Open Access
January, 2017 More on 2-chains with 1-shell boundaries in rosy theories
SunYoung KIM, Junguk LEE
J. Math. Soc. Japan 69(1): 93-109 (January, 2017). DOI: 10.2969/jmsj/06910093


In [4], B. Kim, and the authors classified 2-chains with 1-shell boundaries into either RN (renamable)-type or NR (non renamable)-type 2-chains up to renamability of support of subsummands of a 2-chain and introduced the notion of chain-walk, which was motivated from graph theory: a directed walk in a directed graph is a sequence of edges with compatible condition on initial and terminal vertices between sequential edges. We consider a directed graph whose vertices are 1-simplices whose supports contain $0$ and edges are plus/minus of $2$-simplices whose supports contain $0$. A chain-walk is a 2-chain induced from a directed walk in this graph. We reduced any 2-chains with 1-shell boundaries into chain-walks having the same boundaries.

In this paper, we reduce any 2-chains of 1-shell boundaries into chain-walks of the same boundary with support of size $3$. Using this reduction, we give a combinatorial criterion determining whether a minimal 2-chain is of RN- or NR-type. For a minimal RN-type 2-chains, we show that it is equivalent to a 2-chain of Lascar type (coming from model theory) if and only if it is equivalent to a planar type 2-chain.


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SunYoung KIM. Junguk LEE. "More on 2-chains with 1-shell boundaries in rosy theories." J. Math. Soc. Japan 69 (1) 93 - 109, January, 2017.


Published: January, 2017
First available in Project Euclid: 18 January 2017

zbMATH: 06701584
MathSciNet: MR3597548
Digital Object Identifier: 10.2969/jmsj/06910093

Primary: 03C45
Secondary: 05E45 , 55N35

Keywords: 2-chain having a 1-shell boundary , homology groups , Lascar 2-chains , matrix expression , RN/NR-type 2-chains , rosy theories

Rights: Copyright © 2017 Mathematical Society of Japan

Vol.69 • No. 1 • January, 2017
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