Order topology in vector lattices with closure by one step

An order topology in vector lattices and Boolean algebras is studied under the additional condition of “closure by one step” that generalizes the well-known “regularity” property of Boolean algebras and K-spaces. It is proved that in a vector lattice or a Boolean algebra possessing such a property there exists a basis of solid neighborhoods of zero with respect to an order topology. An example of a Boolean algebra without basis of solid neighborhoods of zero (an algebra of regular open subsets of the interval (0, 1)) is given. Bibliography: 3 titles. Translated fromProblemy Matematicheskogo Analiza, No. 15 1995, pp. 213–220.     

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.     

Notes on piecewise-Koszul algebras

The relationships between piecewise-Koszul algebras and other “Koszul-type” algebras are discussed. The Yoneda-Ext algebra and the dual algebra of a piecewise-Koszul algebra are studied, and a sufficient condition for the dual algebra A ^! to be piecewise-Koszul is given. Finally, by studying the trivial extension algebras of the path algebras of Dynkin quivers in bipartite orientation, we give explicit constructions for piecewise-Koszul algebras with arbitrary “period” and piecewise-Koszul algebras with arbitrary “jump-degree”.     

Complexity of Boolean algebras and their Scott rank

We deal with problems associated with Scott ranks of Boolean algebras. The Scott rank can be treated as some measure of complexity of an algebraic system. Our aim is to propound and justify the procedure which, given any countable Boolean algebra, will allow us to construct a Boolean algebra of a small Scott rank that has the same natural algebraic complexity as has the initial algebra. In particular, we show that the Scott rank does not always serve as a good measure of complexity for the class of Boolean algebras. We also study into the question as to whether or not a Boolean algebra of a big Scott rank can be decomposed into direct summands with intermediate ranks. Examples are furnished in which Boolean algebras have an arbitrarily big Scott rank such that direct summands in them either have a same rank or a fixed small one, and summands of intermediate ranks are altogether missing. This series of examples indicates, in particular, that there may be no nontrivial mutual evaluations for the Scott and Frechet ranks on a class of countable Boolean algebras. Supported by RFFR grant No. 99-01-00485, by a grant for Young Scientists from SO RAN, 1997, and by the Federal Research Program (FRP) “Integration”. Translated fromAlgebra i Logika, Vol. 38, No. 6, pp. 643–666, November–December, 1999.     

Some models in differential geometry

Some variants of the axiomatics of the algebras of “vector fields” in models of noncommutative differential geometry are considered. In the case of a commutative model (the de Rham complex) a matrix analogue of the Kadomtsev-Petviashvili hierarchy is constructed. The corresponding Sato system is presented. The method of deformations of D-modules is used. Bibliography: 14 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 205, 1993, pp. 122–153. Translated by A. M. Nikitin.     

A propos du lemme fondamental pondéré tordu

In order to use the trace formula of Arthur–Selberg in the twisted case, we need to prove the “twisted weighted fundamental lemma”, that is a sophisticated version of the fundamental lemma. Here, we prove that this twisted weighted fundamental lemma follows from two others lemmas, where the torsion has disappeared: the weighted fundamental lemma for Lie algebras and a “non-standard weighted fundamental lemma”, concerning Lie algebras too.     

More constructions for Boolean algebras

 We construct Boolean algebras with prescribed behaviour concerning depth for the free product of two Boolean algebras over a third, in ZFC using pcf; assuming squares we get results on ultraproducts. We also deal with the family of cardinalities and topological density of homomorphic images of Boolean algebras (you can translate it to topology - on the cardinalities of closed subspaces); and lastly we deal with inequalities between cardinal invariants, mainly . Received: 9 September 1998 / Published online: 7 May 2002     

Relation algebras and function semigroups

The concept of relation algebra unifies many familiar notions from algebra (especially those of systems having “natural” models as groups, Boolean algebras etc.). The fundamental theorem on relation algebras asserts the existence of simple conditions which characterize any given class of relation algebras. The idea of relation algebra is very useful for the study of function and transformation semigroups, which is the central part of the theory of semigroups. It provides a general outlook, permits one to formulate many natural problems, and ensures that these problems possess non-trivial solutions. A number of examples illustrate this point.     

Generalized Robba rings

We prove that any projective coadmissible module over the locally analytic distribution algebra of a compact p-adic Lie group is finitely generated. In particular, the category of coadmissible modules does not have enough projectives. In the Appendix a “generalized Robba ring” for uniform pro-p groups is constructed which naturally contains the locally analytic distribution algebra as a subring. The construction uses the theory of generalized microlocalization of quasi-abelian normed algebras that is also developed there. We equip this generalized Robba ring with a selfdual locally convex topology extending the topology on the distribution algebra. This is used to show some results on coadmissible modules.     

A quantum analog of the Poincare-Birkhoff-Witt theorem

We reduce the basis construction problem for character Hopf algebras to a study of special elements, called “super-letters,” which are defined by Shirshov standard words. It is shown that character Hopf algebras having not more than finitely many “hard” super-letters share some of the properties of universal envelopings of finite-dimensional lie algebras. The background for our proofs is the construction of a filtration such that the associated graded algebra is obtained by iterating the skew polynomials construction, possibly followed with factorization. Supported by the National Society of Researchers, México (SNI, exp. 18740, 1997–2000) Translated fromAlgebra i Logika, Vol. 38, No. 4, pp. 476–507, July–August, 1999.     

Notes on δ-Koszul Algebras

This paper is a continuous work of δ-Koszul algebras, which were first introduced by Green and Marcos in 2005 (see Green and Marcos, Commun Algebra 33(6):1753–1764, 2005). Let K^d(A)\mathcal{K}^{\delta}(A) be the category of δ-Koszul modules. It is proved that K^d(A)\mathcal{K}^{\delta}(A) preserves kernels of epimorphisms if and only if the “minimal Horseshoe Lemma” (“MHL” for short) holds. Further, a special class of δ-Koszul algebras named periodic δ -algebras are introduced, which have close connection with Koszul algebras and provide answers to the questions raised by Green and Marcos (Commun Algebra 33(6):1753–1764, 2005). Finally, we construct new periodic δ-algebras from the given ones in terms of one-point extension and sum-extension.     

The $(1-\mathbb{E})$-transform in combinatorial Hopf algebras

We extend to several combinatorial Hopf algebras the endomorphism of symmetric functions sending the first power-sum to zero and leaving the other ones invariant. As a “transformation of alphabets”, this is the (1-\mathbbE)(1-\mathbb{E})-transform, where \mathbbE\mathbb{E} is the “exponential alphabet,” whose elementary symmetric functions are e_n=\frac1n!e_{n}=\frac{1}{n!}. In the case of noncommutative symmetric functions, we recover Schocker’s idempotents for derangement numbers (Schocker, Discrete Math. 269:239–248, 2003). From these idempotents, we construct subalgebras of the descent algebras analogous to the peak algebras and study their representation theory. The case of WQSym leads to similar subalgebras of the Solomon–Tits algebras. In FQSym, the study of the transformation boils down to a simple solution of the Tsetlin library in the uniform case.     

Invariants of the stable equivalence of symmetric special biserial algebras

The present paper is one in a series of papers devoted to the classification of some classes of tame algebras up to stable category equivalence. In this paper, we study symmetric algebras (their stable categories have a structure of triangulated categories) and the simplest class of tame algebras-the class of special biserial algebras (SB-algebras). In the paper, we give a relevant version of the “diagrammatic method” and study the structure of the triangulated category “in a neighborhood” of the periodic part (with respect to Ω) of the stable category. Thus we prove the invariance of the collection of lengths of G-cycles under equivalence of stable categories (see Theorem 2.12). Then we use the invariance stated above, together with some properties of the Cartan matrix of a symmetric SB-algebra, to prove that the number of A-cycles (but not their lengths!) is also an invariant of stable equivalence. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 330, 2006, pp. 5–28.     

Weyl algebras over quantum groups

The term “Weyl algebras” is proposed for differential algebras associated with dual pairs of Hopf algebras. The principle of complete reducibility for the category of “admissible” modules over Weyl algebras is proved. Comodule structures that connect Weyl algebras with the Drinfeld quantum double are investigated. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 118, No. 2, pp. 190–204, February, 1999.     

Non-embeddable simple relation algebras

This paper presents solutions or partial solutions for several problems in the theory of relation algebras. In a simple relation algebra an element x satisfying the condition (a) must be an atom of . It follows that x must also be an atom in every simple extension of . Andréka, Jónsson and Németi [1, Problem 4] (see [12, Problem P5]) asked whether the converse holds: if x is an atom in every simple extension of a simple relation algebra, must it satisfy (a)? We show that the answer is “no”.? The only known examples of simple relation algebras without simple proper extensions are the algebras of all binary relations on a finite set. Jónsson proposed finding all finite simple relation algebras without simple proper extensions [12, Problem P6]. We show how to construct many new examples of finite simple relation algebras that have no simple proper extensions, thus providing a partial answer for this second problem. These algebras are also integral and nonrepresentable.? Andréka, Jónsson, Németi [1, Problem 2] (see [12, Problem P7]) asked whether there is a countable simple relation algebra that cannot be embedded in a one-generated relation algebra. The answer is “yes”. Givant [3, Problem 9] asked whether there is some k such that every finitely generated simple relation algebra can be embedded in a k-generated simple relation algebra. The answer is “no”. Received November 27, 1996; accepted in final form July 3, 1997.     

Order Continuity of Locally Compact Boolean Algebras

The main purpose of this paper is to exhibit the decisive role that order continuity plays in the structure of locally compact Boolean algebras as well as in that of atomic topological Boolean algebras. We prove that the following three conditions are equivalent for a topological Boolean algebra B: (1) B is compact; (2) B is locally compact, Boolean complete, order continuous; (3) B is Boolean complete, atomic and order continuous. Note that under the discrete topology any Boolean algebra is locally compact.     

On algebras satisfying $x^2x^2=N(x)x$

The commutative algebras satisfying the “adjoint identity”: , where N is a cubic form, are shown to be related to a class of generically algebraic Jordan algebras of degree at most 4 and to the pseudo-composition algebras. They are classified under a nondegeneracy condition. As byproducts, the associativity of the norm of any pseudo-composition algebra is proven and the unital commutative and power-associative algebras of degree are shown to be Jordan algebras. Received January 26, 1999; in final form August 26, 1999 / Published online July 3, 2000     

The Loomis–Sikorski Theorem revisited

We prove, constructively, that the Loomis–Sikorski Theorem for σ-complete Boolean algebras follows from a representation theorem for Archimedean vector lattices and a constructive representation of Boolean algebras as spaces of Carathéodory place functions. We also prove a constructive subdirect product representation theorem for arbitrary partially ordered vector spaces. Received August 10, 2006; accepted in final form May 30, 2007.