Amalgamation Property for Varieties of BL-algebras Generated by One Chain with Finitely Many Components

Transfer theorems for finitely subdirectly irreducible algebras

“…The equivalence of (1) and ( 2) is well-known ([3, Lemma 1.2]), and the equivalence of ( 1) and (3), and the implication from (3) to (4) follow from Theorem 2.3 and Lemma 2.6, respectively. Hence it remains to show that (4) implies (2). Suppose that V FSI has the EP and let B ∈ V FSI .…”

“…The non-trivial subvarieties of MV, G, and P that have the AP are the varieties generated by Γ(Z, n) or Γ(Z × lex Z, n, 0 ) (n ∈ N >0 ), MV, G, G 3 , and P [10]. Also, BL has the AP [22] and a precise characterization is given in [2] of the varieties of BL-algebras that have the AP that are generated by a single totally BL-algebra built as an ordinal sum of finitely many totally ordered MV-algebras and product algebras. Here, we consider subvarieties of BL 1 := MV ∨G∨P, each of which is generated by a class of "one-component" totally ordered BL-algebras, i.e., members of MV FSI , G FSI , and P FSI .…”

“…BL-algebras, introduced by Hájek in [14] as an algebraic semantics for his basic fuzzy logic of continuous t-norms, have been studied intensively over the past twenty five years, mostly in the framework of residuated lattices (see, e.g. [1,2,16,21,22]). In this section, we use the tools developed in previous sections to contribute to the still incomplete picture of varieties of BL-algebras that have the AP.…”

“…Note also that in this case, V FSI is a positive universal class if and only if Con A is a chain (totally ordered set) for each A ∈ V FSI . 2 An algebra B is congruence-distributive if Con B is distributive, and has the congruence extension property (for short, CEP) if for any subalgebra A of B and Θ ∈ Con A, there exists a Φ ∈ Con B such that Φ ∩ A 2 = Θ. A class of algebras is said to have one of these properties if each of its members has the property.…”