Boolean topological graphs of semigroups: the lack of first-order axiomatization

The graph of an algebra A is the relational structure G(A) in which the relations are the graphs of the basic operations of A. For a class ?? of algebras let G(??)={G(A)∣A∈??}. Assume that ?? is a class of semigroups possessing a nontrivial member with a neutral element and let ? be the universal Horn class generated by G(??). We prove that the Boolean core of ?, i.e., the topological prevariety generated by finite members of ? equipped with the discrete topology, does not admit a first-order axiomatization relative to the class of all Boolean topological structures in the language of ?. We derive analogous results when ?? is a class of monoids or groups with a nontrivial member.     

Fully invariant and verbal congruence relations

A congruence relation θ on an algebra A is fully invariant if every endomorphism of A preserves θ. A congruence θ is verbal if there exists a variety ${\mathcal{V}}$ such that θ is the least congruence of A such that ${{\bf A}/\theta \in \mathcal{V}}$ . Every verbal congruence relation is known to be fully invariant. This paper investigates fully invariant congruence relations that are verbal, algebras whose fully invariant congruences are verbal, and varieties for which every fully invariant congruence in every algebra in the variety is verbal.     

Monadic MV-algebras I: a study of subvarieties

In this paper, we study and classify some important subvarieties of the variety of monadic MV-algebras. We introduce the notion of width of a monadic MV-algebra and we prove that the equational class of monadic MV-algebras of finite width k is generated by the monadic MV-algebra [0, 1] ^k . We describe completely the lattice of subvarieties of the subvariety ${\mathcal{V}([{\bf 0}, {\bf 1}]^k)}$ generated by [0, 1] ^k . We prove that the subvariety generated by a subdirectly irreducible monadic MV-algebra of finite width depends on the order and rank of ?A, the partition associated to A of the set of coatoms of the boolean subalgebra B(A) of its complemented elements, and the width of the algebra. We also give an equational basis for each proper subvariety in ${\mathcal{V}([{\bf 0}, {\bf 1}]^k)}$ . Finally, we give some results about subvarieties of infinite width.     

Finitely Related Clones and Algebras with Cube Terms

Aichinger et al. (2011) have proved that every finite algebra with a cube-term (equivalently, with a parallelogram-term; equivalently, having few subpowers) is finitely related. Thus finite algebras with cube terms are inherently finitely related??every expansion of the algebra by adding more operations is finitely related. In this paper, we show that conversely, if A is a finite idempotent algebra and every idempotent expansion of A is finitely related, then A has a cube-term. We present further characterizations of the class of finite idempotent algebras having cube-terms, one of which yields, for idempotent algebras with finitely many basic operations and a fixed finite universe A, a polynomial-time algorithm for determining if the algebra has a cube-term. We also determine the maximal non-finitely related idempotent clones over A. The number of these clones is finite.     

MV-closures of Wajsberg hoops and applications

In this paper we construct, given a Wajsberg hoop A, an MV-algebra MV(A) such that the underlying set A of A is a maximal filter of MV(A) and the quotient MV(A)/A is the two element chain. As an application we provide a topological duality for locally finite Wajsberg hoops based on a previously known duality for locally finite MV-algebras. We also give another duality for k-valued Wajsberg hoops based on a different representation of k-valued MV-algebras and show the relation to the first duality. We also apply this construction to give a topological representation for free k-valued Wajsberg hoops.     

Euler method vs. GESS method for dynamical systems

In this paper we introduce GESS method and show that dynamics of the systemy′ =A(s,t,y)y is more faithfully approximated by GESS method than by Euler method. Numerical experiments are given for the comparison of GESS method with Euler method.     

Growth rates of algebras,III: finite solvable algebras

We investigate how the behavior of the function d_A(n), which gives the size of a least size generating set for Aⁿ, influences the structure of a finite solvable algebra A.     

Hankel operators on planar domains

Let Ω be a bounded domain in the plane whose boundary consists of a finite number of disjoint analytic simple closed curves LetA denote the space of analytic functions on Ω which are square integrable over Ω with respect to area measure and letP denote the orthogonal projection ofL ²(Ω,dA) ontoA. A functionb inA induces a Hankel operator (densely defined) onA by the ruleH _b (g)=(I?P)bg. This paper continues earlier investigations of the authors and others by determining conditions under whichH _b is bounded, compact, or lies in the Schatten-von Neumann idealS _p , 1<p<∞     

The properties of orthogonal matrix-valued wavelet packets in higher dimensions

In the paper matrix-valued multiresolution analysis and matrix-valued wavelet packets of spaceL ²(R ⁿ ,C ^{s x s}) are introduced. A procedure for constructing a class of matrix-valued wavelet packets in higher dimensions is proposed. The properties for the matrix-valued multivariate wavelet packets are investigated by using integral transform, algebra theory and operator theory. Finally, a new orthonormal basis ofL ²(R ⁿ ,C ^{s x s}) is derived from the orthogonal multivariate matrix-valued wavelet packets.     

BGP-reflection functors and Lusztig’s symmetries of modified quantized enveloping algebras

Let U be a quantized enveloping algebra and  its modified form.Lusztig gives some symmetries on U and.In view of the realization of U by the reduced Drinfeld double of the Ringel-Hall algebra,one can apply the BGP-reflection functors to the double Ringel-Hall algebra to obtain Lusztig's symmetries on U and their important properties,for instance,the braid relations.In this paper,we define a modified form ■ of the Ringel-Hall algebra and realize the Lusztig's symmetries on  by applying the BGP-reflection functors to ■.     

Weakly Koszul-like modules

The Koszul-like property for any finitely generated graded modules over a Koszul-like algebra is investigated and the notion of weakly Koszul-like module is introduced. We show that a finitely generated graded module M is a weakly Koszul-like module if and only if it can be approximated by Koszul-like graded submodules, which is equivalent to the fact that G(M) is a Koszul-like module, where G(M) denotes the associated graded module of M. As applications, the relationships between minimal graded projective resolutions of M and G(M), and Koszul-like submodules are established. Moreover, the Koszul dual of a weakly Koszul-like module is proved to be generated in degree 0 as a graded E(A)-module.     

Sequences of Commutator Operations

Given the congruence lattice ${{\mathbb{L}}}$ of a finite algebra A with a Mal’cev term, we look for those sequences of operations on ${{\mathbb{L}}}$ that are sequences of higher commutator operations of expansions of A. The properties of higher commutators proved so far delimit the number of such sequences: the number is always at most countably infinite; if it is infinite, then ${{\mathbb{L}}}$ is the union of two proper subintervals with nonempty intersection.     

The Homomorphism Lattice Induced by a Finite Algebra

Each finite algebra A induces a lattice L _A via the quasi-order → on the finite members of the variety generated by A, where B →C if there exists a homomorphism from B to C. In this paper, we introduce the question: ‘Which lattices arise as the homomorphism lattice L _A induced by a finite algebra A?’ Our main result is that each finite distributive lattice arises as L _Q , for some quasi-primal algebra Q. We also obtain representations of some other classes of lattices as homomorphism lattices, including all finite partition lattices, all finite subspace lattices and all lattices of the form L ⊕1, where L is an interval in the subgroup lattice of a finite group.     

Modules over group rings of solvable groups with rank restrictions on subgroups

Let A be an R G-module over a commutative ring R, where G is a group of infinite section p-rank (0-rank), C _G (A) = 1, A is not a Noetherian R-module, and the quotient A/C _A (H) is a Noetherian R-module for every proper subgroup H of infinite section p-rank (0-rank). We describe the structure of solvable groups G of this type.     

Some classes of pre-Hilbert algebras with norm-one central idempotent

In this paper we first characterize the pre-Hilbert algebras with a norm-one central idempotent e such that ‖ex‖ = ‖x‖ for any x ∈ A. This generalizes a well-known theorem by Ingelstam asserting that every alternative pre-Hilbert algebra with a unit 1 such that ‖1‖ = 1 is isomorphic to ?, ?, ? or $\mathbb{O}$ . We also show that every power-associative pre-Hilbert algebra satisfying ‖x ²‖ = ‖x‖² for every element has a unique nonzero idempotent, which is a unit element. In fact, the same conclusion will be proved in a more general setting. As application we give some conditions characterizing when a real algebra A, which is a prehilbert space, is isomorphic to one of the Hilbert algebras ?, ?, ? or $\mathbb{O}$ .     

Sylow-metacyclic groups andQ-admissibility

A finite groupG isQ-admissible if there exists a division algebra finite dimensional and central overQ which is a crossed product forG. AQ-admissible group is necessarily Sylow-metacyclic (all its Sylow subgroups are metacyclic). By means of an investigation into the structure of Sylow-metacyclic groups, the inverse problem (is every Sylow-metacyclic groupQ-admissible?) is essentially reduced to groups of order 2 ^a 3 ^b and to a list of known “almost simple” groups.     

Iterated vanishing cycles

If A ^? is a bounded, constructible complex of sheaves on a complex analytic space X, and ${f : X \rightarrow \mathbb{C}}$ and ${g : X \rightarrow \mathbb{C}}$ are complex analytic functions, then the iterated vanishing cycles φ _g [?1](φ _f [?1]A ^?) are important for a number of reasons. We give a formula for the stalk cohomology H*(φ _g [?1]φ _f [?1]A ^?) _x in terms of relative polar curves, algebra, and Morse modules of A ^?.     

Weighted norm inequalities for the vectorvalued Hardy-Littlewood maximal functions of one parameter rectangles

Let N denote the Hardy-Littlewood maximal operator for the familyR of one parameter rectangles. In this paper, we obtain that for 1 _w ^p (l^r) to L _W ^P (l^r) if and only if w ∈ A_P(R); for 1≤p<∞, N is bounded from L _W ^P (l^r) to weak L _W ^P (l^r) if and only if W ∈ A_P(R). Here we say W∈A_p (1), if $$\begin{gathered} \mathop {sup}\limits_{R \in R} \left( {\tfrac{1}{{|R|}}\smallint _r wdx} \right)\left( {\tfrac{1}{{|R|}}\smallint _R w^{ - 1/(p - 1)} dx} \right)^{p - 1}< \infty ,1< p< \infty , \hfill \\ (Nw)(x) \leqslant Cw(x)a.e.,p = 1 \hfill \\ \end{gathered} $$ ,     

A uniform Birkhoff theorem

Birkhoff’s HSP theorem characterizes the classes of models of algebraic theories as those being closed with respect to homomorphic images, subalgebras, and products. In particular, it implies that an algebra B satisfies all equations that hold in an algebra A of the same signature if and only if B is a homomorphic image of a subalgebra of a (possibly infinite) direct power of A. The former statement is equivalent to the existence of a natural map sending term functions of the algebra A to those of B—the natural clone homomorphism. The study of continuity properties of natural clone homomorphisms has been initiated recently by Bodirsky and Pinsker for locally oligomorphic algebras.Revisiting the argument of Bodirsky and Pinsker, we show that for any algebra B in the variety generated by an algebra A, the induced natural clone homomorphism is uniformly continuous if and only if every finitely generated subalgebra of B is a homomorphic image of a subalgebra of a finite power of A. Based on this observation, we study the question as to when Cauchy continuity of natural clone homomorphisms implies uniform continuity. We introduce the class of almost locally finite algebras, which encompasses all locally oligomorphic as well as all locally finite algebras, and show that, in case A is almost locally finite, then the considered natural homomorphism is uniformly continuous if (and only if) it is Cauchy-continuous. In particular, this provides a locally finite counterpart of the result by Bodirsky and Pinsker. Along the way, we also discuss some peculiarities of oligomorphic permutation groups on uncountable sets.