Perfect Algebras Research Articles

Behaviour of the Frobenius map in a noncommutative world

We study various properties of the Frobenius map φ(x)=xp on a noncommutative algebra R over a field F of positive characteristic p. We call R perfect whenever φ is surjective. More generally, we say that R is additively (respectively, multiplicatively) p-power-closed if, for every x,y∈R, there exists a positive integer n such that xpn+ypn (respectively, xpnypn) is a p-power in R. If, for example, φn were a ring homomorphism (as when R is commutative), then R would be additively and multiplicatively p-power-closed. We show that a finite-dimensional unital algebra R is Lie nilpotent if and only if R is additively (respectively, multiplicatively) p-power-closed and each of its simple homomorphic images has Schur index 1. Since not all finite-dimensional perfect division algebras are Lie nilpotent, the Schur index condition cannot be omitted. We deduce that similar results hold for the class of all finitely generated PI-algebras. Moreover, for this same class, we give a positive solution to the following problem reminiscent of the problems of Kurosh and Levitzki: is every finitely generated perfect algebra finite-dimensional?

Thin and Thinner: Hypothesis-driven Research and the Study of Humans

AbstractSome say that researchers who study humans are locked in to frameworks of epistemic assumptions from which there can be no escape. We explain, on the contrary, how researchers who disagree may nevertheless reconcile their differences. Freedom from the epistemic dungeon is made possible by practices that convert beliefs into testable hypotheses, which are then tested. Such practices are the engines of scientific progress. To clarify misunderstandings about practices of hypothesis testing, we discuss Bayes’ rule, a mathematically perfect algebra for belief revision. To illustrate both the benefits and inevitable limitations of scientific research on religion, we work through the details of a recent national questionnaire study that revealed five different types of supernatural believers.

Countable dimensional right perfect ring

We prove that a right self right perfect algebra which is at most countable dimensional modulo their Jacobson radical is right artinian.

Graded Lie algebras and q-commutative and r-associative parameters

We study graded Lie algebras whose transformation parameters are graded q-commutativive and r-associative. We study first some graded algebras over a field, with no zero divisors at the level of monomials in their graded algebra generators. These generators are q-commutative and r-associative. We address the cohomology of the q-function and r-functions, in particular we study quaternions and octonions. We then define algebras whose transformation parameters are q-commutative and r-associative. We address a generalization of a theorem by Scheunert on its relation to Lie (super)algebras. We show finally that for the cases studied by Scheunert there is always a real and faithful transformation parameter basis with the required q-commutativity. We use this basis to perform a transformation on the graded Lie algebra that relates it to a plain Lie (super)algebra while respecting the self-adjoint character of generators and preserving the group grading. Keywords: Graded Lie (super)algebras, Color Lie (super)algebras, noncommutative algebras, nonassociative algebras, cohomology of deformation parameters, perfect algebra. AMS-MSC: 17B70, 17B75, 22E60, 17A99, 17D99, 13D03, 20J06.

Stability of the Blok Theorem

W. Blok proved that varieties of boolean algebras with a single unary operator uniquely determined by their class of perfect algebras (i.e., duals of Kripke frames) are exactly those which are intersections of conjugate varieties of splitting algebras. The remaining ones share their class of perfect algebras with uncountably many other varieties. This theorem is known as the Blok dichotomy or the Blok alternative. We show that the Blok dichotomy holds when perfect algebras in the formulation are replaced by ω-complete algebras, atomic algebras with completely additive operators or algebras admitting residuals. We also generalize the Blok dichotomy for lattices of varieties of boolean algebras with finitely many unary operators. In addition, we answer a question posed by W. Dziobiak and show that classes of lattice-complete algebras or duals of Scott-Montague frames in a given variety are not determined by their subdirectly irreducible members.

Homology with Trivial Coefficients and Universal Central Extensions of Algebras with Bracket

A 5-term exact sequence and the interpretation of low dimensional groups of homology with trivial coefficients of algebras with bracket are obtained. The endofunctor in the category of algebras with bracket which assigns to a perfect algebra with bracket its universal central extension is constructed and the characterization of the universal central extension by means of the low-dimensional homology groups is done. The conditions required to lift an automorphism or a derivation of A to A′ in a covering f:A′ ↠ A are analyzed.

Perfect effect algebras are categorically equivalent with Abelian interpolation po-groups

AbstractWe introduce perfect effect algebras and we show that every perfect algebra is an interval in the lexicographical product of the group of all integers with an Abelian directed interpolation po-group. To show this we introduce prime ideals of effect algebras with the Riesz decomposition property (RDP). We show that the category of perfect effect algebras is categorically equivalent to the category of Abelian directed interpolation po-groups. Moreover, we prove that any perfect effect algebra is a subdirect product of antilattice effect algebras with the RDP.

On Some Varieties of MTL-algebras

The study of perfect, local and bipartite IMTL-algebras presented in [29] is generalized in this paper to the general non-involutive case, i. e. to MTL-algebras. To this end we describe the radical of MTL-algebras and characterize perfect MTL-algebras as those for which the quotient by the radical is isomorphic to the two-element Boolean algebra, and a special class of bipartite MTL-algebras, BP0, as those for which the quotient by the radical is a Boolean algebra. We prove that BP0 is the variety generated by all perfect MTL-algebras and give some equational bases for it. We also introduce a new way to build MTL-algebras by adding a negation fixpoint to a perfect algebra and also by adding some set of points whose negation is the fixpoint. Finally, we consider the varieties generated by those algebras, giving equational bases for them, and we study which of them define a fuzzy logic with standard completeness theorem.

Perfect and bipartite IMTL-algebras and disconnected rotations of prelinear semihoops

IMTL logic was introduced in [12] as a generalization of the infinitely-valued logic of Lukasiewicz, and in [11] it was proved to be the logic of left-continuous t-norms with an involutive negation and their residua. The structure of such t-norms is still not known. Nevertheless, Jenei introduced in [20] a new way to obtain rotation-invariant semigroups and, in particular, IMTL-algebras and left-continuous t-norm with an involutive negation, by means of the disconnected rotation method. In order to give an algebraic interpretation to this construction, we generalize the concepts of perfect, bipartite and local algebra used in the classification of MV-algebras to the wider variety of IMTL-algebras and we prove that perfect algebras are exactly those algebras obtained from a prelinear semihoop by Jenei's disconnected rotation. We also prove that the variety generated by all perfect IMTL-algebras is the variety of the IMTL-algebras that are bipartite by every maximal filter and we give equational axiomatizations for it.

Gorenstein algebras, symmetric matrices, self-linked ideals, and symbolic powers

Inspired by recent work in the theory of central projections onto hypersurfaces, we characterize self-linked perfect ideals of grade 2 2 as those with a Hilbert–Burch matrix that has a maximal symmetric subblock. We also prove that every Gorenstein perfect algebra of grade 1 1 can be presented, as a module, by a symmetric matrix. Both results are derived from the same elementary lemma about symmetrizing a matrix that has, modulo a nonzerodivisor, a symmetric syzygy matrix. In addition, we establish a correspondence, roughly speaking, between Gorenstein perfect algebras of grade 1 1 that are birational onto their image, on the one hand, and self-linked perfect ideals of grade 2 2 that have one of the self-linking elements contained in the second symbolic power, on the other hand. Finally, we provide another characterization of these ideals in terms of their symbolic Rees algebras, and we prove a criterion for these algebras to be normal.