Existence of partial transposition means representability in cylindric algebras

We show that the representability of cylindric algebras by relativized set algebras depends on the scope of the operation transposition which can be defined on the algebra. The existence of “partial transposition” assures this kind of representability of the cylindric algebra (while the existence of transposition assures polyadic representation). Further we characterize those cylindric algebras in which the operator transposition can be introduced (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The class of infinite dimensional neat reducts of quasi‐polyadic algebras is not axiomatizable

SC, CA, QA and QEA denote the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasi‐polyadic algebras and quasi‐polyadic equality algebras, respectively. Let ω ≤ α < β and let K ∈ {SC,CA,QA,QEA}. We show that the class of α ‐dimensional neat reducts of algebras in K_β is not elementary. This solves a problem in [3]. Also our result generalizes results proved in [2] and [3]. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the representability of neatly embeddable CA’s by cylindric relativized algebras

Neat embedding theorems yield an abstract algebraic characterization for the representability of a given class of algebras by set algebras. Resek and Thompson’s theorem called attention to a new kind of representation in the theory of cylindric algebras, to the representation by cylindric relativised set algebras. In this paper, we present the algebraic characterization of this kind of representation; we formulate neat embedding theorems for this representation.     

Cylindric algebras and finite polyadic algebras

In this survey paper the short history of cylindric and finitary polyadic algebras (term-definitionally equivalent to quasi-polyadic algebras) is sketched, and the two concepts are compared. Roughly speaking, finitary polyadic algebras constitute a subclass of cylindric algebras that include a transposition operator being strong enough. We discuss the following question: should the definition of cylindric algebras include a transposition operator? Results confirm that the existence of a transposition operator ensures representability (by relativised set algebras). The different variants of cylindric algebras including a transposition operator play an important role in the theory of cylindric-like algebras.     

Comparing decision problem for various paradigms of algebraic logic

We show that in many cases the decision problems for varieties of cylindric algebras are much harder than those for the corresponding relation algebra reducts. We also give examples of varieties of cylindric and relation algebras which are algorithmically more complicated than the subvarieties of their representable algebras.     

PBW deformations of quadratic monomial algebras

A result of Braverman and Gaitsgory from 1996 gives necessary and sufficient conditions for a filtered algebra to be a Poincaré-Birkhoff-Witt (PBW) deformation of a Koszul algebra. The main theorem in this paper establishes conditions equivalent to the Braverman-Gaitsgory Theorem to efficiently determine PBW deformations of quadratic monomial algebras. In particular, a graphical interpretation is presented for this result, and we discuss circumstances under which some of the conditions of this theorem need not be checked. Several examples are also provided. Finally, with these tools, we show that each quadratic monomial algebra admits a nontrivial PBW deformation.     

On neat embeddings of cylindric algebras

We show that certain properties of dimension complemented cylindric algebras, concerning neat embeddings, do not generalize much further. Let α ≥ ω. There are non‐isomorphic representable cylindric algebras of dimension α each of which is a generating subreduct of the same β dimensional cylindric algebra. We also show that there exists a representable cylindric algebra ?? of dimension α, such that ?? is a generating subreduct of ?? and ??′, both in CA_{α +ω} , however ?? and ??′ are not isomorphic. This settle questions raised by Henkin, Monk and Tarski (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The number of one-generated diagonal-free cylindric set algebras of finite dimension greater than two

S. Ulam asked about the number of non-isomorphic projective algebras withk generators. This paper answers his question for projective algebras of finite dimension at least three and shows that there are the maximum possible number, continuum many, of non-isomorphic one-generated structures of finite dimensionn, wheren is at least three, of the following kinds: projective set algebras, projective algebras, diagonal-free cylindric set algebras, diagonal-free cylindric algebras, and cylindric algebras.     

Factorisations des extensions galoisiennes

ABSTRACT

In general, Clifford algebras of quadratic forms are finite dimensional; therefore, their representations are easy to describe. However, for homogenous polynomial forms of degree dbm > 2, the situation is different because their Clifford algebras are infinite dimensional. In this article, we get a finite set of pairwise orthogonal idempotents of sum 1 in these algebras. This permits us to obtain interesting properties for d-dimensional representations of polynomial forms of degree d; for example, we show that the image C of the Clifford algebra by such representation is an endomorphism algebra of finitely generated projective Z(C)-module of d-rank, direct sum of finitely generated projective Z(C)-module of 1-rank. Before establishing this, we give a new proof of the Poincaré-Birkhoff-Witt theorem for these algebras with the help of a general composition lemma. At the end of this work, we give a linearization of diagonal binary and ternary forms of degree dbm > 3.     

A note on substitutions in representable cylindric algebras

We show that it is impossible to define a substitution operator for arbitrary representable cylindric algebras that agrees in its basic properties with the notion of substitutions introduced for dimension complemented algebras (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Notes on “A survey of fuzzy implication algebras and their axiomatization”

In this note, we present a correction of Theorem 2 in the paper [1], and show that all the conditions (I1)–(I5) in the definition of fuzzy implication algebras are independent of each other. In addition, we prove that the class of all commutative fuzzy implication algebras forms an equational algebra class.     

Equations des algèbres Lie triple qui sont des algèbres train

In this paper, we consider equations of Lie triple algebras that are train algebras. We obtain two different types of equations depending on assuming the existence of an idempotent or a pseudo-idempotent.In general Lie triple algebras are not power-associative. However we show that their train equation with an idempotent is similar to train equations of power-associative algebras that are train algebras and we prove that Lie triple algebras that are train algebras of rank $\le 4$ with an idempotent are Jordan algebras.Moreover, the set of non-trivial idempotents has the same expression in Peirce decomposition as that of $e$-stable power-associative algebras.We also prove that the algebra obtained by $2$-gametization process of a Lie triple algebra is a Lie triple one.     

On Quasi-Bicrossed Products of Weak Hopf Algebras

The aim of this paper is to study quasi-bicrossed products and a especially quantum quasi-doubles. Firstly, we construct one new kind of quasi-bicrossed products by weak Hopf algebras and then devote a brief discussion to this matter. And, we discuss the conditions for quasi-bicrossed products to possess the structure of almost weak Hopf algebras, containing the case of a special smash product. At the end, we give some properties on the quantum quasi-double, respectively on the quasi-R-isomorphism, the representation-theoretic interpretation and the regularity of the quasi-R-matrix.     

Full field algebras, operads and tensor categories

We study the operadic and categorical formulations of (conformal) full field algebras. In particular, we show that a grading-restricted R×R-graded full field algebra is equivalent to an algebra over a partial operad constructed from spheres with punctures and local coordinates. This result is generalized to conformal full field algebras over VL⊗VR, where VL and VR are two vertex operator algebras satisfying certain finiteness and reductivity conditions. We also study the geometry interpretation of conformal full field algebras over VL⊗VR equipped with a nondegenerate invariant bilinear form. By assuming slightly stronger conditions on VL and VR, we show that a conformal full field algebra over VL⊗VR equipped with a nondegenerate invariant bilinear form exactly corresponds to a commutative Frobenius algebra with a trivial twist in the category of VL⊗VR-modules. The so-called diagonal constructions [Y.-Z. Huang, L. Kong, Full field algebras, arXiv: math.QA/0511328] of conformal full field algebras are given in tensor-categorical language.     

Axiomatizability of reducts of algebras of relations

In this paper, we prove that any subreduct of the class of representable relation algebras whose similarity type includes intersection, relation composition and converse is a non-finitely axiomatizable quasivariety and that its equational theory is not finitely based. We show the same result for subreducts of the class of representable cylindric algebras of dimension at least three whose similarity types include intersection and cylindrifications. A similar result is proved for subreducts of the class of representable sequential algebras. Received October 7, 1998; accepted in final form September 10, 1999.     

Gelfand–Kirillov dimension of generalized Weyl algebras (In memory of Guenter Rudolf Krause (1941–2015))

Given a generalized Weyl algebra A of degree 1 with the base algebra D, we prove that the difference of the Gelfand–Kirillov dimension of A and that of D could be any positive integer or infinity. Under mild conditions, this difference is exactly 1. As applications, we calculate the Gelfand–Kirillov dimensions of various algebras of interest, including the (quantized) Weyl algebras, ambiskew polynomial rings, noetherian (generalized) down-up algebras, iterated Ore extensions, quantum Heisenberg algebras, universal enveloping algebras of Lie algebras, quantum GWAs, etc.     

Monoidal Hom–Hopf Algebras

Hom-structures (Lie algebras, algebras, coalgebras, Hopf algebras) have been investigated in the literature recently. We study Hom-structures from the point of view of monoidal categories; in particular, we introduce a symmetric monoidal category such that Hom-algebras coincide with algebras in this monoidal category, and similar properties for coalgebras, Hopf algebras, and Lie algebras.     

On the category of hyper MV‐algebras

In this paper we study the category of hyper MV‐algebras and we prove that it has a terminal object and a coequalizer. We show that Jia's construction can be modified to provide a free hyper MV‐algebra by a set. We use this to show that in the category of hyper MV‐algebras the monomorphisms are exactly the one‐to‐one homomorphisms. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

广义弱群像代数

本文首先介绍了广义弱群像代数的概念, 并构造了一系列具体的例子, 通过例子可以看出有限群胚代数和群像代数都可以看作是广义弱群像代数; 接着本文证明了广义弱群像代数也可以看作是一类广义弱双Frobenius代数, 并探讨了广义弱双Frobenius代数成为有限群胚代数的条件; 最后本文给出了低维广义弱群像代数的具体结构.     

标度广义效应代数和标度效应代数的结构

研究了标度广义效应代数与标度效应代数的代数结构,给出了比较完整的结果.通过引入全标度广义代数的概念,本文证明了区间[0,1)上的标度广义效应代数和单位区间[0,1]上的标度效应代数完全由单位区间上的阿基米德余模确定,标度广义效应代数恰同构于全标度广义代数的下集.若标度广义代数满足局部有限条件,则它同构于实数加法群的子群代数.满足(S)条件的标度效应代数同构于实数加法群的子群代数和全标度广义代数的字典序乘积的子代数.