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The lattice of balanced equivalence relations of a coupled cell network
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Stewart, Ian, 1945. (2007) The lattice of balanced equivalence relations of a coupled cell network. Cambridge Philosophical Society. Mathematical Proceedings, Vol.143 (No.1). pp. 165183. ISSN 03050041

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Official URL: http://dx.doi.org/10.1017/S0305004107000345
Abstract
A coupled cell system is a collection of dynamical systems, or ‘cells’, that are coupled together. The associated coupled cell network is a labelled directed graph that indicates how the cells are coupled, and which cells are equivalent. Golubitsky, Stewart, Pivato and Török have presented a framework for coupled cell systems that permits a classification of robust synchrony in terms of the concept of a ‘balanced equivalence relation’, which depends solely on the network architecture. In their approach the network is assumed to be finite. We prove that the set of all balanced equivalence relations on a network forms a lattice, in the sense of a partially ordered set in which any two elements have a meet and a join. The partial order is defined by refinement. Some aspects of the theory make use of infinite networks, so we work in the category of networks of ‘finite type’, a class that includes all locally finite networks. This context requires some modifications to the standard framework. As partial compensation, the lattice of balanced equivalence relations can then be proved complete. However, the intersection of two balanced equivalence relations need not be balanced, as we show by a simple example, so this lattice is not a sublattice of the lattice of all equivalence relations with its usual operations of meet and join. We discuss the structure of this lattice and computational issues associated with it. In particular, we describe how to determine whether the lattice contains more than the equality relation. As an example, we derive the form of the lattice for a linear chain of identical cells with feedback.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Mathematics 
Library of Congress Subject Headings (LCSH):  Dynamics, Equivalence relations (Set theory), Lattice theory 
Journal or Publication Title:  Cambridge Philosophical Society. Mathematical Proceedings 
Publisher:  Cambridge University Press 
ISSN:  03050041 
Official Date:  24 July 2007 
Volume:  Vol.143 
Number:  No.1 
Page Range:  pp. 165183 
Identification Number:  10.1017/S0305004107000345 
Status:  Peer Reviewed 
Access rights to Published version:  Open Access 
Funder:  Engineering and Physical Sciences Research Council (EPSRC) 
References:  [1] J.ALDIS. A polynomial time algorithm to determine maximal balanced equivalence relations. Internat 
URI:  http://wrap.warwick.ac.uk/id/eprint/651 
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