Products of coalgebras

We prove that the category of F-coalgebras is complete, that is products and equalizers exist, provided that the type functor F is bounded or preserves mono sources. This generalizes and simplifies a result of Worrell ([Wor98]). We also describe the relationship between the product and the largest bisimulation between and and find an example of two finite coalgebras whose product is infinite. Received January 11, 2000; accepted in final form October 16, 2000.     

State classification for a class of measure-valued branching diffusions in a Brownian medium

Summary. The spatial structure of a new class of measure-valued diffusions which arise as limits in distribution of a sequence of interacting branching particle systems is investigated. We obtain the following criterion of state classification for these superprocesses: their effective state space is contained in the set of purely atomic measures or the set of absolutely continuous measures according as ε=0 or ε≠0, when the coefficient of the motion generator is a smooth function. Received: 15 December 1995 / In revised form: 24 March 1997     

Grothendieck-like duality for Heyting algebras

In this paper we shall use the multipliers in order to define a sheaf on the prime spectrum of a Heyting algebra. This construction allows us to obtain a Grothendieck-style duality for Heyting algebras. Received March 3, 2000; accepted in final form February 16, 2001.     

The complexity of checking identities for finite matrix rings

We complete the investigations into the word-problem for finite matrix rings. Namely we prove that M₂ (Z₃), the ring of 2 x 2 matrices over Z₃, has a coNP-complete term-equivalence (or identity checking) problem.     

The free orthomodular lattice on countably many generators is a subalgebra of the free orthomodular lattice on three generators

We show every at most countable orthomodular lattice is a subalgebra of one generated by three elements. As a corollary we obtain that the free orthomodular lattice on countably many generators is a subalgebra of the free orthomodular lattice on three generators. This answers a question raised by Bruns in 1976 [2] and listed as Problem 15 in Kalmbach's book on orthomodular lattices [6]. Received April 12, 2001; accepted in final form May 6, 2002.     

The M3[D] construction and n-modularity

In 1968, Schmidt introduced the M ₃[D] construction, an extension of the five-element modular nondistributive lattice M ₃ by a bounded distributive lattice D, defined as the lattice of all triples satisfying . The lattice M ₃[D] is a modular congruence-preserving extension of D.? In this paper, we investigate this construction for an arbitrary lattice L. For every n > 0, we exhibit an identity such that is modularity and is properly weaker than . Let M _n denote the variety defined by , the variety of n-modular lattices. If L is n-modular, then M ₃[L] is a lattice, in fact, a congruence-preserving extension of L; we also prove that, in this case, Id M ₃[L] M ₃[Id L]. ? We provide an example of a lattice L such that M ₃[L] is not a lattice. This example also provides a negative solution to a problem of Quackenbush: Is the tensor product of two lattices A and B with zero always a lattice. We complement this result by generalizing the M ₃[L] construction to an M ₄[L] construction. This yields, in particular, a bounded modular lattice L such that M ₄ L is not a lattice, thus providing a negative solution to Quackenbush’s problem in the variety M of modular lattices.? Finally, we sharpen a result of Dilworth: Every finite distributive lattice can be represented as the congruence lattice of a finite 3-modular lattice. We do this by verifying that a construction of Gr?tzer, Lakser, and Schmidt yields a 3-modular lattice. Received May 26, 1998; accepted in final form October 7, 1998.     

Pappian implies Arguesian for modular lattices

We prove the claim in the title, using Hessemberg's analog theorem for projective planes, and the characterization of non-Arguesian modular lattices provided by Day and Jónsson [5,6]. Received January 22, 1999; accepted in final form May 6, 1999.     

The probability of triviality

Given a variety of algebras, what is the probability that for an arbitrary identity p q the only algebra in that satisfies p q is the trivial algebra? More generally, if is a subvariety of what is the probability that p q together with the identities of forms an equational basis for ? We consider these questions for various and and we provide criteria that allow for explicit determination of these probabilities. Received April 14, 1997; accepted in final form January 19, 1998.     

A Liouville theorem for a class of superlinear elliptic equations on cones

We give sufficient conditions for non-existence of positive solutions of the equation on a cone of We further analyze the existence of positive solutions in the radial, subcritical case, and show that under suitable conditions on the coefficients, every radial solution whose value in 0 is sufficiently large must vanish. Received April 2000     

On the cofinalities of Boolean algebras and the ideal of null sets

We will show that if the cofinality of the ideal of Lebesgue measure zero sets is equal to then there exists a Boolean algebra B of cardinality which is not a union of strictly increasing -sequence of its subalgebras. This generalizes a result of Just and Koszmider who showed that it is consistent with that such an algebra exists. Received November 21; accepted in final form April 12, 2001.     

The factor of a subdirectly irreducible algebra through its monolith

A nontrivial algebra with at least one at least binary operation is isomorphic to the factor of a subdirectly irreducible algebra through its monolith if and only if the intersection of all its ideals is nonempty. Received October 6, 2000; accepted in final form September 14, 2001.     

Well-foundedness conditions connected with left-distributivity

We state a problem about free left-distributive algebras. If u and w are members of such an algebra write u < _L w of for some . A conjecture about left-division in such algebras is given; it entails a normal form and that for every w the set of left divisors of w is well-ordered under < _L . Received March 25, 2001; accepted in final form April 10, 2001.     

Mailbox¶Generalising congruence regularity for varieties

No Abstract. Received September 14, 2001; accepted in final form June 9, 2002.     

The lattice of quasivarieties of undirected graphs

In this note, it is shown that the quasivariety of undirected graphs is Q-universal. Received March 3, 2000; accepted in final form January 31, 2001.     

Distributive lattices with a decidable monadic second order theory

A class of finite distributive lattices has a decidable monadic second order theory iff (a) the join irreducible elements of its members have a decidable monadic second order theory, and (b) the width of the lattices is bounded. Similar results are obtained for the monadic chain and the monadic antichain theory where quantification is restricted to chains and antichains, resp. Furthermore, there is no (up to finite difference) maximal set of finite distributive lattices with a decidable monadic (chain or antichain, resp.) theory. Received December 6, 2000; accepted in final form May 30, 2002.     

A criterion for a finite endoprimal algebra to be endodualisable

In this note the strategy for finding endoprimal algebras due to B. A. Davey and J. G. Pitkethly is further explored in the finite case. The Retraction Test Algebra Lemma is used as a tool to show that, in many quasivarieties, endoprimality is equivalent to endodualisability for finite algebras which are suitably related to finitely generated free algebras. Received September 2, 1998; accepted in final form June 3, 1999.     

Lattice orders on matrix algebras

We construct all the lattice orders (up to isomorphism) on a full matrix algebra over a subfield of the field of real numbers so that it becomes a lattice-ordered algebra. Received June 26, 2001; accepted in final form February 9, 2002.     

On order-polynomial completeness of lattices

In this note we prove that the cardinality of an infinite order-polynomially complete lattice (if such a lattice exists) must be greater than each , where is defined such that and for . This strengthens the result in [4]. Received November 10, 1995; accepted in final form April 6, 1998.     

The join of two minimal clones and the meet of two maximal clones

We construct two minimal clones on any finite set such that the join of the two clones contains all operations. Dually, we exhibit two maximal clones on any finite set with at least three elements such that the intersection of the two clones is the trivial clone containing projections only. Received October 20, 1998; accepted in final form December 16, 1998.     

Correction to "Some properties of complete congruence lattices"

Received April 26, 2000; accepted in final form July 19, 2001.