Splittings in the variety of residuated lattices Tomasz Kowalski, Hiroakira Ono Japan Advanced Institute of Science and Technology 1–1 Asahidai, Tatsunokuchi, Ishikawa 923–12, Japan {kowalski,ono}@jaist.ac.jp 14 September 1999 (revised 1 June 2000) A b s t r a c t. It is shown that the only algebra that splits the lattice of subvarieties of the variety R of residuated lattices is the two element boolean algebra. 1. Introduction A residuated lattice is an algebra A = 〈A; ∨, ∧, ·, →, 0, 1〉, such that: (1) 〈A; ∨, ∧, 0, 1〉 is a bounded lattice with the greatest element 1 and smallest 0; (2) 〈A; ·, 1〉 is a commutative monoid; (3) for all x, y, z ∈ A: x · y ≤ z iff x ≤ y → z . The operation ‘·’, often called fusion is distributive over join. In finite residuated lattices, fusion and join determine residuation uniquely, although residuation cannot be defined equationally from other operations. The class R of residuated lattices is a variety. It is arithmetical, has CEP, and is generated by its finite members (cf. [7]). It is also congruence 1-regular, i.e., for any congruence θ, the coset of 1 determines θ uniquely. Cosets of 1 are called congruence filters. Our interest in R stems from the fact that residuated lattices provide an algebraic semantics for logics without contraction (see, e.g., [8], [9] and [10], for details), also known as resource sensitive logics, which have recently enjoyed increasing popularity. The lattice of these logics is dual to the lattice of subvarieties of R. Among important subvarieties of R are the varieties E n , defined relative to R, for any n ∈ ω, by the identity: x n = x n+1 (E n ) where x n stands for x · ... · x    n times . These correspond to logics with restricted contraction, and can easily be shown (cf. [5]) to have equationally definable principal congruences (EDPC). For a thorough discussion of varieties with EDPC, see [1], [2] and [3]. The support from Japan Society for the Promotion of Science, Grant-in-Aid for Exploratory Research No. 11874016 and for Scientific Research (B)(1) No. 10440027, is gratefully acknowledged. 1