On clones, transformation monoids, and finite Boolean algebras

We prove that, for a finite Boolean algebra, there exist only finitely many clones which consist of polynomial functions of the algebra and contain the monoid of its unary polynomial functions. Received January 7, 2000; accepted in final form December 13, 2000.     

There do not exist minimal algebras between C*(X) and C(X) with prescribed real maximal ideal spaces

Let C(X) be the algebra of all real-valued continuous functions on a completely regular Hausdorff space X, and C*(X) the subalgebra of bounded functions. We prove that for any intermediate algebra A between C*(X) and C(X), other than C*(X), there exists a smaller intermediate algebra with the same real maximal ideals as in A. The space X is called A-compact if any real maximal ideal in A corresponds to a point in X. It follows that, for a noncompact space X, there does not exist any minimal intermediate algebra A for which A is A-compact. This completes the answer to a question raised by Redlin and Watson in 1987.     

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.     

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.     

Existence of maximal and minimal solutions to a boundary value problem for the equation of the bending of an elastic beam

By introducing a partial order and using the Mönch fixed point theorem, we establish the existence of maximal and minimal solutions in Banach spaces to a boundary value problem for the equation of the bending of an elastic beam.     

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.     

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.     

Endodualisable and endoprimal finite double Stone algebras

The finite endodualisable double Stone algebras are characterised, and every finite endoprimal double Stone algebra is shown to be endodualisable. Received February 24, 1999; accepted in final form May 10, 1999.     

Congruence join semidistributivity is equivalent to a congruence identity

We show that a locally finite variety is congruence join semidistributive if and only if it satisfies a congruence identity that is strong enough to force join semidistributivity in any lattice. Received February 9, 2000; accepted in final form November 23, 2000.     

Notes on R1-ideals in partial abelian monoids

In the present paper, we deal with a class of R ₁-ideals of cancellative positive partial abelian monoids (CPAMs). We prove that, for I being an R ₁-ideal of a CPAM P, P/I is a CPAM. The lattice of congruence relations associated with R ₁-ideals is a sublattice of the lattice of all equivalence relations. Finally, we prove that an intersection of two Riesz ideals is a Riesz ideal and that the lattice of Riesz ideals is a sublattice of the lattice of all ideals. Received March 19, 1999; accepted in final form December 16, 1999.     

Generating primitive positive clones

Suppose is a set of operations on a finite set A. Define PPC() to be the smallest primitive positive clone on A containing . For any finite algebra A, let PPC#(A) be the smallest number n for which PPC(CloA) = PPC(Clo _n A). S. Burris and R. Willard [2] conjectured that PPC#(A) ≤|A| when CloA is a primitive positive clone and |A| > 2. In this paper, we look at how large PPC#(A) can be when special conditions are placed on the finite algebra A. We show that PPC#(A) ≤|A| holds when the variety generated by A is congruence distributive, Abelian, or decidable. We also show that PPC#(A) ≤|A| + 2 if A generates a congruence permutable variety and every subalgebra of A is the product of a congruence neutral algebra and an Abelian algebra. Furthermore, we give an example in which PPC#(A) ≥|A| - 1)² so that these results are not vacuous. Received August 30, 1999; accepted in final form April 4, 2000.     

Functional monadic Heyting algebras

We show every monadic Heyting algebra is isomorphic to a functional monadic Heyting algebra. This solves a 1957 problem of Monteiro and Varsavsky [9]. Received May 18, 2001; accepted in final form October 18, 2001.     

On the Markov equivalence of maximal ancestral graphs

Necessary and sufficient conditions for a maximal ancestral graph (MAnG) to be Markov equivalent to another MAnG and to a DAG are provided respectively. Also a polynomial-time algorithm for converting a MAnG into its equivalent DAG is given for the first time.     

A non-dualisable entropic algebra

We give an example of a five-element unary algebra which is entropic but does not admit a natural duality. Received September 26, 2000; accepted in final form April 4, 2001.     

Representability and local representability of algebraic theories

A finitary monosorted algebraic theory is called locally representable (or representable) in a category with finite products if every its initial segment is the domain of a full faithful finite-products-preserving functor into (or if itself is the domain of such a functor). The question of when local representability implies representability is discussed, and theories for which local representability always implies representability are fully characterized. Received December 20, 1996; accepted in final form March 19, 1998.     

A relationship between two almost f-algebra products

Let A be an uniformly complete almost f-algebra. Then is a positively generated ordered vector subspace of A with as a positive cone. If is a positive linear operator, we put the linear operator defined by with for all is the algebra of all order bounded linear operators of A). Let denote the range of and let's define a new product by putting for all . It is easily checked that if then , this shows that if it happens that the product is associative then A is an almost f- algebra with respect to this new product. It turns out that a necessarily and sufficient condition in order that be an associative product is that is a commutative subalgebra of . We find necessarily and sufficient conditions on T in order that is an almost f-algebra (respect.; d-algebra, f-algebra) product. Such conditions are described in terms of the algebraic and order structure of the algebra .?The converse problem is also studied. More precisely, let A be an uniformly complete almost f-algebra and assume that is another almost f-algebra product on A. The aim is to find sufficient conditions in order that there exist such that for all . It will be showed that a sufficient condition is that A is a d-algebra with respect to the initial product. An example is produced which shows that the condition "A is a d-algebra with respect to the initial product" can not be weakened. Received November 8, 1999; accepted in final form February 14, 2000.     

The modular group algebras of p-groups of maximal class II

The modular isomorphism problem is settled for 3-groups of maximal class but two families of groups. Moreover, the conjecture that the ideals belonging to the lower central series of a group base are determined by the structure of the group algebra is refuted in greatest generality by virtue of a single group of order 81 of maximal class, and it is proved that the nilpotency class is determined by the structure of the group algebra for p-groups of maximal class.     

Collapsing permutation groups

It is shown in [3] that any nonregular quasiprimitive permutation group is collapsing. In this paper we describe a wider class of collapsing permutation groups. Received June 6, 2000; accepted in final form August 11, 2000.