Canonical extensions and ultraproducts of polarities

Jónsson and Tarski’s notion of the perfect extension of a Boolean algebra with operators has evolved into an extensive theory of canonical extensions of lattice-based algebras. After reviewing this evolution we make two contributions. First it is shown that the failure of a variety of algebras to be closed under canonical extensions is witnessed by a particular one of its free algebras. The size of the set of generators of this algebra can be made a function of a collection of varieties and is a kind of Hanf number for canonical closure. Secondly we study the complete lattice of stable subsets of a polarity structure, and show that if a class of polarities is closed under ultraproducts, then its stable set lattices generate a variety that is closed under canonical extensions. This generalises an earlier result of the author about generation of canonically closed varieties of Boolean algebras with operators, which was in turn an abstraction of the result that a first-order definable class of Kripke frames determines a modal logic that is valid in its so-called canonical frames.     

Atom structures and Sahlqvist equations

This paper addresses the question for which varieties of boolean algebras with operators membership of an atomic algebra is determined by its atom structure . We prove a positive answer for conjugated Sahlqvist varieties; we also show that the conjugation condition is necessary. As a corollary to the positive result and a recent result by I. Hodkinson, we prove that the variety RRA of representable relation algebras, although canonical, cannot be axiomatized by Sahlqvist equations. Received February 21, 1996; accepted in final form October 1, 1997.     

Amalgamation in finite dimensional cylindric algebras

For every finite n > 1, the embedding property fails in the class of all n-dimensional cylindric type algebras which satisfy the following. Their boolean reducts are boolean algebras and two of the cylindrifications are normal, additive and commute. This result also holds for all subclasses containing the representable n-dimensional cylindric algebras. This considerably strengthens a result of S. Comer on CA _n and provides a strong counterexample for interpolation in finite variable fragments of first order logic. We provide a new modern proof, using an argument inspired by modal logic. February 22, 1999.     

Twisted Hamiltonian Lie algebras and their multiplicity-free representations

We construct a class of new Lie algebras by generalizing the one-variable Lie algebras generated by the quadratic conformal algebras (or corresponding Hamiltonian operators) associated with Poisson algebras and a quasi-derivation found by Xu. These algebras can be viewed as certain twists of Xu’s generalized Hamiltonian Lie algebras. The simplicity of these algebras is completely determined. Moreover, we construct a family of multiplicity-free representations of these Lie algebras and prove their irreducibility.     

Relation algebras as expanded FL-algebras

This paper studies generalizations of relation algebras to residuated lattices with a unary De Morgan operation. Several new examples of such algebras are presented, and it is shown that many basic results on relation algebras hold in this wider setting. The variety qRA of quasi relation algebras is defined and shown to be a conservative expansion of involutive FL-algebras. Our main result is that equations in qRA and several of its subvarieties can be decided by a Gentzen system, and that these varieties are generated by their finite members.     

State operators on generalizations of fuzzy structures

We enlarge the language of R?-monoids, which are a non-commutative generalizations of both MV algebras and BL algebras, by adding a unary operation that describes algebraic properties of a state (= an analog of probability measures). The resulting algebras are called stateR?-monoids and state-morphismR?-monoids. We present basic properties of such algebras. We describe subdirectly irreducible algebras, some generators of the varieties of state-morphism R?-monoids, and an interplay between states and state operators.     

Abelian Logic and the Logics of Pointed Lattice-Ordered Varieties

We consider the class of pointed varieties of algebras having a lattice term reduct and we show that each such variety gives rise in a natural way, and according to a regular pattern, to at least three interesting logics. Although the mentioned class includes several logically and algebraically significant examples (e.g. Boolean algebras, MV algebras, Boolean algebras with operators, residuated lattices and their subvarieties, algebras from quantum logic or from depth relevant logic), we consider here in greater detail Abelian ℓ-groups, where such logics respectively correspond to: i) Meyer and Slaney’s Abelian logic [31]; ii) Galli et al.’s logic of equilibrium [21]; iii) a new logic of “preservation of truth degrees”. This paper was written while the second author was a Visiting Professor in the Department of Education at the University of Cagliari. The facilities and assistance provided by the University and by the Department are gratefully acknowledged.     

New Hardy Spaces Associated with Herz Spaces and Beurling Algebras on Homogeneous Groups

The author introduces the Hardy spaces associated with the Herz spaces and the Beurling algebras on homogeneous groups and establishes their atomic decomposition characterizations. As the applications of this decomposition, the duals of these Hardy spaces and the boundedness of the central δ-Calderón-Zygmund operators on these Hardy spaces are studied. Received January 13, 2000, Accepted July 26, 2000     

On free products in varieties of associative algebras

Varieties of associative algebras over a field of characteristic zero are considered. Belov recently proved that, in any variety of this kind, the Hilbert series of a relatively free algebra of finite rank is rational. At the same time, for three important varieties, namely, those of algebras with zero multiplication, of commutative algebras, and of all associative algebras, a stronger assertion holds: for these varieties, formulas that rationally express the Hilbert series of the free product algebra via the Hilbert series of the factors are well known. In the paper, a system of counterexamples is presented which shows that there is no formula of this kind in any other variety, even in the case of two factors one of which is a free algebra. However, if we restrict ourselves to the class of graded PI-algebras generated by their components of degree one, then there exist infinitely many varieties for each of which a similar formula is valid. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 693–702, May, 1999.     

On the Partially Ordered Semigroup Generated by the Class Operators I,R,H,S,P

Let I, H, S, P be the usual class operators on universal algebras. For a class K of universal algebras of the same type, let R({K}) be the class of all algebras isomorphic to a retract of a member of K and let R denote the corresponding class operator. In this paper the semigroup generated by class operators I, R, H, S, P and the corresponding partially ordered set are described. Also the standard semigroups of the above operators are determined for some varieties.     

The self-injective cluster-tilted algebras

We are going to determine the self-injective cluster-tilted algebras. All are of finite representation type and special biserial. There are two different classes. The first class are the self-injective serial (or Nakayama) algebras with n ≥ 3 simple modules and Loewy length n–1. The second class of algebras has an even number 2m of simple modules; m indecomposable projective modules have length 3, the remaining m have length m + 1. Received: 28 May 2007     

Varieties of Algebras without the Amalgamation Property

Let α be an ordinal and κ be a cardinal, both infinite, such that κ ≤ |α|. For τ ∈^αα, let sup(τ) = {i ∈ α: τ(i) ≠ i}. Let G _κ = {τ ∈^αα: |sup(τ)| < κ}. We consider variants of polyadic equality algebras by taking cylindrifications on Γ ? α, |Γ| < κ and substitutions restricted to G _κ. Such algebras are also enriched with generalized diagonal elements. We show that for any variety V containing the class of representable algebas and satisfying a finite schema of equations, V fails to have the amalgamation property. In particular, many varieties of Halmos’ quasi-polyadic equality algebras and Lucas’ extended cylindric algebras (including that of the representable algebras) fail to have the amalgamation property.     

Endoprimal algebras

An algebra A is endoprimal if, for all the only maps from A ^k to A which preserve the endomorphisms of A are its term functions. One method for finding finite endoprimal algebras is via the theory of natural dualities since an endodualisable algebra is necessarily endoprimal. General results on endoprimality and endodualisability are proved and then applied to the varieties of sets, vector spaces, distributive lattices, Boolean algebras, Stone algebras, Heyting algebras, semilattices and abelian groups. In many classes the finite endoprimal algebras turn out to be endodualisable. We show that this fails in general by proving that , regarded as either a bounded semilattice or upper-bounded semilattice is dualisable, endoprimal but not endodualisable. Received May 16, 1997; accepted in final form November 6, 1997.     

Piecewise-Koszul algebras

It is a small step toward the Koszul-type algebras.The piecewise-Koszul algebras are, in general,a new class of quadratic algebras but not the classical Koszul ones,simultaneously they agree with both the classical Koszul and higher Koszul algebras in special cases.We give a criteria theorem for a graded algebra A to be piecewise-Koszul in terms of its Yoneda-Ext algebra E(A),and show an A_∞-structure on E(A).Relations between Koszul algebras and piecewise-Koszul algebras are discussed.In particular,our results are related to the third question of Green-Marcos.     

Monotone clones and the varieties they determine

A finite, nontrivial algebra is order-primal if its term functions are precisely the monotone functions for some order on the underlying set. We show that the prevariety generated by an order-primal algebra P is relatively congruence-distributive and that the variety generated by P is congruence-distributive if and only if it contains at most two non-ismorphic subdirectly irreducible algebras. We also prove that if the prevarieties generated by order-primal algebras P and Q are equivalent as categories, then the corresponding orders or their duals generate the same order variety. A large class of order-primal algebras is described each member of which generates a variety equivalent as a category to the variety determined by the six-element, bounded ordered set which is not a lattice. These results are proved by considering topological dualities with particular emphasis on the case where there is a monotone near-unanimity function.This research was carried out while the third author held a research fellowship at La Trobe University supported by ARGS grant B85154851. The second author was supported by a grant from the NSERC.     

Varieties of affine Kac-Moody algebras

In this paper the identities of the complex affine Kac-Moody algebras are studied. It is proved that the identities of twisted affine algebras coincide with those of the corresponding nontwisted algebras. Moreover, in the class of nontwisted affine Kac-Moody algebras, each of these algebras is uniquely defined by its identities. It is shown that the varieties of affine algebras, as well as the varieties defined by finitely generated three-step solvable Lie algebras, have exponential growth. Translated fromMatematicheskie Zametki, Vol. 62 No. 1, pp. 95–102, July 1997. Translated by A. I. Shtern     

Convex Potentials and Optimal Shift Generated Oblique Duals in Shift Invariant Spaces

We introduce extensions of the convex potentials for finite frames (e.g. the frame potential defined by Benedetto and Fickus) in the framework of Bessel sequences of integer translates of finite sequences in \(L^2(\mathbb {R}^k)\). We show that under a natural normalization hypothesis, these convex potentials detect tight frames as their minimizers. We obtain a detailed spectral analysis of the frame operators of shift generated oblique duals of a fixed frame of translates. We use this result to obtain the spectral and geometrical structure of optimal shift generated oblique duals with norm restrictions, that simultaneously minimize every convex potential; we approach this problem by showing that the water-filling construction in probability spaces is optimal with respect to submajorization (within an appropriate set of functions) and by considering a non-commutative version of this construction for measurable fields of positive operators.     

On Fomin–Kirillov algebras for complex reflection groups

In this note, we apply classification results for finite-dimensional Nichols algebras to generalizations of Fomin–Kirillov algebras to complex reflection groups. First, we focus on the case of cyclic groups where the corresponding Nichols algebras are only finite-dimensional up to order four, and we include results about the existence of Weyl groupoids and finite-dimensional Nichols subalgebras for this class. Second, recent results by Heckenberger–Vendramin [ArXiv e-prints, 1412.0857 (December 2014)] on the classification of Nichols algebras of semisimple group type can be used to find that these algebras are infinite-dimensional for many non-exceptional complex reflection groups in the Shephard–Todd classification.