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Noncommutative network models

Published online by Cambridge University Press:  11 November 2019

Joe Moeller Open the ORCID record for Joe Moeller [Opens in a new window]
Joe Moeller*
Affiliation:
Department of Mathematics, University of California, Riverside, Riverside, CA, USA
*
Email: [email protected]
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Abstract

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Network models, which abstractly are given by lax symmetric monoidal functors, are used to construct operads for modeling and designing complex networks. Many common types of networks can be modeled with simple graphs with edges weighted by a monoid. A feature of the ordinary construction of network models is that it imposes commutativity relations between all edge components. Because of this, it cannot be used to model networks with bounded degree. In this paper, we construct the free network model on a given monoid, which can model networks with bounded degree. To do this, we generalize Green’s graph products of groups to pointed categories which are finitely complete and cocomplete.

Keywords

graph productuniversal algebraKneser graphmonoidal categorynetwork
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 30 , Issue 1 , January 2020 , pp. 14 - 32
Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - SA
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike licence (http://creativecommons.org/licenses/by-nc-sa/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is included and the original work is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use.
Copyright
© The Author(s) 2019

References

Adámek, J., Lawvere, F.W. and Rosický, J. (2003). On the duality between varieties and algebraic theories. Algebra Universalis 49 (1) 35–49.CrossRefGoogle Scholar
Baez, J.C., Foley, J., Moeller, J. and Pollard, B.S. (2018). Network models. arXiv:1711.00037.Google Scholar
Burris, S.N. and Sankappanavar, H.P. (1981). A Course in Universal Algebra, New York, Springer-Verlag.CrossRefGoogle Scholar
Fountain, J. and Kambites, M. (2009). Graph products of right cancellative monoids. Journal of the Australian Mathematical Society 87 (2) 227–252.CrossRefGoogle Scholar
Green, E.R. (1990). Graph Products of Groups. Phd thesis, University of Leeds.Google Scholar
Hungerford, T.W. (1974). Algebra, Graduate Texts in Mathematics, vol. 73, New York, Springer-Verlag.Google Scholar
Joyal, A. and Street, R. (1993). Braided tensor categories. Advances in Mathematics 102 (1) 20–78.CrossRefGoogle Scholar
Lawvere, F.W. (1963). Functorial Semantics of Algebraic Theories. Phd thesis, Columbia University.CrossRefGoogle ScholarPubMed
Lawvere, F.W. (1989). Display of graphics and their applications, as exemplified by 2-categories and the hegelian “taco”. In: Proceedings of the First International Conference on Algebraic Methodology and Software Technology, University of Iowa, 51–74.Google Scholar
Lovász, L. (1978). Kneser’s conjecture, chromatic number, and homotopy. Journal of Combinatorial Theory, Series A, 25 (3) 319–324.CrossRefGoogle Scholar
Mac Lane, S. (1998). Categories for the Working Mathematician. Berlin, Springer.Google Scholar
Margolis, S., Saliola, F. and Steinberg, B. (to appear). Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry. Memoirs of the American Mathematical Society.Google Scholar
Quillen, D. (1967). Homotopical Algebra, Lecture Notes in Mathematics, vol. 43, Berlin, Heidelberg, Springer-Verlag.CrossRefGoogle Scholar
Selinger, P. (2011). A survey of graphical languages for monoidal categories. In: Coecke, B. (ed.) New Structures for Physics, Berlin, Heidelberg, Springer-Verlag, 289–355.Google Scholar
Veloso da Costa, A. (2001). Graph products of monoids. Semigroup Forum 63 247–277.CrossRefGoogle Scholar