Paolo Agliano

On subtractive varieties II: General properties

As a sequel to [23] we investigate ideal properties focusing on subtractive varieties. After having listed a few basic results, we give several characterizations of the commutator of ideals and prove, for example, that it commutes with finite direct products. We deal with the ideal extension property (IEP) and with related commutator properties, showing for instance that IEP implies that the commutator commutes with restriction to subalgebras. Then we characterize varieties with distributive ideal lattices and relate this property with (a form of) equationally definable principal ideals and with IEP. Then, at the other extreme, we deal with Abelian and Hamiltonian properties (of ideals and congruences), giving for example a purely ideal theoretic characterization of varieties of Abelian groups with linear operations. To finish with, we present a few examples aiming at vindicating our work.

Prime spectra in modular varieties

By using the concept of modular commutator, prime congruences are defined for algebras in modular varieties. Then the prime spectrum of an algebra is defined and various spectral properties are discussed. In particular some conditions are given for the spectrum of an algebra to be homeomorphic to a ring spectrum.

Congruence semimodular varieties I: Locally finite varieties

The lattice of closed subsets of a set under such a closure operator is semimodular. Perhaps the best known example of a closure operator satisfying the exchange principle is the closure operator on a vector space W where for X ___ W we let C(X) equal the span of X. The lattice of C-closed subsets of W is isomorphic to Con(W) in a natural way; indeed, if Y _~ W x W and Cg(Y) denotes the congruence on W generated by Y, then the closure operator Cg satisfies the exchange principle. For another example, let C denote the closure operator on the set A x A where C(X) equals the equivalence relation generated by X ~_ A 2. This closure operator satisfies the exchange principle, so the lattice of all equivalence relations on A is semimodular. Equivalently, the congruence lattice of any set is semimodular. The preceding examples suggest to us that semimodularity may be a natural congruence condition worth investigating. Research into varieties of algebras with modular congruence lattices had led to the development of a deep structure theory for them. We wonder: how much of the structure involved in congruence modular varieties exists for congruence semimodular varieties? How much more diversity is permitted? This paper may be considered to be an attack on the former question

The one-block property in varieties of semigroups

The one-block property is stated and it is proved that, whenever it holds for a class closed under homomorphic images, it implies congruence semimodularity for the whole class. An equational characterization of regular varieties having the one-block property is obtained. Some characterization is obtained also for irregular varieties of semigroups having the one-block property.

Cosets in universal algebra

The aim of this paper is to show that a close investigation of congruence classes of an algebra may give useful information on varieties to which the algebra happens to belong. In principle our approach is quite distinct from that of Wille [9] since we are interested in giving a “natural” structure to the set of congruence classes containing a given element of the algebra, and to relate the properties of this structure to general features of the variety generated by the algebra. It comes out that such kind of results is better understood if we consider the set of congruence classes containing a given element as a subset of a generally richer family of subset of the algebra--we will call these “cosets” of the algebra-which in fact is endowed with the very natural structure of an algebraic lattice. It is easy to see that in many classical cases cosets are very-well-known animals: in the case of groups we get cosets (with respect to normal subgroups), in rings we get ideals (when the fixed element is zero), etc. One main result sounds as follows: for a variety V the fact that, for each algebra JZZ E V, for each LZE A the natural map a/ which sends a congruence R E Con(,&‘) into a/R is a lattice homomorphism is equivalent to congruence permutability. Our notation is, more or less, standard; for general background in Universal Algebra we refer to [ 1, 41.