An integral domain (or a commutative cancellative monoid) is atomic if every nonzero nonunit element factors into atoms, and it satisfies the ACCP if every ascending chain of principal ideals ...eventually stabilizes. The interplay between these two properties has been investigated since the 1970s. An atomic domain (or monoid) satisfies the finite factorization property (FFP) if every element has only finitely many factorizations, and it satisfies the bounded factorization property (BFP) if for each element there is a bound for the number of atoms in each of its factorizations. These two properties have been systematically studied since they were introduced by Anderson, Anderson, and Zafrullah in 1990. Noetherian domains satisfy the BFP, while Dedekind domains satisfy the FFP. It is well known that, for commutative cancellative monoids (and, in particular, for multiplicative monoids of integral domains), FFP ⇒ BFP ⇒ ACCP ⇒ atomic. For n≥2, we show that each of these four properties transfers back and forth between an information semialgebra S (certain commutative cancellative semiring) and the multiplicative monoid Tn(S)• consisting of n×n upper triangular matrices over S. We also show that a similar transfer behavior takes place if one replaces Tn(S)• by its submonoid Un(S) consisting of upper triangular matrices with units along their main diagonals. As a consequence, we find that the atomic chain FFP ⇒ BFP ⇒ ACCP ⇒ atomic also holds for the two classes comprising the noncommutative monoids Tn(S)• and Un(S). Finally, we construct various rational information semialgebras to verify that, in general, none of the established implications is reversible.
Let K be a field (finite or infinite) of char(K)≠2 and let UT2(K) be the 2×2 upper triangular matrix algebra over K. If ⋅ is the usual product on UT2(K) then with the new product a∘b=(1/2)(a⋅b+b⋅a) ...we have that UT2(K) is a Jordan algebra, denoted by UJ2=UJ2(K). In this paper, we describe the set I of all polynomial identities of UJ2 and a linear basis for the corresponding relatively free algebra. Moreover, if K is infinite we prove that I has the Specht property. In other words I, and every T-ideal containing I, is finitely generated as a T-ideal.
Let F be an algebraically closed field of characteristic different from 2. We show that the images of multilinear ⁎-graded polynomials on UT2 are homogeneous vector spaces. An analogous result holds ...for UT3 endowed with non-trivial grading. We further show that these results are optimal, in the following sense: there exist multilinear ⁎-graded polynomials whose image on UTn (n≥3) with the trivial grading is not a vector space, and whose image on UTn (n≥4) with the natural Zn-grading is also not a vector space. In particular, an analog of the L'vov-Kaplansky conjecture can not be expected in the setting of algebras with (graded) involutions.
Maximum flag-rank distance codes Alfarano, Gianira N.; Neri, Alessandro; Zullo, Ferdinando
Journal of combinatorial theory. Series A,
October 2024, 2024-10-00, Letnik:
207
Journal Article
Recenzirano
Odprti dostop
In this paper we extend the study of linear spaces of upper triangular matrices endowed with the flag-rank metric. Such metric spaces are isometric to certain spaces of degenerate flags and have been ...suggested as suitable framework for network coding. In this setting we provide a Singleton-like bound which relates the parameters of a flag-rank-metric code. This allows us to introduce the family of maximum flag-rank distance codes, that are flag-rank-metric codes meeting the Singleton-like bound with equality. Finally, we provide several constructions of maximum flag-rank distance codes.
Let A be an associative algebra over a fixed field F of characteristic zero. In this paper we focus our attention on those algebras A graded by Z2, the cyclic group of order 2. In this case A is said ...to be a superalgebra and it can be decomposed in the direct sum of homogeneous subspaces: A=A0⊕A1. The main goal of this paper is to prove tight relations between some graded linear maps that can be defined on superalgebras, namely involutions, superinvolutions and pseudoinvolutions. Along the way, we shall present a classification of the pseudoinvolutions that one can define on the algebra UTn(F) of n×n upper-triangular matrices. In the final part of the paper we shall also give some consequences of these results in the context of the theory of polynomial identities.
Let K be a finite field and let UTn(K) be the algebra of n×n upper triangular matrices over K. In this paper we describe the set of all G-graded polynomial identities of UTn(K), where G is any group. ...Moreover, we describe a linear basis for the corresponding relatively free graded algebra.
Let F be a field of characteristic 0 and let UTm(F) be the algebra of upper triangular matrices of order m with entries from F. In this paper we give a description of the Y-proper graded cocharacters ...of UTm(F) equipped with any elementary G-grading, where G is a finite group.
We determine the differential polynomial identities of 3×3 upper triangular matrices over a base field of characteristic zero, under the action of its full Lie algebra of derivations. We compute the ...exact differential codimension sequence of the multilinear ones and describe their Sn-structure by means of an explicit decomposition of the Sn-cocharacter of their proper part.
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