Bounded and Well-Placed Theories in the Lattice of Equational Theories

Among the earliest publications of Evelyn Nelson are four papers which appeared in 1971—Nelson [1971a, 1971b] and Burris and Nelson [1971/72, 1971]. These papers contributed to our understanding of the intricate structural diversity of the lattice of equational theories. In particular, these papers revealed that large partition lattices occur as intervals in the lattice of equational theories of semigroups. In the present paper, we focus on some contrasting features of the lattice of equational theories. We identify a filter in the lattice of equational theories, called the filter of bounded theories. This filter is countably infinite and fairly tractable, especially for similarity types which provide only one operation symbol. Every equational theory can be viewed as a fully invariant congruence relation on the term algebra of the appropriate similarity type. From this viewpoint, certain classes of equational theories emerge from the imposition of various cardinality restrictions on the equivalence classes of terms induced by these theories. At one extreme are the term-finite equational theories—those theories each of whose equivalence classes is finite. At another extreme, one should put those theories which induce only finitely many equivalence classes of terms. However, it is easy to see that, for each similarity type, there is only one such theory: the trivial theory based on x ≈ y. (Officially, we take equations to be ordered pairs of terms, but we frequently use the suggestive notation s ≈ t in place of the official 〈s, t〉.) A slightly more elaborate notion proves more suitable. Observe that for any equational theory T , the image of any T -equivalence class of terms with respect to any automorphism of the term algebra is again a T -equivalence class. So the automorphism group of the term algebra partitions the T -equivalence classes into orbits. We say that an equational theory T is bounded iff the automorphism group of the term algebra partitions the T -equivalence classes into finitely many orbits.