On Jónsson's theorem (Q797617)

This paper discusses applications and generalizations of Jónsson's famous theorem, stating that in a congruence distributive variety V(K), generated by a class K, the subdirectly irreducible members remain in \(HSP_ U(K)\), where \(P_ U(K)\) denotes the class of ultraproducts of members of K. With the discovery of a commutator operation in congruence modular varieties several generalizations to such varieties were developed by several authors, best collected in Hrushovskij's theorem: Let S be subdirectly irreducible in V(K) with monolith (smallest nontrivial congruence) \(\mu\). Let \(\alpha\) be maximal with \([\alpha,\mu]=0\). Then \(S/\alpha \in HSP_ U(K).\) Freese and McKenzie have in particular studied the case when V(K) is locally finite, obtaining that \(S/\alpha \in SP_ UHS(K)\). Moreover, if A is a finite algebra and \(B\in V(K)\) is simple, then \(| B| \leq | A|\). They also give conditions when two algebras A and B, generating the same variety, have to be isomorphic. The paper discusses those results and includes a proof of Hrushovskij's theorem.