# Simple axiomatization of reticulations on residuated lattices

Document Type : Research Paper

Author

Department of mathematics‎, ‎School of System Design and Technolodgy‎, ‎Tokyo Denki University‎, ‎Japan.

Abstract

‎We give a simple and independent axiomatization of reticulations on residuated lattices‎, ‎which were axiomatized by five conditions in [C‎. ‎Mureşan‎, ‎The reticulation of a residuated lattice‎, ‎Bull‎. ‎Math‎. ‎Soc‎. ‎Sci‎. ‎Math‎. ‎Roumanie‎ ‎51 (2008)‎, ‎no‎. ‎1‎, ‎47--65]‎. ‎Moreover‎, ‎we show that reticulations can be considered as lattice homomorphisms between residuated lattices and bounded distributive lattices‎. ‎Consequently‎, ‎the result proved by Muresan in 2008‎, ‎for any two reticulattions $(L_1‎, ‎\lambda_1)‎, ‎(L_2‎, ‎\lambda_2)$ of a residuated lattice $X$ there exists an isomorphism $f‎: ‎L_1 \to L_2$ such that $f\circ \lambda_1 = \lambda_2$‎, ‎can be considered as a homomorphism theorem‎.

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#### References

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### History

• Receive Date: 14 October 2015
• Revise Date: 19 April 2016
• Accept Date: 19 April 2016