Minimal varieties of residuated lattices

Abstract

Abstract. In this paper we investigate the atomic level in the lattice of subvarieties of residuated lattices. In particular, we give infinitely many commutative atoms and construct continuum many non-commutative, representable atoms that satisfy the idempotent law; this answers Problem 8.6 of [12]. Moreover, we show that there are only two commutative idempotent atoms and only two cancellative atoms. Finally, we study the connections with the subvariety lattice of residuated bounded-lattices. We modify the construction mentioned above to obtain a continuum of idempotent, representable minimal varieties of residuated bounded-lattices and illustrate how the existing construction provides continuum many covers of the variety generated by the three-element non-integral residuated boundedlattice.

Generalized ordinal sums and translations

“…In this case we say that K is admissible by L. This is a slight generalization of the definition in [6]. A class K is admissible by a class L, if every algebra of K is admissible by every algebra of L.…”

“…Examples of FL o -algebras that are admissible by all residuated lattices are given in Figure 2 and they include To n , for n a positive natural number, and N w , for w an infinite or bi-infinite word; see [6] for the definitions. We will be interested in To 1 , which is the unique 3-element non-integral FL o -algebra.…”

“…[6] Assume that L is a class of residuated lattices and that K is a finite strictly simple residuated lattice admissible by L. Then,…”

“…The congruence lattice of K[L] is closely related to those of K and L. Moreover, the 'operator' K is functorial and commutes with homomorphic images, subalgebras and ultraproducts. The paper makes crucial use of the results in [6] and [5]. A partial preview of the results was given in [7].…”

Commutative idempotent residuated lattices

“…In fact, N. Galatos proved a stronger result in [8]: principal congruences in commutative n-potent residuated lattices are equationally definable. This result is indeed more complicated.…”