Modular subalgebra lattices

Abstract

Varieties in which every algebra has a modular subalgebra lattice were studied by Trevor Evans and Bernhard Ganter [1]. They proved that every such variety is Hamiltonian (i.e., any subalgebra is a congruence class of a suitable congruence), and if, in addition, the variety is idempotent (i.e., every one element subset is a subalgebra), then it is a variety of trivial algebras in the sense that every operation is a projection. Moreover, they gave a characterization of the subalgebra-rnodular varieties through a condition for all ternary terms of the variety. We will restate this result as Theorem 10 below. Our experience with groups taught us that while the subgroup lattice is modular for groups of quite different structure (see Iwasawa [2]), the modularity of the subgroup lattice of the direct square G x G implies that G is Abelian (Lukfics and Pfilfy [41). This simple observation inspired the general question, what can be deduced if the modularity of the subalgebra lattice Sub (A x A) is assumed. It turned out that the results of Evans and Ganter can be generalized. Namely, it follows that A is Hamiltonian and whenever A is idempotent it must be trivial, with the exception of one two-element algebra. In order to prove that the algebra is Abelian (in the sense that it satisfies the Term Condition (TC)), we had to assume that Sub (A 4) is modular. Furthermore, the distributivity of Sub (A 3) implies that the algebra is strongly Abelian, i.e., it satisfies the strong Term Condition (TC*) introduced by McKenzie [5]. We consider modularity and distributivity of the subalgebra lattice for direct products, because these properties are inherited by factor algebras and also, trivially, by subalgebras. However, we give a simple example where the subalgebra lattice of some factor algebra A/O does not belong to the lattice variety generated by Sub A.