A note on homomorphic images, subalgebras and various products

Let I, H, S, P, P_fin, P_f , P_u, P_s be the usual operators on classes of algebras of the same type (P, P_fin, P_f, P_u, P_s are respectively for direct, finite, filtered, ultra and subdirect products). The partially ordered monoid generated by the operators H, S, P with respect to composition of operators, I as an identity element, and natural ordering between operators is described by Pigozzi (Algebra Universalis 2 (1972), 346–353). The aim of this note is to describe the partially ordered monoids generated by H, S, P_u and by H, S, P_s and as well to summarize known results on the partially ordered monoids of operators generated by H, S and some of the above introduced products.Dedicated to the memory of Aleksandar PopoviReceived April 1, 2001; accepted in final form July 11, 2004.     

Partially Ordered Monoids Generated by Operators H , S , P , and by H , S , P f Are Isomorphic

Let I , H , S , P , P _f be the usual operators on classes of algebras of the same type (P _f for filtered products). The partially ordered monoid generated by the operators H , S , P with respect to composition of operators, I as an identity element, and a natural ordering between operators is described by Pigozzi (Algebra Universalis 2 (1972), 346—353). Let us denote by \cal M =\langle H, S, P\rangle and by \cal M _f =\langle H, S, P _f \rangle the partially ordered monoids generated by {H, S, P} and by {H, S, P _f } respectively. The aim of this paper is to prove that \cal M is isomorphic to \cal M _f . October 29, 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 inflation class operator

We consider the inflation class operator, denoted by F, where for any class K of algebras, F(K) is the class of all inflations of algebras in K. We study the interaction of this operator with the usual algebraic operators H, S andP, and describe the partially-ordered monoid generated by H, S, P andF (with the isomorphism operator I as an identity). Received February 3, 2004; accepted in final form January 3, 2006.     

On ideal extensions of partial monounary algebras

In the present paper we introduce the notion of an ideal of a partial monounary algebra. Further, for an ideal (I, f _I ) of a partial monounary algebra (A, f _A ) we define the quotient partial monounary algebra (A, f _A )/(I, f _I ). Let (X, f _X ), (Y, f _Y ) be partial monounary algebras. We describe all partial monounary algebras (P, f _P ) such that (X, f _X ) is an ideal of (P, f _P ) and (P, f _P )/(X, f _X ) is isomorphic to (Y, f _Y ). This work was supported by the Slovak VEGA Grant No. 1/3003/06 and by the Science and Technology Assistance Agency under the contract No. APVT-20-004104.     

Deddens Algebras and Shift

We consider Deddens algebras associated to operators of the form S−λI, where S is the unilateral shift and λ is a complex number. We show that such an algebra properly contains the commutant of S and that it is always weakly dense in L(H){{\mathcal L}({\mathcal H})}. Yet, it contains no rank one operators, unless λ = 0, in which case it equals L(H){{\mathcal L}({\mathcal H})}.     

On Symmetric Differential Operators Associated with Sobolev Orthogonal Polynomials: A Characterization

Given the Sobolev bilinear form (f, g) _S =>u ₀, f g< + >u ₁, f " g "<, with u ₀ and u ₁ linear functionals, a characterization of the linear second–order differential operators with polynomial coefficients, symmetric with respect to (, ) _S in terms of u ₀ and u ₁ is obtained. In particular, several interesting functionals u ₀ and u ₁ are considered, recovering as particular cases of our study, results already known in the literature.     

The Forcing Convexity Number of a Graph

For two vertices u and v of a connected graph G, the set I(u,v) consists of all those vertices lying on a u-v geodesic in G. For a set S of vertices of G, the union of all sets I(u,v) for u, v S is denoted by I(S). A set S is a convex set if I(S) = S. The convexity number con(G) of G is the maximum cardinality of a proper convex set of G. A convex set S in G with |S| = con(G) is called a maximum convex set. A subset T of a maximum convex set S of a connected graph G is called a forcing subset for S if S is the unique maximum convex set containing T. The forcing convexity number f(S, con) of S is the minimum cardinality among the forcing subsets for S, and the forcing convexity number f(G, con) of G is the minimum forcing convexity number among all maximum convex sets of G. The forcing convexity numbers of several classes of graphs are presented, including complete bipartite graphs, trees, and cycles. For every graph G, f(G, con) con(G). It is shown that every pair a, b of integers with 0 a b and b is realizable as the forcing convexity number and convexity number, respectively, of some connected graph. The forcing convexity number of the Cartesian product of H × K ₂ for a nontrivial connected graph H is studied.     

Hankel operators in Schatten ideals

We study membership to Schatten ideals S _E , associated with a monotone Riesz–Fischer space E, for the Hankel operators H _f defined on the Hardy space H ²(∂D). The conditions are expressed in terms of regularity of its symbol: we prove that H _f ∈S _E if and only if f∈B _E , the Besov space associated with a monotone Riesz–Fischer space E(dλ) over the measure space (D,dλ) and the main tool is the interpolation of operators. Received: December 17, 1999; in final form: September 25, 2000?Published online: July 13, 2001     

The influence of variables in product spaces

LetX be a probability space and letf: X ⁿ → {0, 1} be a measurable map. Define the influence of thek-th variable onf, denoted byI _f (k), as follows: Foru=(u ₁,u ₂,…,u _n−1) ∈X ⁿ⁻¹ consider the setl _k (u)={(u ₁,u ₂,...,u _k−1,t,u _k ,…,u _n−1):t ∈X}. More generally, forS a subset of [n]={1,...,n} let the influence ofS onf, denoted byI _f (S), be the probability that assigning values to the variables not inS at random, the value off is undetermined. Theorem 1:There is an absolute constant c ₁ so that for every function f: X ⁿ → {0, 1},with Pr(f ⁻¹(1))=p≤1/2,there is a variable k so that Theorem 2:For every f: X ⁿ → {0, 1},with Prob(f=1)=1/2, and every ε>0,there is S ⊂ [n], |S|=c ₂(ε)n/logn so that I _f (S)≥1−ε. These extend previous results by Kahn, Kalai and Linial for Boolean functions, i.e., the caseX={0, 1}. Work supported in part by grants from the Binational Israel-US Science Foundation and the Israeli Academy of Science.     

When isf(x1, x2, ..., xn) =u1(x1) +u2(x2) + ... +un(xn)?

We discuss subsetsS of ℝⁿ such that every real valued functionf onS is of the formf(x₁, x₂, ..., x_n) =u ₁(x₁) +u ₂(x₂) +...+u _n(x_n), and the related concepts and situations in analysis.     

Study of global solutions of parabolic equations via a priori estimates. Part I: Equations with principal elliptic part equal to the laplacian

We study global solutions of u_t=Δu + f(u, Du) with zero boundary data. Under suitable hypotheses on f, we show that an energy, which is roughly the H¹ norm of u, cannot change too rapidly. When f depends only on u, this estimate implies that all global solutions are bounded.     

A Composition Theorem for Multiple Summing Operators

We prove that the composition S(u₁, …, u_n) of a multilinear multiple 2-summing operator S with 2-summing linear operators u_j is nuclear, generalizing a linear result of Grothendieck.     

Sur l’annulation du deuxième foncteur de (co)homologie d’André-Quillen

For a given idealI of a commutative ringA, B=A/I, the vanishing of the second André-Quillen (co)homology functorH ₂ (A, B, δ) is characterized in terms of the canonical homomorphism α:S(I)→R(I) from the symmetric algebra of the idealI onto its Rees algebra. This is done by introducing a Koszul complex that characterizes commutative graded algebras which are symmetric algebras.

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This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag.     

A Composition Theorem for Multiple Summing Operators

We prove that the composition S(u₁, …, u_n) of a multilinear multiple 2-summing operator S with 2-summing linear operators u_j is nuclear, generalizing a linear result of Grothendieck. Both authors were partially supported by DGICYT grant BMF2001-1284.     

On subsemigroup lattices without non-trivial identities

A latticeL is called discriminating if for any free latticeF and for any finite number of elementsu ₁,u ₂, ...,u _nF, there exists a homomorphismf:FL such thatf(u _i )f(u _j ) wheneveru _i u _j (1i, jn). In this paper it is proved that the subsemigroup lattice SubS of a commutative semigroupS does not satisfy a non-trivial identity if and only if SubS is discriminating. In particular, in this case every finite projective lattice can be embedded into SubS. It should be noted that the most important examples of semigroups whose subsemigroup lattices satisfy no non-trivial identity and therefore have the discriminating property are the following: the infinite cyclic semigroup, the free semilattice of countable rank, any commutative nilsemigroup which is not nilpotent and so on.Presented by V. A. Gorbunov.The author thanks Prof. L. N. Shevrin and Dr. M. V. Volkov for a number of useful remarks.     

Subnormal operators with compact selfcommutator

For an arbitrary subnormal operator we estimate the essential norm and trace of commutators of the form [T _u, S], whereT _u is a Toeplitz operator with continuous symbol. In particular, we obtain criteria for the compactness of [S ^*,S]. The trace estimates apply to multiplication operators on Hardy spaces over general domains.     

More on the Hochschild and Cyclic Homologies of Crossed Modules of Algebras

We investigate the Hochschild and cyclic homologies of crossed modules of algebras in some special cases. We prove that the cotriple cyclic homology of a crossed module of algebras (I, A, ρ) is isomorphic to HC _*(ρ): HC _*(I) → HC _*(A), provided I is H-unital and the ground ring is a field with characteristic zero. We also calculate the Hochschild and cyclic homologies of a crossed module of algebras (R, 0, 0) for each algebra R with trivial multiplication. At the end, we give some applications proving a new five term exact sequence.