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The structure of idempotent residuated chains

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Abstract

In this paper we study some special residuated lattices, namely, idempotent residuated chains. After giving some properties of Green’s relation \( \mathcal{D} \) on the monoid reduct of an idempotent residuated chain, we establish a structure theorem for idempotent residuated chains. As an application, we give necessary and sufficient conditions for a band with an identity to be the monoid reduct of some idempotent residuated chain. Finally, based on the structure theorem for idempotent residuated chains, we obtain some characterizations of subdirectly irreducible, simple and strictly simple idempotent residuated chains.

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Authors and Affiliations

  1. Department of Mathematics, Northwest University Xi’an, Shaanxi, 710069, China

    Wei Chen & Xianzhong Zhao

Authors
  1. Wei Chen
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  2. Xianzhong Zhao
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Correspondence to Wei Chen.

Additional information

This work is supported by a grant of NSF, China # 10471112 and a grant of Shaanxi Provincial Natural Science Foundation # 2005A15.

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Chen, W., Zhao, X. The structure of idempotent residuated chains. Czech Math J 59, 453–479 (2009). https://doi.org/10.1007/s10587-009-0031-5

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  • DOI: https://doi.org/10.1007/s10587-009-0031-5

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