Congruence semimodular varieties. I: Locally finite varieties
[Part II is reviewed below.] The authors try to answer the question: How much of the structure involved in congruence modular varieties exists for congruence semimodular (CSM) varieties? They examine locally finite CSM varieties. For any finite algebra \(A\) in a CSM variety, natural congruences \(\underset {T}\sim\) are defined which play a role similar to that played by the commutator for algebras in a congruence modular variety. It is shown that if \(\underset {T} \sim\) is a congruence on the finite algebras of some locally finite CSM variety \(V\), then \(\underset {T} \sim\) induces a congruence \(\underset {T}\simeq\) on the subvariety lattice \(L_ V\) of \(V\). Furthermore, the quotient lattice \(L_ V/\underset {T} \simeq\) is both algebraic and dually algebraic and the natural map of \(L_ V\) onto \(L_ V/\underset {T}\simeq\) is complete (Theorem 4.3). The existence of irredundant subdirect product decomposition in CSM varieties is investigated. Further, it is proved that if \(V\) is a variety with the Congruence Extension Property, then \(V\) is CSM iff the 5-generated free \(V\)-algebra is CSM (Theorem 7.3). Theorem 8.5 states that a locally finite variety \(V\) is geometric iff \(V\) is a varietal product of a strongly Abelian geometric variety and an affine variety whose corresponding ring is finite and semi-simple.
scientific article; zbMATH DE number 679604
scientific article