Failure of n‐uniqueness: a family of examples

In this paper, the connections between model theory and the theory of infinite permutation groups (see 11 ) are used to study the n‐existence and the n‐uniqueness for n‐amalgamation problems of stable theories. We show that, for any n ? 2, there exists a stable theory having (k + 1)‐existence and k‐uniqueness, for every k ? n, but has neither (n + 2)‐existence nor (n + 1)‐uniqueness. In particular, this generalizes the example, for n = 2, due to Hrushovski given in 3 . © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Netlike partial cubes III. The median cycle property

A graph G has the Median Cycle Property (MCP) if every triple (u₀,u₁,u₂) of vertices of G admits a unique median or a unique median cycle, that is a gated cycle C of G such that for all i,j,k∈{0,1,2}, if x_i is the gate of u_i in C, then: {x_i,x_j}⊆I_G(u_i,u_j) if i≠j, and d_G(x_i,x_j)<d_G(x_i,x_k)+d_G(x_k,x_j). We prove that a netlike partial cube has the MCP if and only if it contains no triple of convex cycles pairwise having an edge in common and intersecting in a single vertex. Moreover a finite netlike partial cube G has the MCP if and only if G can be obtained from a set of even cycles and hypercubes by successive gated amalgamations, and equivalently, if and only if G can be obtained from K₁ by a sequence of special expansions. We also show that the geodesic interval space of a netlike partial cube having the MCP is a Pash-Peano space (i.e. a closed join space).     

Amalgamation in finite dimensional cylindric algebras

For every finite n > 1, the embedding property fails in the class of all n-dimensional cylindric type algebras which satisfy the following. Their boolean reducts are boolean algebras and two of the cylindrifications are normal, additive and commute. This result also holds for all subclasses containing the representable n-dimensional cylindric algebras. This considerably strengthens a result of S. Comer on CA _n and provides a strong counterexample for interpolation in finite variable fragments of first order logic. We provide a new modern proof, using an argument inspired by modal logic. February 22, 1999.     

Some Categorical Algebraic Properties: Counter-Examples for Functor Categories

This work is a complement to the authors earlier papers, where it is shown that a functor category inherits from such properties as amalgamation, transferability and congruence extension if has either products or certain pushouts. A general scheme is given for constructing counter-examples which show that the latter condition on is essential. In particular, it is shown that the functor categories , , ( resp.) do not satisfy the amalgamation (congruence extension resp.) property in general. Moreover, one class of categories is described, where the condition of the existence of certain pushouts is not only sufficient, but also necessary for to preserve the considered properties of .Mathematics Subject Classifications (2000) 18A25, 18A32, 18B99, 08B26.Dali Zangurashvili: The support rendered by INTAS Grant 97 31961 is gratefully acknowledged.     

Algorithmic Aspects of Tree Amalgamation

The amalgamation of leaf-labeled trees into a single (super)tree that “displays” each of the input trees is an important problem in classification. We discuss various approaches to this problem and show that a simple and well-known polynomial-time algorithm can be used to solve this problem whenever the input set of trees contains a minimum size subset that uniquely determines the supertree. Our results exploit a recently established combinatorial property concerning the structure of such collections of trees.     

Craig interpolation for semilinear substructural logics

The Craig interpolation property is investigated for substructural logics whose algebraic semantics are varieties of semilinear (subdirect products of linearly ordered) pointed commutative residuated lattices. It is shown that Craig interpolation fails for certain classes of these logics with weakening if the corresponding algebras are not idempotent. A complete characterization is then given of axiomatic extensions of the “R‐mingle with unit” logic (corresponding to varieties of Sugihara monoids) that have the Craig interpolation property. This latter characterization is obtained using a model‐theoretic quantifier elimination strategy to determine the varieties of Sugihara monoids admitting the amalgamation property.     

Extending regular edge-colorings of complete hypergraphs

A coloring (partition) of the collection $(\genfrac{}{}{0.0pt}{}{X}{h})$ of all $h$-subsets of a set $X$ is $r$-regular if the number of times each element of $X$ appears in each color class (all sets of the same color) is the same number $r$. We are interested in finding the conditions under which a given $r$-regular coloring of $(\genfrac{}{}{0.0pt}{}{X}{h})$ is extendible to an $s$-regular coloring of $(\genfrac{}{}{0.0pt}{}{Y}{h})$ for $s⩾r$ and $Y⊋X$. The case $\text{◂,▸}h=2,r=s=1$ was solved by Cruse, and due to its connection to completing partial symmetric latin squares, many related problems are extensively studied in the literature, but very little is known for $h⩾3$. The case $r=s=1$ was solved by Häggkvist and Hellgren, settling a conjecture of Brouwer and Baranyai. The cases $h=2$ and $h=3$ were solved by Rodger and Wantland, and Bahmanian and Newman, respectively. In this paper, we completely settle the cases $h=4,\text{◂⩾▸}|Y|⩾4|X|$ and $h=5,\text{◂⩾▸}|Y|⩾5|X|$.     

Determination of total mercury in nuts at ultratrace level

Total mercury, at μg kg⁻¹ level, was determined in different types of nuts (cashew nut, Brazil nuts, almond, pistachio, peanut, walnut) using a direct mercury analyser after previous sample defatting and by cold vapour atomic fluorescence spectrometry. There is not enough sensitivity in the second approach to determine Hg in previously digested samples due to the strong matrix effect. Mercury levels in 25 edible nut samples from Brazil and Spain were found in the range from 0.6 to 2.7 μg kg⁻¹ by using the pyrolysis of sample after the extraction of the nut fat. The accuracy of the proposed method was confirmed by analysing certified reference materials of Coal Fly Ash-NIST SRM 1633b, Fucus-IAEA 140 and three unpolished Rice Flour NIES-10. The observed results were in good agreement with the certified values. The recoveries of different amounts of mercury added to nut samples ranged from 94 to 101%. RSD values corresponding to three measurements varied between 2.0 and 14% and the limit of detection and quantification of the method were 0.08 and 0.3 μg kg⁻¹, respectively.     

Amalgamation of Ortholattices

We show that the variety of ortholattices has the strong amalgamation property and that the variety of orthomodular lattices has the strong Boolean amalgamation property, i.e. that two orthomodular lattices can be strongly amalgamated over a common Boolean subalgebra. We give examples to show that the variety orthomodular lattices does not have the amalgamation property and that the variety of modular ortholattices does not even have the Boolean amalgamation property. We further show that no non-Boolean variety of orthomodular lattices which is generated by orthomodular lattices of bounded height can have the Boolean amalgamation property.