### Working paper

## A semigroup identity for tropical 3x3 matrices

In March 2011 scholars met in Prague at the conference Interculturalism, Meaning and Identity. This event revitalised this important theme related to Diversity and Recognition. The terms 'interculturalism' and 'integration' are experiencing a renaissance. As the extent of human movement between nations increases attempts are made to balance cultural difference and social cohesion. In some contexts immigration and settlement policies are becoming more draconian in response. Because of this, interculturalism can take on many meanings. However, pivotal to the way interculturalism is understood is identification. As the relationship between nation, ethnicity and language becomes more complex so too do the ways in which people represent them selves. The cultural resources drawn on and the processes used to form identities are examined in this truly international collection. So too are the implications of these developments for how we theorise culture, meaning and identity.

The paper treats the issue of identity of the ego, which constitutes the central problem of personology. The skeptical approach to this problem, which sees it as not being subject to be resolved by means of science, began with D. Hume's work. Contemporary personologists (P. Ricoeur and others) approach this problem through study of culture, which imparts the ego with «narrative identity». Cultural historic psychology is a «Bridge of interpretations», upon which philosophy of culture meets psychology, and psychological data associated with «personality» are interpreted on the basis of some specific cultural philosophic theory. The «conflict of interpretations» plays and essential role in personology, which participates in the processes of emergence and overcoming of the cultural crisis. Philosophical and methodological problems that define the near term perspective development of personology are formulated: whether there are any «ego invariants» that remain regardless of any possible cultural determination; whether the ego possesses any rigidity in relation to cultural determination and, if it does, what is the nature of this rigidity; whether ego identity is destroyed when cultural determination diminishes or ceases, etc.

This paper investigates the language situation in Moscow schools with an ethnocultural component – a new form of national schools. The analysis is based on interviews which were recorded in 2007, in two Moscow schools, one of them with Armenian ethno-cultural component, and the other, with Azeri. The sample included ten students from each school (five boys and five girls).

In the paper the process of linguistic integration of Azeri and Armenian children into modern Russian society is analyzed. The comparison between these two groups is particularly appealing, because the effects of Soviet Russification, and the language situations in general, were different in Armenia and in Azerbaijan. I show that this difference influences the use of language by Azeri and Armenian children.

This chapter proposes an unfolding view of the EU as a sort of post-modern neo-medieval empire, in which narratives of othering towards Central and Eastern Europe preserve their salience.

In this paper we introduce distinction between “ontologically non-fregean” logics and “pragmatically non-fregean” ones; by means of such distinction a classification of non-fregean logics is presented as well. We believe that NFL must be considered as a many-leveled structure; each level taken separately may vary in different way – from classical to non-classical. It is not these levels themselves that we should call “fregean” or “non-fregean”, but the ways they are stuck together within the whole system. The more levels a system has, the more kinds of “fregean” and “non-fregean” we can find in it.

In article features of national and confessional self-identification of the Russian youth as parts of the title nation are considered. Ethnic and national consciousness are analyzed as significant components of process of individual and group self-identification. Research covers the studying and working youth which is arrived and which initially living in the city. The youth is the object which studying allows to predict regularities of social development in the future. Consideration of a problem considers multi-confessional, multi-ethnic and boundary in the geographical relation character of Ural as region. The emphasis is placed on specifics of behavior of representatives of title nation, as youth considerably defining a social portrait. The concept of the big city is used as steady, allocated with a number of characteristic features. Authors establish the reasons of the reduced interest to a religious and ethnic identification of with group at the young people belonging to different social groups and united by residence in the large city. The conditions necessary for an intensification of process of identification are defined. Means of updating of processes of formation of identity of youth are offered.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.