Natural dualities for semilattice-based algebras

While every finite lattice-based algebra is dualisable, the same is not true of semilattice-based algebras. We show that a finite semilattice-based algebra is dualisable if all its operations are compatible with the semilattice operation. We also give examples of infinite semilattice-based algebras that are dualisable. In contrast, we present a general condition that guarantees the inherent non-dualisability of a finite semilattice-based algebra. We combine our results to characterise dualisability amongst the finite algebras in the classes of flat extensions of partial algebras and closure semilattices. Throughout, we emphasise the connection between the dualisability of an algebra and the residual character of the variety it generates. Presented by R. Willard.     

Dualizability of automatic algebras

We make a start on one of George McNulty's Dozen Easy Problems  : “Which finite automatic algebras are dualizable?” We give some necessary and some sufficient conditions for dualizability. For example, we prove that a finite automatic algebra is dualizable if its letters act as an abelian group of permutations on its states. To illustrate the potential difficulty of the general problem, we exhibit an infinite ascending chain _A1?_A2?_A3??${\mathbf{A}}_1?{\mathbf{A}}_2?{\mathbf{A}}_3??$ of finite automatic algebras that are alternately dualizable and non-dualizable.     

On lattices of congruences of relational systems and universal algebras

Let \(\mathfrak{X}\) =〈X;R〉 be a relational system.X is a non-empty set andR is a collection of subsets ofX ^α, α an ordinal. The system of equivalence relations onX having the substitution property with respect to members ofR form a complete latticeC( \(\mathfrak{X}\) ) containing the identity but not necessarilyX×X. It is shown that for any relational system (X;R) there is a groupoid definable onX whose congruence lattice isC( \(\mathfrak{X}\) )U{X×X} . Theorem 2 and Corollary 2 contain some interesting combinatorial pecularities associated with oriented complete graphs and simple groupoids.     

Graded decomposition numbers for cyclotomic Hecke algebras

In recent joint work with Wang, we have constructed graded Specht modules for cyclotomic Hecke algebras. In this article, we prove a graded version of the Lascoux-Leclerc-Thibon conjecture, describing the decomposition numbers of graded Specht modules over a field of characteristic zero.     

A Birkhoff theorem for partial algebras via completion

An equational theory (a Birkhoff theorem) for functorial partial algebras is established via the corresponding theory for functorial total algebras.This work was done with partial support of the DFG (BRD).Presented at the European Colloquium of Category Theory, Tours, France, 25–31 July 1994.     

Finite lattices as relative congruence lattices of finite algebras

Let A be a finite algebra and a quasivariety. By A is meant the lattice of congruences θ on A with . For any positive integer n, we give conditions on a finite algebra A under which for any n-element lattice L there is a quasivariety such that . The author was supported by INTAS grant 03-51-4110.     

Strong duality of finite algebras that generate the same quasivariety

We show that if two finite algebras generate the same quasivariety and one is strongly dualizable, then the other is also strongly dualizable.     

Spin Hecke algebras of finite and affine types

We introduce the spin Hecke algebra, which is a q-deformation of the spin symmetric group algebra, and its affine generalization. We establish an algebra isomorphism which relates our spin (affine) Hecke algebras to the (affine) Hecke-Clifford algebras of Olshanski and Jones-Nazarov. Relation between the spin (affine) Hecke algebra and a nonstandard presentation of the usual (affine) Hecke algebra is displayed, and the notion of covering (affine) Hecke algebra is introduced to provide a link between these algebras. Various algebraic structures for the spin (affine) Hecke algebra are established.     

Graded cellular bases for the cyclotomic Khovanov-Lauda-Rouquier algebras of type A

This paper constructs an explicit homogeneous cellular basis for the cyclotomic Khovanov-Lauda-Rouquier algebras of type A.     

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. The authors wish to express their gratitude to B. A. Davey and T. Katriňák for their helpful remarks and to J. G. Pitkethly for her assistance with the pictures. A support by Slovak grants VEGA 1/4057/97, 1/3026/06 and APVV-51-009605 is acknowledged by the first author who also wishes to thank the Mathematical Institute of the University of Oxford and the School of Mathematical and Statistical Sciences of La Trobe University for their hospitality.     

Weights of multipartitions and representations of Ariki-Koike algebras

We consider representations of the Ariki-Koike algebra, a q-deformation of the group algebra of the complex reflection group C_r?S_n. The representations of this algebra are naturally indexed by multipartitions of n, and for each multipartition λ we define a non-negative integer called the weight of λ. We prove some basic properties of this weight function, and examine blocks of small weight.     

On some developments in the representation theory of cylindric-like algebras

In this survey paper we recall the representation problem of certain classes of algebras of relations. Two specific recent research directions in algebraic logic are surveyed: the one is connected with the representation theory of cylindric relativized set algebras including Resek’s representation theorem and merry-go-round axioms, the other discusses the representation problem in some non-well-founded set theories where the axiom of foundation is replaced by Boffa’s weak anti-foundation axiom. This paper is dedicated to Walter Taylor. Received September 8, 2005; accepted in final form January 12, 2006. The first author was supported by Hungarian National Foundation for Scientific Research grants T43242 and T035192. The second author was supported by Hungarian National Foundation for Scientific Research grants D042177, T43242 and T035192.     

The representations of cyclotomic BMW algebras

In this paper, we classify the irreducible ?_r,n-modules over a field.     

Some of our primal algebras are missing: the canonical primal theory

It has been proven elsewhere that every variety has associated with it a unique canonical theory, where idempotent morphisms split. This article exhibits models of the canonical theory associated with any primal variety, for example, Boolean algebras. One such variety of models is generated by the several-sorted algebra with carriers of all prime cardinalities and with a clone of all finitary operations ω on and between carriers. This primal algebra was unknown. There are more. Presented by R. McKenzie. Received December 20, 2005; accepted in final form May 2, 2006.