We are pleased to announce additional seminar talks starting at 2:30 pm TODAY prior to Prof. Rhodes' lecture: UNIVERSITY OF HERTFORDSHIRE MATHEMATICAL COMPUTER SCIENCE & APPLICATIONS SEMINAR followed by the COMPUTER SCIENCE RESEARCH COLLOQUIUM Wright Building, Room F326 Hatfield, College Lane Campus * * * Preceding Prof Rhodes' talk will be an impromptu * * * *** Mathematical Computer Science & Applications Seminar **** "Algebra and Computation: Analogues of the Decimal Expansion for Automata, Semigroups, and Groups with Applications to Computer-Generated Coordinate Systems for Understanding" by Prof. C. L. Nehaniv & Dr. Attila Egri-Nagy (Algorithms, Adaptive Systems & BioComputation Research Groups, School of Computer Science, University of Hertfordshire) Main Colloquium Lecture 4-5+ pm: "Prime Decomposition Theorem for Finite Idempotent Semirings using the Triangular Product of B. I. Plotkin" Prof. John L. Rhodes (Mathematics Department, University of California, Berkeley, U.S.A.) Coffee/tea and biscuits will be available prior to the 4pm lecture Everyone is Welcome to Attend Abstracts follow: Content of Talks by Prof. Nehaniv and Dr. Egri-Nagy (2:30 pm): -------------------------------------------------------------- Computer algebra tools have been and are being developed at the University of Hertfordshire to compute Krohn-Rhodes and related group-theoretic coordinate systems. Applications include artificial intelligence, machine intelligence (automatic solving of symmetry structures such as Rubik's cube, without learning), as well as understanding of complex systems such as biochemical and genetic regulatory networks via apply the tools to automata models (Petri nets, reaction graphs, Boolean networks, etc.). We briefly explain and overview how Krohn-Rhodes Theory for finite semigroups and automata and Frobenius-Lagrange embeddings for finite groups can be viewed as providing *coordinate systems* for understanding and manipulating these structures. These, and analogues of these theorems proved for all, possibly infinite, semigroups and groups, can be considered analogues of the decimal expansion in which different coordinates correspond to divisors of the original structure. These are related to level-transitive spherically homogeneous actions on order-theoretic rooted trees. In the case of countably generated semigroups and groups the existence of such an action on a finitely branching rooted tree of order type of the natural numbers is equivalent to residual finiteness, and relates to the study of infinite groups (such as the so-called Ukrainian automata groups of Grigorchuk, Sidki, et al. and Burnside groups). Abstract of Main Lecture by Prof. Rhodes (starting 4 pm): -------------------------------------------------------------- A Prime Decomposition Theorem for finite idempotent semirings is proved using the triangular product of Plotkin adapted to semirings. A pair of results referred to as the Triangular Decomposition Theorem and the Ideal Decomposition Theorem are presented. Applying these in the context of idempotent semirings yields the decomposition half of the Prime Decomposition Theorem for idempotent semirings. Further portions of the talk are devoted to proving matrix algebras over the power set of a finite group are irreducible with respect to the triangular product. A moral of the talk is that much more of ring theory works over semirings than one would except. Applications to computing group complexity of the power set of a finite semigroup are given. This is new joint research with Ben Steinberg and is covered in Chapter 9 of our recent book 'The q-theory of Finite Semigroups' Springer 2008. The talk will be adapted to a computer science audience. -------------------------------------------------- Hertfordshire Computer Science Research Colloquium http://homepages.feis.herts.ac.uk/~nehaniv/colloq Mathematical Computer Science & Applications Seminar http://homepages.feis.herts.ac.uk/~comqcln/mcsa/