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AbstractWe confirm a conjecture, about neat embeddings of cylindric algebras, made in 1969 by J. D. Monk, and a later conjecture by Maddux about relation algebras obtained from cylindric algebras. These results in algebraic logic have the following consequence for predicate logic: for every finite cardinal a= 3 there is a logically valid sentence X, in a first-order language Lwith equality and exactly one nonlogical binary relation symbol E, such that Xcontains only 3 variables (each of which may occur arbitrarily many times), Xhas a proof containing exactly a+ 1 variables, but Xhas no proof containing only avariables. This solves a problem posed by Tarski and Givant in 1987.

Extensive studies on invertebrates from Ny-Ålesund, Spitsbergen, Svalbard and more limited data on aphids from Abisko, Sweden, produced the following main conclusions: (1) The population response to raised summer temperatures differed between the above and the below ground species, both in terms of speed and magnitude. (2) Similar animal communities responded differently to similar temperature manipulations on sites with different vegetation cover and composition. (3) For soil animals the between-year and between-site variations in population densities, were greater than the differences produced by the temperature manipulation experiments at any one site in any year. (4) Infrequent extreme climatic events strongly influence long-term trends in population density and community composition. (5) The population response of invertebrates to climate warming is greatest and most rapid at the coldest sites. (6) The spatial distribution of the above ground insect herbivores on their host plant is temperature limited. (7) The numerical abundance of flying predators/parasitoids of the above-ground herbivores is low. (8) The spatial distribution of some predators may be thermally restricted and less extensive than that of their prey. (9) Habitat temperature is the driving variable determining the flight activity patterns of insects. (10) Increased summer temperatures may alter or disrupt the seasonal patterns of insect emergence, particularly in species where the life cycle is cued into the seasonal rhythm. (11) The common species of arctic soil mites and Collembola are well adapted to survive enhanced summer temperatures, providing that moisture is not limited. (12) Water availability during the summer growing period is probably of greater significance than temperature in determining the survival and success of many arctic soil invertebrate groups. (13) Arctic soil microarthropod species are well adapted to survive and operate at subzero and low positive summer temperatures. (14) Freeze-thaw events represent critical points in the life history of the microarthropods. (15) Supercooling points are sometimes poor indicators of the capacity of arctic soil microarthopods to survive low temperatures. From these findings predictions are made as to how high arctic communities will respond to predicted changes in climate.

The distributions of host-specific herbivorous insects along latitudinal and altitudinal gradients, particularly within arctic and alpine environments, provide useful analogs for predicted future changes that are likely to occur over time at any one location, given a gradually changing thermal environment. It is suggested that selected examples of these insect/plant systems can serve as highly responsive sensors of changing climatic temperatures. Distributions of insects that show a restricted occurrence withinthe overall range of their host plant are predicted to respond more rapidly than the distribution of the host plants themselves. Examples are given to show how these insect-plant systems might be used to monitor long-term temperature change. These examples, drawn from both our published and unpublished work on Homoptera (Insecta), include the following common and widely distributed host plant species within the Northern Hemisphere: Dryas octopetala, Chamerion angustifolium, dwarf Salixspecies including S. lapponum, and Calluna vulgaris/Erica cinerea.

We give a criterion involving existence of many generic sequences of automorphisms for a countable structure to have the small index property. We use it to show that (i) any ω‐stable ω‐categorical structure, and (ii) the random graph have the small index property. We also show that the automorphism group of such a structure is not the union of a countable chain of proper subgroups.

AbstractA boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any fixed dimension (at least 3) are elementary.

AbstractWe consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterized according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Finte relation algebras with homogeneous representations are characterized by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is ?-categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable.An important open problem from algebraic logic is addressed by devising another two-player game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras.Other instances of this approach are looked at, and include the step by step method.

The paper is in two parts. In Part I we describe a construction of a certain kind of subdirect product of a family of rings. We endow the index set of the family with the partial order structure of an SFP domain, as introduced by Plotkin, and provide a commuting system of homomorphisms between those rings whose indices are related in the ordering. We then take the subdirect product consisting of those elements of the direct product having finite support in the sense of this domain structure. In the special case where the homomorphisms are isomorphisms of a fixed ring S, our construction reduces to taking the Boolean power of Sby a Boolean algebra canonically associated with the SFP domain. We examine the ideals of a ring obtainable in this way, showing for instance that each ideal is determined by its projections onto the factor rings. We give conditions on the underlying SFP domain that ensure that the ring is atomless. We examine the relationship between the L∞ω‐theory of the ring and that of the SFP domain. In Part II we prove a ‘non‐structure theorem’ by exhibiting 2ℵ1pairwise non‐embeddable L∞ω‐equivalent rings of cardinality ℵ1with various higher‐order properties. The construction needs only ZFC, and uses Aronszajn trees to build many different SFP domains with bases of cardinality ℵ1.

Extensive studies on invertebrates from Ny-Alesund, Spitsbergen, Svalbard and more limited data on aphids from Abisko, Sweden, produced the following main conclusions: (1) The population response to raised summer temperature differed between the above- and below-ground species, both in terms of speed and magnitude. (2) Similar animal communities responded differently to similar temperature manipulations on sites with different vegetation cover and composition. (3) For soil animals the between-year and between-site variations in population densities were greater than the differences produced by temperature manipulation experiments at any one site in any year. (4) Infrequent extremeclimatic events strongly influence long-term trends in population density and community composition. (5) The population response of invertebrates to climate warming is greatest and most rapid at the coldestsites. (6) The spatial distribution of the above ground insect herbivores on their host plant is temperature limited. (7) The numerical abundance of flying predators/parasitoids of the above-ground herbivores is low. (8) The spatial distribution of some predators may be thermally restricted and less extensive than that of their prey. (9) Habitat temperature is the driving variable determining the flight activity patterns of insects. (10) Increased summer temperatures may alter or disrupt the seasonal patterns of insect emergence, particularly inspecies where the life cycles is cued into the seasonal rhythm. (11)The common species of arctic soil mites and Collembola are well adapted to survive enhanced summer temperatures, providing that moisture is not limited. (12) Water availability during the summer growing period is probably of greater significance than temperature in determining the survival and success of many arctic soil invertebrate groups. (13) Arctic soil microarthopod species are well adapted to survive and operate at subzero and low positive summer temperatures. (14) Freeze-th [ABSTRACT FROM AUTHOR]

Meta-languages are vital to the development and usage of formal systems, and yet the nature of meta-languages and associated notions require clarification. Here we attempt to provide a clear definition of the requirements for a language to be a meta-language, together with consideration of issues of proof theory, model theory and interpreters for such a language.

The paper is in two parts. In Part I we describe a construction of a certain kind of subdirect product of a family of rings. We endow the index set of the family with the partial order structure of an SFP domain, as introduced by Plotkin, and provide a commuting system of homomorphisms between those rings whose indices are related in the ordering. We then take the subdirect product consisting of those elements of the direct product having finite support in the sense of this domain structure. In the special case where the homomorphisms are isomorphisms of a fixed ring S, our construction reduces to taking the Boolean power of S by a Boolean algebra canonically associated with the SFP domain. We examine the ideals of a ring obtainable in this way, showing for instance that each ideal is determined by its projections onto the factor rings. We give conditions on the underlying SFP domain that ensure that the ring is atomless. We examine the relationship between the L∞ω-theory of the ring and that of the SFP domain. In Part II we prove a ‘non-structure theorem’ by exhibiting 2ℵ1 pairwise non-embeddable L∞ω-equivalent rings of cardinality ℵ1 with various higher-order properties. The construction needs only ZFC, and uses Aronszajn trees to build many different SFP domains with bases of cardinality ℵ1.