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

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

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.