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Matrix identities involving multiplication and transposition

Karl Auinger, Igor Dolinka, Michael V. Volkov (2012)

Journal of the European Mathematical Society

We study matrix identities involving multiplication and unary operations such as transposition or Moore–Penrose inversion. We prove that in many cases such identities admit no finite basis.

Metric abstract elementary classes with perturbations

Åsa Hirvonen, Tapani Hyttinen (2012)

Fundamenta Mathematicae

We define an abstract setting suitable for investigating perturbations of metric structures generalizing the notion of a metric abstract elementary class. We show how perturbation of Hilbert spaces with an automorphism and atomic Nakano spaces with bounded exponent fit into this framework, where the perturbations are built into the definition of the class being investigated. Further, assuming homogeneity and some other properties true in the example classes, we develop a notion of independence for...

Metric groups, unitary representations and continuous logic

Aleksander Ivanov (2021)

Communications in Mathematics

We describe how properties of metric groups and of unitary representations of metric groups can be presented in continuous logic. In particular we find L ω 1 ω -axiomatization of amenability. We also show that in the case of locally compact groups some uniform version of the negation of Kazhdan’s property (T) can be viewed as a union of first-order axiomatizable classes. We will see when these properties are preserved under taking elementary substructures.

m-normal theories

Ludomir Newelski (2001)

Fundamenta Mathematicae

Originally, m-independence, ℳ -rank, m-stability and m-normality were defined only for small stable theories. Here we extend the definitions to an arbitrary small countable complete theory. Then we investigate these notions in the new, broader context. As a consequence we show that any superstable theory with < 2 ℵ ₀ countable models is m-normal. In particular, any *-algebraic group interpretable in such a theory is abelian-by-finite.

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