Epimorphisms are not surjective, even in simple algebras
Using techniques developed by Comer on sheaf-theoretic duality for cylindric algebras, we refine a result of Judit Madarasz by showing that epimorphisms are not surjective even in the class of simple infinite dimensional cylindric algebras.
Algebraisable versions of topological predicate logic, Part 1 Toplogical logic via cylindric algebras
Motivated by questions like: which spatial structures may be characterized by means of modal logic, what is the logic of space, how to encode in modal logic different geometric relations, topological logic provides a framework for studying the confluence of the topological semantics for S4 modalities, based on topological spaces rather than Kripke frames, with the S4 modality induced by the interior operator. Following research initiated by Sgro, and further pursued algebraically by Georgescu, we prove an interpolation theorem and an omitting types theorem for various extensions of predicate topological logic and Chang’s modal logic. Our proof is algebraic addressing expansions of cylindric algebras using interior operators and boxes, respectively. Then we proceed like is done in abstract algebraic logic by studing algebraisable extensions of both logics; obtaining a plethora of results on the amalgamation property for various subclasses of their algebraic counterparts, which are varieties. As a sample, we show that the free algebras of infinite dimensions enjoy several weak forms of interpolation, a property equivalent to the fact that the class of simple algebras have the amalgamation property, but they fail the usual Craig interpolation property, because the whole variety fails to have the amalgamation property. Such interpolation properties fail for finite dimensions > 1. Notions like atom-canonicity and complete representations are approached for finite dimensional topological cylindric algebras. The logical consequences of our algebraic results are carefully worked out for infinitary extensions of Chang’s predicate modal logic and finite versions thereof, by restricting to n variables, n finite, viewed as a propositional multi-dimensional modal logic, and n products of bimodal whose frames are of the form (U,U × U,R) where R is a pre-order, endowed with diagonal constants. We show that for any finite n > 2 such modal
Phase diagram and magnetic properties of the diluted fcc system NipMg1-pO.
Starting from Ni and Mg nitrates, about 20 samples of ${\mathrm{Ni}}_{\mathit{p}}$${\mathrm{Mg}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$O samples (with 0.06\ensuremath{\le}p\ensuremath{\le}0.86) were prepared and x-ray-diffraction studies showed the samples to have the NaCl structure with the lattice constant fitting the expression a(p)=4.2115-0.0340p \AA{}. Temperature-dependent magnetic-susceptibility (\ensuremath{\chi}) studies of the samples were carried out between 1.8 and 600 K using a superconducting-quantum-interference-device magnetometer and N\'eel temperatures ${\mathit{T}}_{\mathit{N}}$'s were determined from the peak in \ensuremath{\partial}(\ensuremath{\chi}T)/\ensuremath{\partial}T. The variation of t=${\mathit{T}}_{\mathit{N}}$(p)/${\mathit{T}}_{\mathit{N}}$(1) vs p is compared with that of ${\mathrm{Co}}_{\mathit{p}}$${\mathrm{Mg}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$O. For both systems, the variations for pg0.31 are found to fit the predicted values for a simple-cubic Heisenberg antiferromagnet and a theoretical basis for this anomalous result is advanced. The experimental percolation threshold ${\mathit{p}}_{\mathit{c}}$=0.15\ifmmode\pm\else\textpm\fi{}0.01 and for ${\mathit{p}}_{\mathit{c}}$p0.31, \ensuremath{\chi} below ${\mathit{T}}_{\mathit{N}}$ shows irreversible behavior for the zero-field-cooled and field-cooled cases, suggestive of spin-glass-like behavior also observed in other diluted fcc antiferromagnets, such as ${\mathrm{Co}}_{\mathit{p}}$${\mathrm{Mg}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$O and ${\mathrm{Eu}}_{\mathit{p}}$${\mathrm{Sr}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$Te. It is suggested that the differences in the t-vs-p variations for p0.31 in ${\mathrm{Ni}}_{\mathit{p}}$${\mathrm{Mg}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$O, ${\mathrm{Co}}_{\mathit{p}}$${\mathrm{Mg}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$O, and ${\mathrm{Eu}}_{\mathit{p}}$${\mathrm{Sr}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$Te may be related to the differences in the ratio of the next-nearest-neighbor to nearest-neighbor exchange constants in these systems.