A STRONG PROPERTY OF THE WEAK SUBALGEBRA LATTICE FOR LOCALLY FINITE ALGEBRAS OF FINITE TYPE

The aim of this paper is to show that the weak subalgebra lattice uniquely determines the subalgebra lattice for locally finite algebras of a fixed finite type. However, this algebraic result turns out to be a very particular case of the following hypergraph result (which is interesting itself): A total directed hypergraph D of finite type is uniquely determined, in the class of all the directed hypergraphs of this type, by its skeleton up to the orientation of some pairwise edge-disjoint directed hypercycles and hyperpaths. The skeleton of D is a hypergraph obtained from D by omitting the orientation of all edges.

Identities with Generalized Derivations on Prime Rings and Banach Algebras

Let R be a prime ring, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R, L a non-central Lie ideal of R, and H, G nonzero generalized derivations of R. Suppose that there exists an integer n ≥ 1 such that H(un)un + unG(un) ∈ C for all u ∈ L, then either there exists a ∈ U such that H(x) = xa and G(x) = -ax, or R satisfies the standard identity s4 and one of the following holds: (i) char (R) = 2; (ii) n is even and there exist a′ ∈ U, α ∈ C and derivations d, δ of R such that H(x) = a′ x + d(x) and G(x) = (α-a′)x + δ(x); (iii) n is even and there exist a′ ∈ U and a derivation δ of R such that H(x)=xa′ and G(x) = -a′ x + δ(x); (iv) n is odd and there exist a′, b′ ∈ U and α, β ∈ C such that H(x) = a′ x + x(β-b′) and G(x) = b′ x+x(α-a′); (v) n is odd and there exist α, β ∈ C and a derivation d of R such that H(x) = α x+d(x) and G(x) = β x + d(x); (vi) n is odd and there exist a′ ∈ U and α ∈ C such that H(x) = xa′ and G(x) = (α - a′)x. As an application of this purely algebraic result, we obtain some range inclusion results of continuous or spectrally bounded generalized derivations H and G on Banach algebras R satisfying the condition H(xn)xn + xnG(xn) ∈ rad (R) for all x ∈ R, where rad (R) is the Jacobson radical of R.

CURVATURE STRUCTURE OF SELF-DUAL 4-MANIFOLDS

We show the existence of a modified Cliff(1,1)-structure compatible with an Osserman 0-model of signature (2,2). We then apply this algebraic result to certain classes of pseudo-Riemannian manifolds of signature (2,2). We obtain a new characterization of the Weyl curvature tensor of an (anti-)self-dual manifold and we prove some new results regarding (Jordan) Osserman manifolds.

Pairs of derivations on rings and Banach algebras

We give a generalization of Vukman's theorem concerning a pair of derivations on rings. Then applying this purely algebraic result we obtain several range inclusion results of pair of derivations on Banach algebras.

COMPACTNESS OF THE SPACE OF LEFT ORDERS

A left order on a magma (e.g. semigroup) is a total order of its elements that is left invariant under the magma operation. A natural topology can be introduced on the set of all left orders of an arbitrary magma. We prove that this topological space is compact. Interesting examples of nonassociative magmas, whose spaces of right orders we analyze, come from knot theory and are called quandles. Our main result establishes an interesting connection between topological properties of the space of left orders on a group, and the classical algebraic result by Conrad [4] and Łoś [13] concerning the existence of left orders.

On Catalan Trees and the Jacobian Conjecture

New combinatorial properties of Catalan trees are established and used to prove a number of algebraic results related to the Jacobian conjecture. Let $F=(x_1+H_1,x_2+H_2,\dots,x_n+H_n)$ be a system of $n$ polynomials in $C[x_1,x_2,\dots,x_n]$, the ring of polynomials in the variables $x_1,x_2, \dots, x_n$ over the field of complex numbers. Let $H=(H_1,H_2,\dots,H_n)$. Our principal algebraic result is that if the Jacobian of $F$ is equal to 1, the polynomials $H_i$ are each homogeneous of total degree 2, and $({{\partial H_i}\over {\partial x_j}})^3=0$, then $H\circ H\circ H=0$ and $F$ has an inverse of the form $G=(G_1,G_2,\dots,G_n)$, where each $G_i$ is a polynomial of total degree $\le6$. We prove this by showing that the sum of weights of Catalan trees over certain equivalence classes is equal to zero. We also show that if all of the polynomials $H_i$ are homogeneous of the same total degree $d\ge2$ and $({{\partial H_i}\over {\partial x_j}})^2=0$, then $H\circ H=0$ and the inverse of $F$ is $G=(x_1-H_1,\dots,x_n-H_n)$.

Solidification of pressure-driven flow in a finite rigid channel with application to volcanic eruptions

Competition between conductive cooling and advective heating occurs whenever hot fluid invades a cold environment. Here the solidification of hot viscous flow driven by a fixed pressure drop through an initially planar or cylindrical channel embedded in a cold rigid solid is analysed. At early times, or far from the channel entrance, the flow starts to solidify and block the channel, thus reducing the flow rate. Close to the channel entrance, and at later times, the supply of new hot fluid starts to melt back the initial chill. Eventually, either solidification or meltback becomes dominant throughout the channel, and flow either ceases or continues until the source is exhausted. The evolution of the dimensionless system, which is characterized by the initial Péclet number Pe, the Stefan number S and the dimensionless solidification temperature Θ, is calculated numerically and by a variety of asymptotic schemes. The results show the importance of variations along the channel and caution against models based on a single ‘representative’ width. The critical Péclet number Pec, which marks the boundary between eventual solidification and eventual meltback, is determined for a wide range of parameters and found to be much larger for cylindrical channels than for planar channels, owing to the slower rate of decay of the heat flux into the solid in a cylindrical geometry. For a planar channel Pec is given by the simple algebraic result Pec ∼ 0.46[Θ2/(1 - Θ)(S + 2/π)]3 when (1 - Θ)−1 [Gt ] S [Gt ] 1, but in general it requires numerical solution. Similar analyses, in which there is a spatially varying and time-dependent interaction between the rates of solidification and flow, have a range of applications to geological and industrial processes.

Undecidable semiassociative relation algebras

If K is a class of semiassociative relation algebras and K contains the relation algebra of all binary relations on a denumerable set, then the word problem for the free algebra over K on one generator is unsolvable. This result implies that the set of sentences which are provable in the formalism ℒw× is an undecidable theory. A stronger algebraic result shows that the set of logically valid sentences in ℒw× forms a hereditarily undecidable theory in ℒw×. These results generalize similar theorems, due to Tarski, concerning relation algebras and the formalism ℒ×.