Hausdorff properties of topological algebras

Let P be a property of topological spaces. Let [P] be the class of all varieties having the property that any topological algebra in has underlying space satisfying property P. We show that if P is preserved by finite products, and if is preserved by ultraproducts, then [P] is a class of varieties that is definable by a Maltsev condition.?The property that all T ₀ topological algebras in are j-step Hausdor. (Hj) is preserved by finite products, and its negation is preserved by ultraproducts. We partially characterize the Maltsev condition associated to by showing that this topological implication holds in every (2j + 1)-permutable variety, but not in every (2j + 2)-permutable variety.?Finally, we show that the topological implication holds in every k-permutable, congruence modular variety. Received March 1, 2000; accepted in final form October 18, 2001.     

Topological implications in varieties

In 1997, Coleman showed that a variety V is n-permutable for some n iff every T ₀-topological Algebra in V is T ₁. Here we show that the implication " sober" is another such characterization for n-permutability. Other implications of a similar nature are given. For example, an n-permutable variety having a majority term satisfies ". Received July 13, 1998; accepted in final form April 13, 1999     

Nonexpansive algebras

A nonexpansive algebra is a pseudometric algebra in which the operations are all nonexpansive. We study such algebras, particularly in the case of algebras in permutable and n-permutable varieties, leading to new characterizations of such varieties. Free nonexpansive algebras are also investigated. This paper is dedicated to Walter Taylor. Received September 27, 2005; accepted in final form February 4, 2006. The author would like to thank the referee for the many helpful suggestions.     

On subtractive varieties,I

A varietyV is subtractive if it obeys the laws s(x, x)=0, s(x, 0)=x for some binary terms and constant 0. This means thatV has 0-permutable congruences (namely [0]R ºS=[0]S ºR for any congruencesR, S of any algebra inV). We present the basic features of such varieties, mainly from the viewpoint of ideal theory. Subtractivity does not imply congruence modularity, yet the commutator theory for ideals works fine. We characterize i-Abelian algebras, (i.e. those in which the commutator is identically 0). In the appendix we consider the case of a classical ideal theory (comprising: groups, loops, rings, Heyting and Boolean algebras, even with multioperators and virtually all algebras coming from logic) and we characterize the corresponding class of subtractive varieties.Presented by A. F. Pixley.     

A characterization of Hausdorff separation for a special class of varieties

In 1997, Coleman suggested that congruence modularity and n-permutability are necessary and sufficient properties for a variety to have all of its T₀-topological algebras be Hausdorff. The sufficiency part of this statement was later shown by Kearnes and Sequeira. Here we show that necessity holds for a certain subclass of varieties, hence providing a characterization of the property in this particular case. This paper is dedicated to Walter Taylor. Received August 11, 2004; accepted in final form August 30, 2005.     

Distance-Regular Graphs with Strongly Regular Subconstituents

In [3] Cameron et al. classified strongly regular graphs with strongly regular subconstituents. Here we prove a theorem which implies that distance-regular graphs with strongly regular subconstituents are precisely the Taylor graphs and graphs with a ₁ = 0 and a _i {0,1} for i = 2,...,d.     

Closure Operators in Convergence Spaces

We define two closure operators of some well-known topological categories and investigate the relationships between these closure operators and the one that is given in [9]. As a consequence, we characterize separation properties T ₀, T ₁, and T ₂ for these well-known categories and compare them with the ones that are given in [3] and [6]. Finally, we characterize the epimorphisms in the subcategories of these given categories. This revised version was published online in June 2006 with corrections to the Cover Date.     

I n -symmetrical Heyting algebras

In [11, p. 210] A. Monteiro suggested the possibility of generalizing the results of L. Iturrioz [4] to theI _n -symmetrical Heyting algebras which he namedI _n -algebras. We prove that these algebras are semi-simple. We characterize simple algebras and their subalgebras. Finally, we determine the structure of theI _n -algebra with a finite set of free generators and we give an answer to one of the problems posed by A. Monteiro.Presented by W. Taylor.Some of the results of this paper were presented at the Annual Meeting of the Union Matemática Argentina (September, 1987) ([14]).     

n‐linear weakly Heyting algebras

The present paper introduces and studies the variety ????_n of n‐linear weakly Heyting algebras. It corresponds to the algebraic semantic of the strict implication fragment of the normal modal logic K with a generalization of the axiom that defines the linear intuitionistic logic or Dummett logic. Special attention is given to the variety ????₂ that generalizes the linear Heyting algebras studied in [10] and [12], and the linear Basic algebras introduced in [2]. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Equivalence for varieties in general and for $ {\cal BOOL} $ in particular

The varieties equivalent to a given variety are characterized in a purely categorical way. In fact they are described as the models of those Lawvere theories which are Morita equivalent to the Lawvere theory of which therefore are characterized first. Along this way the conceptual meanings of the n-th matrix power construction of a variety and McKenzie's σ-modification of classes of algebras [22] become transparent. Besides other applications not only the well known equivalences between the varieties of Post algebras of fixed orders m and the variety of Boolean algebras are obtained; moreover it can be shown that the varieties are the only varieties equivalent to . The results then are generalized to quasivarieties and more general classes of algebras. Received November 4, 1998; accepted in final form September 15, 1999.     

On <Emphasis Type="Italic">n</Emphasis> × <Emphasis Type="Italic">m</Emphasis>-valued Łukasiewicz-Moisil algebras

n×m-valued Łukasiewicz algebras with negation were introduced and investigated in [20, 22, 23]. These algebras constitute a non trivial generalization of n-valued Łukasiewicz-Moisil algebras and in what follows, we shall call them n×m-valued Łukasiewicz-Moisil algebras (or LM _n×m -algebras). In this paper, the study of this new class of algebras is continued. More precisely, a topological duality for these algebras is described and a characterization of LM _n×m -congruences in terms of special subsets of the associated space is shown. Besides, it is determined which of these subsets correspond to principal congruences. In addition, it is proved that the variety of LM _n×m -algebras is a discriminator variety and as a consequence, certain properties of the congruences are obtained. Finally, the number of congruences of a finite LM _n×m -algebra is computed.      

Every finitely generated regular field extension has a stable transcendence base

A fieldK is called stable if every finitely generaed regular field extensionF/K has a transcendence basex ₁, …,x _n with the following properties: The field extensionF/K(x ₁,…,x _n ) is separable and the Galois hull ofF/K(x ₁,…,x _n ) remains regular overK, i.e.K is algebraically closed in . We prove in this paper thatevery field is stable. This generalizes results from [FJ1] and [GJ] which prove that fields of characteristic 0 and infinite perfect fields are stable, respectively. [G] showed that finite fields are stable in dimension 1, i.e. every finitely generated regular field extension of transcendence degree 1 over a finite field has a stable transcendence base. In the last section of this paper we apply the theorem to the construction of PAC fields with additional properties. A fieldK is called PAC if every absolutely irreducible variety overK has at least oneK-rational point.     

Contractivity properties of Ornstein-Uhlenbeck semigroup for general commutation relations

We study a certain class of von Neumann algebras generated by selfadjoint elements ω_i=a_i+a_i⁺, where a_i, a_i⁺ satisfy the general commutation relations:We assume that operator T for which the constants are matrix coefficients satisfies the braid relation. Such algebras were investigated in [BSp] and [K] where the positivity of the Fock representation and factoriality were shown. In this paper we prove that T-Ornstein-Uhlenbeck semigroup U_t=Γ_T(e^−t), t>0 arising from the second quantization procedure is hyper- and ultracontractive. The optimal bounds for hypercontractivity are also discussed.This paper was partially supported by KBN grant no 2P03A00732 and also by RTN grant HPRN-CT-2002-00279.     

Clone and hypervariety correspondences

The general correspondence between clones and hypervarieties was set out by Walter Taylor in [5], By mapping each varietyV to its cloneClone V Taylor obtained inverse isomorphisms between the lattice of all clone varieties and the lattice of all hypervarieties. Taylor also described a more specific correspondence in [6], using the 1-clone Cl₁ V of a varietyV. Since 1-clones are just monoids, he found a one-to-one correspondence from the uncountably infinite lattice of monoid varieties into the lattice of all hypervarieties. In this paper we show how Taylor's construction may be carried out forn-clones, for anyn 1. Thus we usen-clones to restrict the general correspondence to a family (_n) of correspondences, including the monoid correspondence as the special casen=1. These correspondences pick out certain families of hypervarieties, then-closed hypervarieties (forn 1), and we show that there are at least countably many such hypervarieties for eachn. This represents some progress towards the goal of understanding the structure of the lattice of all hypervarieties.Presented by W. Taylor.Research supported by NSERC of Canada.     

The size-Ramsey number of trees

IfG andH are graphs, let us writeG→(H)₂ ifG contains a monochromatic copy ofH in any 2-colouring of the edges ofG. Thesize-Ramsey number r _e(H) of a graphH is the smallest possible number of edges a graphG may have ifG→(H)₂. SupposeT is a tree of order |T|≥2, and lett ₀,t ₁ be the cardinalities of the vertex classes ofT as a bipartite graph, and let Δ(T) be the maximal degree ofT. Moreover, let Δ₀, Δ₁ be the maxima of the degrees of the vertices in the respective vertex classes, and letβ(T)=T ₀Δ₀+t ₁Δ₁. Beck [7] proved thatβ(T)/4≤r _e(T)=O{β(T)(log|T|)¹²}, improving on a previous result of his [6] stating thatr _e(T)≤Δ(T)|T|(log|T|)¹². In [6], Beck conjectures thatr _e(T)=O{Δ(T)|T|}, and in [7] he puts forward the stronger conjecture thatr _e(T)=O{β(T)}. Here, we prove the first of these conjectures, and come quite close to proving the second by showing thatr _e(T)=O{β(T)logΔ(T)}.     

Ordering trees by algebraic connectivity

LetG be a graph onn vertices. Denote byL(G) the difference between the diagonal matrix of vertex degrees and the adjacency matrix. It is not hard to see thatL(G) is positive semidefinite symmetric and that its second smallest eigenvalue,a(G) > 0, if and only ifG is connected. This observation led M. Fiedler to calla(G) thealgebraic connectivity ofG. Given two trees,T ₁ andT ₂, the authors explore a graph theoretic interpretation for the difference betweena(T ₁) anda(T ₂).Research supported by ONR contract 85K0335     

Mixing and sweeping out

A weakly mixing transformationT and a sequence (d _n) are constructed such thatT is uniformly mixing on (d _n),T is uniformly sweeping out on ([αd _n]) for allα∈(0, 1), and for all rationalα∈(0, 1)T is not mixing on ([αd _n]).     

Characterization of non-matrix varieties of associative algebras

A variety of associative algebras is called a non-matrix variety if it does not contain the algebra of 2 × 2 matrices over the base field K. There are some known characterizations of non-matrix varieties. We give some new characterizations in terms of properties of nilelements. Let V be a variety of associative algebras over an infinite field. Then the following conditions are equivalent: (1) V is a non-matrix variety, (2) any finitely generated algebra A ∈ V satisfies an identity of the form [x ₁, x ₂] … [x _2s−1, x _2s ] ≡ 0, (3) let A ∈ V; then for any nilelements a, b ∈ A, the element a + b is again a nilelement. Let E be the Grassmann algebra in countable many generators. We also give similar characterizations for non-matrix varieties over fields of characteristic zero that do not contain E or E ⊗ E.     

Dilations and Commutant Lifting for Jointly Isometric Operators—A Geometric Approach

A tuple of commuting contractionsT=(T₁, T₂, …, T_n) is called a joint-isometry if ∑ T*_jT_j=I. We give a geometric proof that joint isometries have a regular unitary dilation and that its commutant lifts. We also show thatTis subnormal and that its minimal normal extension is also jointly isometric.