative algebras over a commutative ring $k$ with identity. These notions are applied to the study of pre-Lie $k$-algebras and, more generally, Lie-admissible $k$-algebras. Associating with any algebra $(A,\cdot)$ its sub-adjacent anticommutative algebra $(A,[-,-])$ is a functor from the category of $k$-algebras with pre-morphisms to the category of anticommutative $k$-algebras. We describe the commutator of two ideals of a pre-Lie algebra, showing that the condition (Huq=Smith) holds for pre-Lie algebras. This allows to make use of all the notions concerning multiplicative lattices in the study of the multiplicative lattice of ideals of a pre-Lie algebra. We study idempotent endomorphisms of a pre-Lie algebra $L$, i.e., semidirect-product decompositions of $L$ and bimodules over $L$.
The category of all $k$-algebras with a bilinear form, whose objects are all pairs $(R,b)$ where $R$ is a $k$-algebra and $b\colon R\times R\to k$ is a bilinear mapping, is equivalent to the category of unital $k$-algebras $A$ for which the canonical homomorphism $(k,1)\to(A,1_A)$ of unital $k$-algebras is a splitting monomorphism in the category of $k$-modules. Call the left inverses of this splitting monomorphism "weak augmentations" of the algebra. There is a category isomorphism between the category of $k$-algebras with a weak augmentation and the category of unital $k$-algebras $(A,b_A)$ with a bilinear form $b_A$ compatible with the multiplication of $A$, i.e., such that $b_A(x,y)=b_A(z,w)$ for all $x,y,z,w\in A$ for which $xy=zw$.
The goal of this paper is to deepen the study of multiplicative lattices in the sense of Facchini, Finocchiaro and Janelidze. We provide a sort of Prime Ideal Principle that guarantees that maximal implies prime in a variety of cases (among them the case of commutative rings with identity). This result is used to study the lattice theoretic counterpart of multiplicative closed sets, that of m-systems. The notion of m-system is also studied from the topological point of view.