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On a Connection between Information and Group Lattices
FICO, 3661 Valley Centre Drive, San Diego, CA 92130, USA
Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, CO 80523, USA
* Author to whom correspondence should be addressed.
Received: 19 January 2011; in revised form: 14 March 2011 / Accepted: 14 March 2011 / Published: 18 March 2011
Abstract: In this paper we review a particular connection between information theory and group theory. We formalize the notions of information elements and information lattices, first proposed by Shannon. Exploiting this formalization, we expose a comprehensive parallelism between information lattices and subgroup lattices. Qualitatively, isomorphisms between information lattices and subgroup lattices are demonstrated. Quantitatively, a decisive approximation relation between the entropy structures of information lattices and the log-index structures of the corresponding subgroup lattices, first discovered by Chan and Yeung, is highlighted. This approximation, addressing both joint and common entropies, extends the work of Chan and Yeung on joint entropy. A consequence of this approximation result is that any continuous law holds in general for the entropies of information elements if and only if the same law holds in general for the log-indices of subgroups. As an application, by constructing subgroup counterexamples, we find surprisingly that common information, unlike joint information, obeys neither the submodularity nor the supermodularity law. We emphasize that the notion of information elements is conceptually significant—formalizing it helps to reveal the deep connection between information theory and group theory. The parallelism established in this paper admits an appealing group-action explanation and provides useful insights into the intrinsic structure among information elements from a group-theoretic perspective.
Keywords: information element; entropy; information lattice; subgroup lattice; information inequality; information law; common information; joint information; isomorphism
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MDPI and ACS Style
Li, H.; Chong, E.K.P. On a Connection between Information and Group Lattices. Entropy 2011, 13, 683-708.
Li H, Chong EKP. On a Connection between Information and Group Lattices. Entropy. 2011; 13(3):683-708.
Li, Hua; Chong, Edwin K. P. 2011. "On a Connection between Information and Group Lattices." Entropy 13, no. 3: 683-708.