Positive Sugihara monoids

It is proved that in the variety of positive Sugihara monoids, every finite subdirectly irreducible algebra is a retract of a free algebra. It follows that every quasivariety of positive Sugihara monoids is a variety, in contrast with the situation in several neighboring varieties. This result shows that when the logic R-mingle is formulated with the Ackermann constant t, then its full negation-free fragment is hereditarily structurally complete. Presented by R. W. Quackenbush. Received August 28, 2005; accepted in final form July 31, 2006.     

Special reflexive graphs in modular varieties

We investigate a special kind of reflexive graph in any congruence modular variety. When the variety is Maltsev these special reflexive graphs are exactly the internal groupoids, when the variety is distributive they are the internal reflexive relations. We use these internal structures to give some characterizations of Maltsev, distributive and arithmetical varieties.Received August 6, 2003; accepted in final form June 4, 2004.     

M-solid varieties generated by lattices

Following W. Taylor, we define an identity to be hypersatisfied by a variety V iff, whenever the operation symbols of V are replaced by arbitrary terms (of appropriate arity) in the operations of V, then the resulting identity is satisfied by V in the usual sense. Whenever the identity is hypersatisfied by a variety V, we shall say that is a hyperidentity of V, or a V hyperidentity. When the terms being substituted are restricted to a submonoid M of all the possible choices, is called an M-hyperidentity, and a variety V is M-solid if each identity is an M-hyperidentity. In this paper we examine the solid varieties whose identities are lattice M-hyperidentities. The M-solid varieties generated by the variety of lattices in this way provide new insight on the construction and representation of various known classes of non-commutative lattices. Received October 8, 1999; accepted in final form March 22, 2000.     

Hyperidentities and solid varieties

Let V be a variety of type τ. A type τ hyperidentity of V is an identity of V which also holds in an additional stronger sense: for every substitution of terms of the variety (of appropriate arity) for the operation symbols in the identity, the resulting equation holds as an identity of the variety. Such identities were first introduced by Walter Taylor in [27] in 1981. A variety is called solid if all its identities also hold as hyperidentities. For example, the semigroup variety of rectangular bands is a solid variety. For any fixed type τ, the collection of all solid varieties of type τ forms a complete lattice which is a sublattice of the lattice L(τ) of all varieties of type τ. In this paper we give an overview of the study of hyperidentities and solid varieties, particularly for varieties of semigroups, culminating in the construction of an infinite collection of solid varieties of arbitrary type. This paper is dedicated to Walter Taylor. Received July 16, 2005; accepted in final form January 3, 2006. This paper is an expanded version of a talk presented at the Conference on Algebras, Lattices and Varieties in Honour of Walter Taylor, in Boulder Colorado, August 2004. The author’s research is supported by NSERC of Canada.     

On curves of genus 2 with Jacobian of GL2-type

Ribet [Ri] has generalized the conjecture of Shimura–Taniyama–Weil to abelian varieties defined over Q,giving rise to the study of abelian varieties of GL₂-type. In this context, all curves over Q of genus one have Jacobian variety of GL₂-type. Our aim in this paper is to begin with the analysis of which curves of genus 2 have Jacobian variety of GL₂-type. To this end, we restrict our attention to curves with rational Rosenhain model and non-abelian automorphism group, and use the embedding of this group into the endomorphism algebra of its Jacobian variety to determine if it is of GL₂-type. Received: 31 March 1998 / Revised version: 29 June 1998     

Characterizations of certain completely regular varieties

Abstract. We characterize orthogroups, local orthogroups and (left,right) cryptogroups within completely regular semigroups by means of absence of certain kind of subsemigroups. For each of these varieties V , we determine the complete set of minimal non-V -varieties. For each of the latter varieties, we determine the lattice of its subvarieties. We then give a generating semigroup and a basis of its identities for every variety which occurs in this way. The subvariety lattices are illustrated by three diagrams.     

Irredundant self-dual bases for self-dual lattice varieties

For any finitely based self-dual variety of lattices, we determine the sizes of all equational bases that are both irredundant and self-dual. We make the same determination for {0, 1}-lattice varieties.Received July 11, 2002; accepted in final form August 27, 2004.     

Varieties of finite supersolvable groups with the M. Hall property

The varieties in the title are shown to be precisely the product varieties G_p*Ab(d) for some prime p and some positive integer d dividing p−1. Here G_p denotes the variety of all finite p-groups and Ab(d) the variety of all finite Abelian groups of exponent dividing d. It turns out that these are exactly those varieties H of supersolvable groups for which all finitely generated free pro-H groups are freely indexed in the sense of Lubotzky and van den Dries. Several alternative characterizations of these varieties are presented. Some applications to formal language theory and finite monoid theory are also given. Among these is the determination of all supersolvable solutions H to the equations PH = J*H and J*H = J H which is, to the present date, the most complete solution to a problem raised by Pin. Another consequence of our results is that for each such variety H the monoid variety PH = J*H = J H has decidable membership. The authors gratefully acknowledge the support of NSERC     

Minimal Generators in Varieties

Given a variety V, the free V-algebra F 1 on one generator represents the canonical underlying functor from V to the category Set of sets. Hence one might ask, whether F 1 is, in some sense, a `canonical' generator of V. To make this question precise the notion of `minimal varietal generator' is introduced. It is shown that in many (though not all) varieties F 1 is a generator of this kind, and often even the unique one. The question whether every variety has a (unique) minimal varietal generator, remains on open problem.     

Homology stability for classical regular semisimple varieties

We use techniques from homotopy theory, in particular the connection between configuration spaces and iterated loop spaces, to give geometric explanations of stability results for the cohomology of the varieties of regular semisimple elements in the simple complex Lie algebras of classical type A, B or C, as well as in the group . We show that the cohomology spaces of stable versions of these varieties have an algebraic stucture, which identifies them as “free Poisson algebras” with suitable degree shifts. Using this, we are able to give explicit formulae for the corresponding Poincaré series, which lead to power series identities by comparison with earlier work. The cases of type B and C involve ideas from equivariant homotopy theory. Our results may be interpreted in terms of the actions of a Weyl group on its coinvariant algebra (i.e. the coordinate ring of the affine space on which it acts, modulo the invariants of positive degree; this space coincides with the cohomology ring of the flag variety of the associated Lie group) and on the cohomology of its associated complex discriminant variety. Received August 31, 1998; in final form August 1, 1999 / Published online October 30, 2000     

Subdirect products of certain varieties of unary algebras

J. Płonka in [12] noted that one could expect that the regularization ℛ(K) of a variety K of unary algebras is a subdirect product of K and the variety D of all discrete algebras (unary semilattices), but is not the case. The purpose of this note is to show that his expectation is fulfilled for those and only those irregular varieties K which are contained in the generalized variety TDir of the so-called trap-directable algebras. This work was supported by Grant 1227 of Ministry of Science and Technology, Republic of Serbia.     

A class of formal languages

We deal with languages that are classes of fully invariant congruences on free semigroups of finite rank. The question is posed as to whether a given fully invariant congruence coincides with a syntactic congruence of the language in question. If all classes of a given fully invariant congruence are rational languages, the corresponding variety is then said to be rational. A number of properties of rational varieties is established—in particular, we point to the way in which they are related to finite varieties. Translated fromAlgebra i Logika, Vol. 37, No. 4, pp. 478–492, July–August, 1998.     

On the D-affinity of the flag variety in type B2

The flag varieties in characteristic 0 are well-known to be D-affine. In positive characteristic, however, only those in type A ₁ and A ₂ have been proved to be so. In this paper we will show in type B ₂ the cohomology vanishing of the first term in the p-filtration of the sheaf of differential operators on the flag variety. This is a necessary condition for the variety to be D-affine. Received: 7 February 2000 / Revised version: 30 June 2000     

Optimal Mal’tsev conditions for congruence modular varieties

For varieties, congruence modularity is equivalent to the tolerance intersection property, TIP in short. Based on TIP, it was proved in [5] that for an arbitrary lattice identity implying modularity (or at least congruence modularity) there exists a Mal’tsev condition such that the identity holds in congruence lattices of algebras of a variety if and only if the variety satisfies the corresponding Mal’tsev condition. However, the Mal’tsev condition constructed in [5] is not the simplest known one in general. Now we improve this result by constructing the best Mal’tsev condition and various related conditions. As an application, we give a particularly easy new proof of the result of Freese and Jónsson [11] stating that modular congruence varieties are Arguesian, and we strengthen this result by replacing “Arguesian” by “higher Arguesian” in the sense of Haiman [18]. We show that lattice terms for congruences of an arbitrary congruence modular variety can be computed in two steps: the first step mimics the use of congruence distributivity, while the second step corresponds to congruence permutability. Particular cases of this result were known; the present approach using TIP is even simpler than the proofs of the previous partial results.Dedicated to the memory of Ivan RivalReceived February 12, 2003; accepted in final form August 5, 2004.This revised version was published online in August 2005 with a corrected cover date.     

A superrank of varieties of Lie algebras

A variety of Lie algebras over a field of characteristic 0 has a finite superrank if it is generated by the Grassmann envelope of a finitely generated Lie superalgebra. We prove that every commutator variety not in N_cA has infinite superrank. Consequently, infinite are superranks of all polynilpotent varieties of Lie algebras except N_c and N_cA. Supported by RFFR grant No. 96-01-00146. Translated fromAlgebra i Logika, Vol. 37, No. 4, pp. 394–412, July–August. 1998.     

On the secant varieties to the osculating variety of a Veronese surface

In this paper we study the k-th osculating variety of the order d Veronese embedding of P ⁿ . In particular, for k=n=2 we show that the corresponding secant varieties have the expected dimension except in one case.     

A bound on the plurigenera of projective varieties

We exhibit a sharp Castelnuovo bound for the i-th plurigenus of a smooth variety of given dimension n and degree d in the projective space P ^r , and classify the varieties attaining the bound, when n2, r2n+1, d>>r and i>>r. When n=2 and r=5, or n=3 and r=7, we give a complete classification, i.e. for any i1. In certain cases, the varieties with maximal plurigenus are not Castelnuovo varieties, i.e. varieties with maximal geometric genus. For example, a Castelnuovo variety complete intersection on a variety of dimension n+1 and minimal degree in P ^r , with r>(n ² +3n)/(n–1), has not maximal i-th plurigenus, for i>>r. As a consequence of the bound on the plurigenera, we obtain an upper bound for the self-intersection of the canonical bundle of a smooth projective variety, whose canonical bundle is big and nef. Mathematics Subject Classification (2000):Primary 14J99; Secondary 14N99     

Centrality in Rees-Sushkevich varieties

Denote by RS _n the variety generated by all completely 0-simple semigroups over groups of exponent dividing n. Subvarieties of RS _n are called Rees-Sushkevich varieties and those that are generated by completely simple or completely 0-simple semigroups are said to be exact. For each positive integer m, define C _m RS _n to be the class of all semigroups S in RS _n with the property that if the product of m idempotents of S belongs to some subgroup of S, then the product belongs to the center of that subgroup. The classes C _m RS _n constitute varieties that are the main object of investigation in this article. It is shown that a sublattice of exact subvarieties of C ₂ RS _n is isomorphic to the direct product of a three-element chain with the lattice of central completely simple semigroup varieties over groups of exponent dividing n. In the main result, this isomorphism is extended to include those exact varieties for which the intersection of the core with any subgroup, if nonempty, is contained in the center of that subgroup. The equational property of the varieties C _m RS _n is also addressed. For any fixed n ≥ 2, it is shown that although the varieties C _m RS _n , where m = 1, 2, ... , are all finitely based, their complete intersection (denoted by C _∞ RS _n ) is non-finitely based. Further, the variety C _∞ RS _n contains a continuum of ultimately incomparable infinite sequences of finitely generated exact subvarieties that are alternately finitely based and non-finitely based. Received October 29, 2003; accepted in final form February 11, 2007.