Solvable covers of the boolean variety of unital ℓ-groups

The varieties of solvable lattice-ordered groups covering the abelian variety were shown independently by Gurchenkov, Reilly, and Darnel to be the Scrimger varieties of ?-groups and the three Medvedev representable covers. In this article, the authors give a parallel characterization of varieties of solvable unital ?-groups which cover the minimal nontrivial variety of boolean unital ?-groups.     

A variety of lattice-ordered groups containing all representable covers of the abelian variety

A small variety of representable lattice-ordered groups is constructed, which contains all of the representable covers of the abelian variety.     

Automorphisms and moduli spaces of varieties with ample canonical class via deformations of abelian covers

By a recent result of Viehweg, projective manifolds with ample canonical class have a coarse moduli space, which is a union of quasiprojective varieties.In this paper, we prove that there are manifolds with ample canonical class that lie on arbitrarily many irreducible components of the moduli; moreover, for any finite abelian group G there exist infinitely many components M of the moduli of varieties with ample canonical class such that the generic automorphism group G^Mis equal to G. In order to construct the examples, we use abelian covers. Let Y be a smooth complex projective variety of dimension ? 2. A Galois cover f :X ? Y whose Galois group is finite and abelian is called an abelian cover of Y; by [Pal], it is determined by its building data, i.e. by the branch divisors and by some line bundles on Y, satisfying appropriate compatibility conditions. Natural deformations of an abelian cover are also introduced in [Pal]. In this paper we prove two results about abelian covers:first, that if the building data are sufficiently ample, then the natural deformations surject on the Kuranishi family of X; second, that if the building data are sufficiently ample and generic, then Aut(X)= G.     

Covers of the Abelian Variety of Generalized MV-Algebras

Using the categorical equivalence of the class of generalized MV-algebras with the class of unital ?-groups, we describe all varieties of symmetric top abelian unital ?-groups that cover the variety  _u? of abelian unital ?-groups. Equivalently, we describe all cover varieties of the variety of MV-algebras, ?, within the variety of generalized MV-algebras admitting only one negation and each of whose maximal ideals is normal. In particular, there are continuum many representable varieties of generalized MV-algebras that cover ?.     

Varieties ofl-groups are finitely based

We deal with varieties of lattice-ordered groups {ie149-1} defined by the identity [xⁿ, yⁿ]=e. The structure of subdirectly indecomposable l-groups in the variety {ie149-2} is studied, and we establish that l-varieties satisfying the identity [xⁿ, yⁿ]=e and generated by a finitely generated l-group are finitely based. It is shown that l-varieties {ie149-3} with finite axiomatic rank {ie149-4} also have finite bases of identities. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 268–287, May–June, 1996.     

Countably valued lattice-ordered groups

A countably valued lattice-ordered group is a lattice-ordered group in which every element has only countably many values. Such lattice-ordered groups are proven to be normal-valued and, though not necessarily special-valued, every element in a countably valuedl-group must have a special value. The class of countably valuedl-groups forms a torsion class, and the torsion radical determined by this class is anl-ideal that is the intersection of all maximal countably valued subgroups.Countably valuedl-groups are shown to be closed with respect toeventually constant sequence extensions, and it is shown that many properties of anl-group pass naturally to its eventually constant sequence extension.Presented by M. Henriksen.     

Amalgamations ofl-varieties

It is proved that only o-approximable l-varieties can share the amalgamation property. We construct a formation of rigidly ordered groups that is not amalgamated in the variety of o-approximable l-groups and, in particular, in the variety of rigid l-groups. It is shown that all the known formations of o-approximable l-groups having no amalgams in the variety of o-approximable l-groups are amalgamated in the variety of all lattice-ordered groups. Translated fromAlgebra i Logika, Vol. 35, No. 1, pp. 31–40, January–February, 1996.     

Minimal varieties of residuated lattices

In this paper we investigate the atomic level in the lattice of subvarieties of residuated lattices. In particular, we give infinitely many commutative atoms and construct continuum many non-commutative, representable atoms that satisfy the idempotent law; this answers Problem 8.6 of [12]. Moreover, we show that there are only two commutative idempotent atoms and only two cancellative atoms. Finally, we study the connections with the subvariety lattice of residuated bounded-lattices. We modify the construction mentioned above to obtain a continuum of idempotent, representable minimal varieties of residuated bounded-lattices and illustrate how the existing construction provides continuum many covers of the variety generated by the three-element non-integral residuated bounded-lattice.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived August 1, 2003; accepted in final form April 27, 2004.     

Special-valued l-groups and Abelian covers

This paper presents a new and independent proof of the theorem (proven first by Kopytov and Gurchenkov [7] and again by Reilly [10]) that covers of the Abelian l-variety are either representable or are Scrimger covers. The proof in this paper is based upon the l-Cauchy constructions of Ball [1]; once these are applied to the problem, the proof becomes elementary.     

Completely regular semigroups in the variety generated by the bicyclic semigroup

We shall show that a completely regular semigroup is in the semigroup variety generated by the bicyclic semigroup if and only if it is an orthogroup whose maximal subgroups are abelian. Therefore the lattice of subvarieties of the variety generated by the bicyclic semigroup contains as a sublattice a countably infinite distributive lattice of semigroup varieties, each of which consists of orthogroups with maximal subgroups that are torsion abelian groups. In particular, every band divides a power of the bicyclic semigroup.Presented by B. M. Schein.     

Some notes on feinberg's k-independence problem

A short proof is given of a recent theorem of M. Feinberg on representable matroids. The result is shown not to hold for nonrepresentable matroids of rank 3.     

Invariant Measures in Free MV-Algebras

MV-algebras can be viewed either as the Lindenbaum algebras of ?ukasiewicz infinite-valued logic, or as unit intervals of lattice-ordered abelian groups in which a strong order unit has been fixed. The free n-generated MV-algebra Free _n is representable as an algebra of continuous piecewise-linear functions with integer coefficients over the unit cube [0, 1] ⁿ . The maximal spectrum of Free _n is canonically homeomorphic to [0, 1] ⁿ , and the automorphisms of the algebra are in 1–1 correspondence with the pwl homeomorphisms with integer coefficients of the unit cube. In this article, we prove that the only probability measure on [0, 1] ⁿ which is null on underdimensioned 0-sets and is invariant under the group of all such homeomorphisms is the Lebesgue measure. From the viewpoint of lattice-ordered abelian groups, this fact means that, in relevant cases, fixing an automorphism-invariant strong unit implies fixing a distinguished probability measure on the maximal spectrum. From the viewpoint of algebraic logic, it means that the only automorphism-invariant truth averaging process that detects pseudotrue propositions is the integral with respect to Lebesgue measure.     

Intervals in l-groups as L-algebras

L-algebras are related to algebraic logic and quantum structures. They were introduced by the first author [J. Algebra 320 (2008)], where a self-similar closure S(X) of any L-algebra X was employed to derive a criterion for X to be representable as an interval in a lattice-ordered group. In the present paper, this criterion is improved without using the embedding. It is shown that an L-algebra is representable as an interval in a lattice-ordered group if and only if it is semiregular with a smallest element and bijective negation. Any such L-algebra gives rise to a perfect dual with respect to the inverse of the negation. This is proved by a self-dual characterization of semiregularity.     

Covers in the Lattice of Varieties of m-Groups

We consider some questions on covers in the lattice of varieties of m-groups. We prove the existence of a nonabelian cover of the smallest nontrivial variety of m-groups. We show that there exists an uncountable set of o-approximable varieties of m-groups each of which has continuum many o-approximable covers. In the lattice of o-approximable varieties of m-groups we find a variety that has no covers in this variety and no independent basis of identities.     

Simply presented tag modules

A tag module is a generalization, in any abelian category, of a torsion abelian group. The theory of such modules is developed, it is shown that countably generated tag modules are simply presented, and that Ulm's theorem holds for simply presented tag modules. Zippin's theorem is stated and proved for countably generated tag modules.     

Ordering groups and validity in lattice-ordered groups

An inductive characterization is given of the subsets of a group that extend to the positive cone of a right order on the group. This characterization is used to relate validity of equations in lattice-ordered groups (?-groups) to subsets of free groups that extend to the positive cone of a right order. As a consequence, new proofs are obtained of the decidability of the word problem for free ?-groups and generation of the variety of ?-groups by the ?-group of automorphisms of the real line. An inductive characterization is also given of the subsets of a group that extend to the positive cone of an order on the group. In this case, the characterization is used to relate validity of equations in varieties of representable ?-groups to subsets of relatively free groups that extend to the positive cone of an order.     

Narrowness implies uniformity

This paper is about varietiesV of universal algebras which satisfy the following numerical condition on the spectrum: there are only finitely many prime integersp such thatp is a divisor of the cardinality of some finite algebra inV. Such varieties are callednarrow. The variety (or equational class) generated by a classK of similar algebras is denoted by V(K)=HSPK. We define Pr (K) as the set of prime integers which divide the cardinality of a (some) finite member ofK. We callK narrow if Pr (K) is finite. The key result proved here states that for any finite setK of finite algebras of the same type, the following are equivalent: (1) SPK is a narrow class. (2) V(K) has uniform congruence relations. (3) SK has uniform congruences and (3) SK has permuting congruences. (4) Pr (V(K))= Pr(SK). A varietyV is calleddirectly representable if there is a finite setK of finite algebras such thatV= V(K) and such that all finite algebras inV belong to PK. An equivalent definition states thatV is finitely generated and, up to isomorphism,V has only finitely many finite directly indecomposable algebras. Directly representable varieties are narrow and hence congruence modular. The machinery of modular commutators is applied in this paper to derive the following results for any directly representable varietyV. Each finite, directly indecomposable algebra inV is either simple or abelian.V satisfies the commutator identity [x,y]=x·y·[1,1] holding for congruencesx andy over any member ofV. The problem of characterizing finite algebras which generate directly representable varieties is reduced to a problem of ring theory on which there exists an extensive literature: to characterize those finite ringsR with identity element for which the variety of all unitary leftR-modules is directly representable. (In the terminology of [7], the condition is thatR has finite representation type.) We show that the directly representable varieties of groups are precisely the finitely generated abelian varieties, and that a finite, subdirectly irreducible, ring generates a directly representable variety iff the ring is a field or a zero ring.