The Subquasivariety Lattice of a Discriminator Variety

Abstract

Let V be a discriminator variety such that the class B=[A # V: A is simple and has no trivial subalgebra] is closed under ultraproducts. This property holds, for example, if V is locally finite or if the language is finite. Let v(V) and q(V) denote the lattice of subvarieties and subquasivarieties of V, respectively. We prove that q(V) is modular iff q(V) is distributive iff v(V) satisfies a certain condition where the case in which the language has a constant symbol is``v(V) is a chain or q(V) =v(V).'' We give an isomorphism between q(V) and a lattice constructed in terms of v(V). Via this isomorphism we characterize the completely meet irreducible (prime) elements of q(V) in terms of the completely meet irreducible elements of v(V). We conclude the paper with applications to the varieties of Boolean algebras, relatively complemented distributive lattices, 4ukasiewicz algebras, Post algebras, complementary semigroups of rank k (x n rx)-rings, R 5 lattices (P-algebras, B-algebras), and monadic algebras. Academic Press

Open questions related to the problem of Birkhoff and Maltsev

“…The crucial point in establishing a formula for dm(L n ) lay in the case that n − 1 is a product of powers of two or more primes. When n − 1 is a power of a prime, it was relatively easy to establish the formula for dm(L n ), since this is precisely when the Q-lattice of the variety associated with L n is distributive (see Blanco, Campercholi, and Vaggione [26]). …”

Least V-quasivarieties of MV-algebras

“…Some special quasivarieties of MV-algebras have been studied [3,12,14,15,17]. For instance quasivarieties generated by chains.…”

“…In the case of MV-algebras, a quasivariety is locally finite if and only if it is subquasivariety of a discriminator variety, that is a subquasivariety of a variety of the form V I,∅ . The lattice of subquasivarieties of a discriminator variety have been deeply studied in [3]. Moreover every locally finite quasivariety is generated by its critical algebras [11], where an algebra A is said to be critical iff it is a finite algebra not belonging to the quasivariety generated by all its proper subalgebras.…”

MV-algebras: a variety for magnitudes with archimedean units

“…When K is a variety RCEP coincides with the congruence extension property (CEP for short). 4 Here ∨ is the MV-algebraic sup operation of (2). Proof.…”

“…Trivially, SFMP and FEP are satisfied by every locally finite quasivariety. Locally finite quasivarieties of MV-algebras are studied in [48] and [4], and a proof is given that they are finitely generated and finitely based. On the other hand, there exist strict quasivarieties satisfying SFMP and FEP which are not locally finite.…”