Congruences in residuated lattices
Shuang Feng, Yang Jing-mei · 2015
The aim of this paper is to study congruences in residuated lattices. A congruence in an algebra in a universal sense is an equivalence which preserves all the algebraic operations. In every residuated lattice (L, ∧, V, ⊗, →), we show that an equivalence is a universal congruence, iff it preserves both → and ∧, iff it is respect to both → and ∧. If the residuated lattice is divisible, then an equivalence is a universal congruence iff it preserves both → and ⊗. Further, if the residuated lattice is an MV-algebra, then an equivalence is a universal congruence iff it just preserves →. A potential mistake in [8] is pointed out.