みなさま，

東京大学の蓮尾です．来週は研究室のセミナーに 浦本武雄さん（京都大学数学教室D3）をお招きして，形式言語理論と 圏論的双対性の関連についての結果をお話いただきます． どうかご自由にご参加ください．

どうかよろしくお願いいたします．それでは！ 蓮尾 一郎 研究室ウェブページ： https://www-mmm.is.s.u-tokyo.ac.jp/indexj.html

============ Tue 30 April 2013, 15:30-17:30

Takeo Uramoto (Dept. Mathematics, Kyoto University) Tannaka-style reconstruction theorem for profinite monoids with an application to Eilenberg-Reiterman's variety theory

理学部7号館1階 102教室 Room 102, School of Science Bldg. No. 7 アクセス： https://www-mmm.is.s.u-tokyo.ac.jp/indexj.html （一番下） Access: http://www-mmm.is.s.u-tokyo.ac.jp/ (see bottom)

Eilenberg-Reiterman's variety theory is one of the central theories in algebraic studies of finite automata and regular languages. In its recent development, Rhodes and Steinberg pointed out that the Boolean algebra of regular languages admits a bialgebra structure and revealed its important role in the duality-theoretic formulation of the variety theory. In this talk, we study the structure of the bialgebra of regular languages using a traditional idea from the linear representation theory of groups. In particular, we aim at clarifying a close connection between algebraic structures on the bialgebra of regular languages and geometric structures of finite automata. Our main result claims that there exists a bijective correspondence between bilinear multiplications of regular languages that are compatible with respect to word decompositions and binary products on finite automata that are stable with respect to simulations. This is proved as a corollary of the Tannaka-style reconstruction theorem for profinite monoids, which is an analogue of the classical Tannaka duality theorem for compact groups and indicates a categorical universality of the Boolean algebra of regular languages with respect to finite automata and simulations.