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.

Keywords

Main Subjects


L.P. Belluce, Semisimple algebras of in_nite valued logic and bold fuzzy set theory, Canad. J. Math. 38 (1986), no. 6, 1356--1379.

G. Georgescu, The reticulation of a quantale, Rev. Roumaine Math. Pures Appl. 40 (1995), no. 7-8, 619--631.

L. Leuştean, The prime and maximal spectra and the reticulation of BL-algebras, Central Europen J. Math. 1 (2003), no. 3, 382--397.

C. Mureşan, The reticulation of a residuated lattice, Bull. Math. Soc. Sci. Math. Roumanie 51 (2008), no. 1, 47--65.

H. Simmons, Reticulated rings, J. Algebra 66 (1980), no. 1, 169--192.