Hyperidentities of some generalizations of lattices

In the paper we present bases and hyperbases of hyperidentities of some generalizations of the variety L of all lattices and the variety D of distributive lattices. We describe the form of hyperidentities of some varieties with two binary operations. Received January 22, 1997; accepted in final form January 7, 1998.     

On Euler–Jaczewski sequence and Remmert–Van de Ven problem for toric varieties

Let X be a complete toric variety and Y a smooth projective variety with . We prove that, if is a surjective morphism then . Received: 15 May 2001; in final form: 22 October 2001/ Published online: 4 April 2002     

Equational bases for varieties of Ockham algebras

We consider the variety O of Ockham algebras and its subvarieties of the form P _m,n (m > n ≥0), sometimes with an additional condition. We use Priestley duality and a remarkable theorem of Urquhart to develop a simple method for determining the equational bases of the subvarieties. The axioms that we obtain have the same canonical form and involve few variables. We illustrate our method by the detailed study of the variety MS ₂ and some considerations about P _3,2. Received July 28, 1998; accepted in final form May 25, 2000.     

Completions of GBL-algebras: negative results

We provide a simple sufficient criterion to show that a given variety of GBL-algebras does not admit (local) completions. As corollaries, we obtain that no variety of GBL-algebras containing Chang’s chain, no nontrivial variety of ℓ-groups, nor the variety of product algebras admit completions. The first result strengthens a result of Gehrke and Priestley. Received August 10, 2006; accepted in final form March 8, 2007.     

Mailbox¶A Note on minimal varieties generated by order-primal algebras

We prove that the order-primal algebra of a non-trivial finite connected poset P generates a minimal variety if and only if P is dismantlable. Received November 26, 2002; accepted in final form January 24, 2003.     

Finite modular congruence lattices

We consider the variety of modular lattices generated by all finite lattices obtained by gluing together some M₃’s. We prove that every finite lattice in this variety is the congruence lattice of a suitable finite algebra (in fact, of an operator group). Received February 26, 2004; accepted in final form December 16, 2004.     

Semivarieties of nilpotent groups

Semivarieties of groups are quasivarieties defined by quasi-identities of the form t = 1 → f = 1. It is proved that a set of semivarieties in every variety of class two nilpotent p-groups of finite exponent having a commutator subgroup of exponent p (p is a prime) is at most countable. It is stated that a variety of class two nilpotent groups with commutator subgroup of exponent p contains a set of semivarieties of the cardinality of the continuum.     

Quasi-projective reduction of toric varieties

We define a quasi–projective reduction of a complex algebraic variety X to be a regular map from X to a quasi–projective variety that is universal with respect to regular maps from X to quasi–projective varieties. A toric quasi–projective reduction is the analogous notion in the category of toric varieties. For a given toric variety X we first construct a toric quasi–projective reduction. Then we show that X has a quasi–projective reduction if and only if its toric quasi–projective reduction is surjective. We apply this result to characterize when the action of a subtorus on a quasi–projective toric variety admits a categorical quotient in the category of quasi–projective varieties. Received October 29, 1998; in final form December 28, 1998     

Discriminator or dual discriminator?

Varieties are considered with p(x, y, z), a single ternary operation, which acts as a local discriminator or dual discriminator on the subdirectly irreducible elements. If p(x, y, z) is "global", then all subvarieties are finitely based. In the general case a continuum of non-finitely based subvarieties are presented. A graph theoretical picture leads to a variety of groupoids connecting the left-zero and the right-zero semigroups. For this variety some open problems are presented. Received October 7, 1998; accepted in final form October 4, 1999.     

Topological implications in varieties

In 1997, Coleman showed that a variety V is n-permutable for some n iff every T ₀-topological Algebra in V is T ₁. Here we show that the implication " sober" is another such characterization for n-permutability. Other implications of a similar nature are given. For example, an n-permutable variety having a majority term satisfies ". Received July 13, 1998; accepted in final form April 13, 1999     

Definable principal subcongruences

For varieties of algebras, we present the property of having "definable principal subcongruences" (DPSC), generalizing the concept of having definable principal congruences. It is shown that if a locally finite variety V of finite type has DPSC, then V has a finite equational basis if and only if its class of subdirectly irreducible members is finitely axiomatizable. As an application, we prove that if A is a finite algebra of finite type whose variety V(A) is congruence distributive, then V(A) has DPSC. Thus we obtain a new proof of the finite basis theorem for such varieties. In contrast, it is shown that the group variety V(S ₃ ) does not have DPSC. Received May 9 2000; accepted in final form April 26, 2001.     

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.     

On profinite completions and canonical extensions

We show that if a variety V of monotone lattice expansions is finitely generated, then profinite completions agree with canonical extensions on V. The converse holds for varieties of finite type. This paper is dedicated to Walter Taylor. Received May 14, 2005; accepted in final form September 8, 2005.     

Closure Łukasiewicz algebras

In this paper, the variety of closure n-valued Łukasiewicz algebras, that is, Łukasiewicz algebras of order n endowed with a closure operator, is investigated. The lattice of subvarieties in the particular case in which the open elements form a three-valued Heyting algebra is obtained.     

Pseudocomplemented semilattices are finite-to-finite relatively universal

It is shown that the category of directed graphs is isomorphic to a subcategory of the variety S of all pseudocomplemented semilattices which contains all homomorphisms whose images do not lie in the subvariety B of all Boolean pseudocomplemented semilattices. Moreover, the functor exhibiting the isomorphism may be chosen such that each finite directed graph is assigned a finite pseudocomplemented semilattice. That is to say, it is shown that the variety S of all pseudocomplemented semilattices is finite-to-finite B-relatively universal. This illustrates the complexity of the endomorphism monoids of pseudocomplemented semilattices since it follows immediately that, for any monoid M, there exists a proper class of non-isomorphic pseudocomplemented semilattices such that, for each member S, the endomorphisms of S which do not have an image contained in the skeleton of S form a submonoid of the endomorphism monoid of S which is isomorphic to M. Received June 17, 2006; accepted in final form May 8, 2007.     

Equivalence for varieties in general and for $ {\cal BOOL} $ in particular

The varieties equivalent to a given variety are characterized in a purely categorical way. In fact they are described as the models of those Lawvere theories which are Morita equivalent to the Lawvere theory of which therefore are characterized first. Along this way the conceptual meanings of the n-th matrix power construction of a variety and McKenzie's σ-modification of classes of algebras [22] become transparent. Besides other applications not only the well known equivalences between the varieties of Post algebras of fixed orders m and the variety of Boolean algebras are obtained; moreover it can be shown that the varieties are the only varieties equivalent to . The results then are generalized to quasivarieties and more general classes of algebras. Received November 4, 1998; accepted in final form September 15, 1999.     

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.     

Axiomatizing complex algebras by games

Given a variety , we provide an axiomatization of the class of complex algebras of algebras in . can be obtained effectively from the axiomatization of ; in fact, if this axiomatization is recursively enumerable, then is recursive. Received January 18, 2000; accepted in final form December 18, 2000.