Simple axiomatization of reticulations on residuated lattices

‎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, ‎4765]‎. ‎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 fcirc lambda_1 = lambda_2‎, ‎can be considered as a homomorphism theorem‎.