The lattice of interpretability types of varieties.

*(English)*Zbl 0559.08003
Mem. Am. Math. Soc. 305, 125 p. (1984).

Let V and W be varieties, not necessarily of the same type, and let \({}^-\) be a map relating each operation f of the language of V to a term \(\bar f\) in the language of W. Then \({}^-\) can naturally be extended to terms. V is said to be interpretable in W, written \(V\leq W\), if for each equation \(\sigma =\tau\) of V the corresponding equation \({\bar \sigma}={\bar \tau}\) holds in W. Put differently, V is interpretable in W if each W algebra can be considered as a V algebra by choosing as fundamental operations those \(\bar f\) which come from fundamental V-operations f. Category-theoretically, \(V\leq W\) if there is a functor \(\psi\) :W\(\to V\) preserving underlying sets.

Since \(\leq\) is not antisymmetric, V and W are identified if \(V\leq W\) and \(W\leq V\). In that case V and W are called equivalent, \(V\equiv W\). The \(\equiv\)-classes then form a partial order (L,\(\leq)\). Even though, L is actually a proper class, each subset of L has an infimum and a supremum, thus L is called a lattice. The lattice L was introduced by W. D. Neumann who associated Mal’cev conditions with certain filters in L. This memoir is devoted to a detailed study of L. A wealth of informations is obtained mainly on varieties that have played major roles in equational logic to relate those varieties within L. Finally 7 order diagrams are presented which express in condensed form much of the results of the monograph.

Joins and meets in L are given by coproducts, resp. products of varieties, so it is not too surprising that most of the varieties that have played central roles in equational logic are \(\wedge\)-irreducible, or even \(\wedge\)-prime. A result covering a large number of cases states that a union of a chain of varieties \(\cup V_ i\) where each \(V_ i\) is generated by a finite algebra with a prime number of elements, is \(\wedge\)-prime.

A Mal’cev filter is a filter in L generated by countably many finitely based varieties of finite type. Mal’cev filters represent exactly the classes of varieties in which a given Mal’cev condition is true. Here a Mal’cev filter is associated with every \(\wedge\)-irreducible locally finite variety V, thus yielding new Mal’cev definable classes of varieties. Many Mal’cev filters associated to familiar Mal’cev conditions are shown to be prime. Yet, interestingly enough, the Mal’cev filter of congruence distributive varieties is shown to be a proper intersection of two Mal’cev filters. Besides lattice theoretic notions, other varietal constructions such as k-th power varieties and k-th root filters are studied in connection with the ordering of V. An interesting example of a variety U equivalent to its power \(U^{[2]}\) deserves to be mentioned.

In drawing figures of parts of L, which, by the way, is nonmodular, it is usually hard to show that two varieties V and W are inequivalent. A variety of tools are used for this reason, for example topological spaces which occur as spaces in V but not in W or Mal’cev conditions holding in one variety but not in the other. As a major tool SIN-algebras (\b{s}imple under terpretation) are introduced which, in a sense, are invariant under different interpretations. Often, though, it remains open whether an inclusion is proper. So the monograph closes with a list of 27 open problems. For example, problem 2 asks whether there are any nontrivial covers in L.

Since \(\leq\) is not antisymmetric, V and W are identified if \(V\leq W\) and \(W\leq V\). In that case V and W are called equivalent, \(V\equiv W\). The \(\equiv\)-classes then form a partial order (L,\(\leq)\). Even though, L is actually a proper class, each subset of L has an infimum and a supremum, thus L is called a lattice. The lattice L was introduced by W. D. Neumann who associated Mal’cev conditions with certain filters in L. This memoir is devoted to a detailed study of L. A wealth of informations is obtained mainly on varieties that have played major roles in equational logic to relate those varieties within L. Finally 7 order diagrams are presented which express in condensed form much of the results of the monograph.

Joins and meets in L are given by coproducts, resp. products of varieties, so it is not too surprising that most of the varieties that have played central roles in equational logic are \(\wedge\)-irreducible, or even \(\wedge\)-prime. A result covering a large number of cases states that a union of a chain of varieties \(\cup V_ i\) where each \(V_ i\) is generated by a finite algebra with a prime number of elements, is \(\wedge\)-prime.

A Mal’cev filter is a filter in L generated by countably many finitely based varieties of finite type. Mal’cev filters represent exactly the classes of varieties in which a given Mal’cev condition is true. Here a Mal’cev filter is associated with every \(\wedge\)-irreducible locally finite variety V, thus yielding new Mal’cev definable classes of varieties. Many Mal’cev filters associated to familiar Mal’cev conditions are shown to be prime. Yet, interestingly enough, the Mal’cev filter of congruence distributive varieties is shown to be a proper intersection of two Mal’cev filters. Besides lattice theoretic notions, other varietal constructions such as k-th power varieties and k-th root filters are studied in connection with the ordering of V. An interesting example of a variety U equivalent to its power \(U^{[2]}\) deserves to be mentioned.

In drawing figures of parts of L, which, by the way, is nonmodular, it is usually hard to show that two varieties V and W are inequivalent. A variety of tools are used for this reason, for example topological spaces which occur as spaces in V but not in W or Mal’cev conditions holding in one variety but not in the other. As a major tool SIN-algebras (\b{s}imple under terpretation) are introduced which, in a sense, are invariant under different interpretations. Often, though, it remains open whether an inclusion is proper. So the monograph closes with a list of 27 open problems. For example, problem 2 asks whether there are any nontrivial covers in L.

Reviewer: H.-P.Gumm

##### MSC:

08B05 | Equational logic, Mal’tsev conditions |

08B15 | Lattices of varieties |

08B25 | Products, amalgamated products, and other kinds of limits and colimits |

08B10 | Congruence modularity, congruence distributivity |

08A40 | Operations and polynomials in algebraic structures, primal algebras |