Maximal clones on uncountable sets that include all permutations

We first determine the maximal clones on a set X of infinite regular cardinality κ which contain all permutations but not all unary functions, extending a result of Heindorf’s for countably infinite X. If κ is countably infinite or weakly compact, this yields a list of all maximal clones containing the permutations, since in that case the maximal clones above the unary functions are known. We then generalize a result of Gavrilov’s to obtain on all infinite X a list of all maximal submonoids of the monoid of unary functions which contain the permutations. Received January 8, 2004; accepted in final form December 22, 2004.     

Compact factor congruences imply Boolean factor congruences

We prove that any variety in which every factor congruence is compact has Boolean factor congruences, i.e., for all A in the set of factor congruences of A is a distributive sublattice of the congruence lattice of A.     

Monoidal intervals of clones on infinite sets

Let X be an infinite set of cardinality κ. We show that if L is an algebraic and dually algebraic distributive lattice with at most 2^κ completely join irreducibles, then there exists a monoidal interval in the clone lattice on X which is isomorphic to the lattice 1+L obtained by adding a new smallest element to L. In particular, we find that if L is any chain which is an algebraic lattice, and if L does not have more than 2^κ completely join irreducibles, then 1+L appears as a monoidal interval; also, if λ?2^κ, then the power set of λ with an additional smallest element is a monoidal interval. Concerning cardinalities of monoidal intervals these results imply that there are monoidal intervals of all cardinalities not greater than 2^κ, as well as monoidal intervals of cardinality 2^λ, for all λ?2^κ.     

Many-sorted algebras in congruence modular varieties

We present several basic results on many-sorted algebras, most of them only valid in congruence modular varieties. We describe a connection between the properties of many-sorted varieties and those of varieties of one sort and give some results on functional completeness, the commutator and Abelian algebras.Presented by H. P. Gumm.     

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.     

Algebraic theories, clones and their segments

Isomorphism and elementary equivalence of segments of clones of objects in concrete categories are investigated. A survey of results about the finitary case is presented and a new theorem about the infinitary case is proved.Financial support of the Grant Agency of the Czech Republic under the grant no. 201/93/0950 and of the Grant Agency of the Charles University under the grant GAUK 349 is gratefully acknowledged.     

Precomplete Clones on Infinite Sets which are Closed under Conjugation

We show that on an infinite set, there exist no other precomplete clones closed under conjugation except those which contain all permutations. Since on base sets of some infinite cardinalities, in particular on countably infinite ones, the precomplete clones containing the permutations have been determined, this yields a complete list of the precomplete conjugation-closed clones in those cases. In addition, we show that there exist no precomplete submonoids of the full transformation monoid which are closed under conjugation except those which contain the permutations; the monoids of the latter kind are known.     

Structural entailment

We give a number of characterizations of structural entailment. In particular, we show that an alter ego structurally entails an algebraic relation s on a finite algebra if and only if s can be obtained via a local construct from . We show, via a range of applications, that, whereas entailment is important in the study of dualisability, structural entailment is important in the study of full and strong dualisability. We also give an application to the transfer of strong dualities that connects this paper to our earlier paper [9] on full versus strong duality. Received June 27, 2003; accepted in final form May 12, 2005.     

When is a natural duality ‘good’?

Building on the most current work in the theory of natural dualities, we continue the study of strong dualities for the quasi-variety generated by a finite algebra. We investigate ten different versions of what we would like to mean by a good duality. Each version concerns, among other things, a specific restriction on the type of the structures in the dual category which insures that the dual structures will in a useful sense be simple. Through each investigation we seek a theorem characterizing, in terms of finitely verifiable conditions, those finite algebras generating a quasi-variety which admits a strong duality meeting the given restrictions. Our study includes a careful treatment of coproducts, logarithmic dualities and strong dualities by various unary structures.Dedicated to the memory of Alan DayPresented by J. Sichler.Research supported by a 1992 ARC Grant (Davey).     

Monotone clones and the varieties they determine

A finite, nontrivial algebra is order-primal if its term functions are precisely the monotone functions for some order on the underlying set. We show that the prevariety generated by an order-primal algebra P is relatively congruence-distributive and that the variety generated by P is congruence-distributive if and only if it contains at most two non-ismorphic subdirectly irreducible algebras. We also prove that if the prevarieties generated by order-primal algebras P and Q are equivalent as categories, then the corresponding orders or their duals generate the same order variety. A large class of order-primal algebras is described each member of which generates a variety equivalent as a category to the variety determined by the six-element, bounded ordered set which is not a lattice. These results are proved by considering topological dualities with particular emphasis on the case where there is a monotone near-unanimity function.This research was carried out while the third author held a research fellowship at La Trobe University supported by ARGS grant B85154851. The second author was supported by a grant from the NSERC.     

Rota’s umbral calculus and recursions

Umbral Calculus can provide exact solutions to a wide range of linear recursions. We summarize the relevant theory and give a variety of examples from combinatorics in one, two and three variables.Dedicated to the Memory of Gian-Carlo Rota     

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.     

Full does not imply strong, does it?

We give a duality for the variety of bounded distributive lattices that is not full (and therefore not strong) although it is full but not strong at the finite level. While this does not give a complete solution to the “Full vs Strong” Problem, which dates back to the beginnings of natural duality theory in 1980, it does solve it at the finite level. One consequence of this result is that although there is a Duality Compactness Theorem, which says that if an alter ego of finite type yields a duality at the finite level then it yields a duality, there cannot be a corresponding Full Duality Compactness Theorem. Received October 1, 2002; accepted in final form November 10, 2004.     

Relations compatible with near unanimity operations

Given a system of k-ary relations on a finite set A which are compatible with a (k + 1)-ary near unanimity operation on A, we provide a characterization of when is the system of all k-ary subuniverses of an algebra A on A.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived August 26, 2003; accepted in final form July 10, 2004.     

Endoprimality without duality

This paper investigates endoprimal algebras using techniques from universal algebra but not from duality theory, and thereby exposes quite directly how endoprimality occurs.     

Concerning functional equations of the generalized Bol-moufang type

Aequationes mathematicae -     

Hybrid bases for varieties of semigroups

We consider the lower part of the lattice of varieties of semigroups. We present finite bases of hybrid identities for the varieties of normal bands, commutative bands and abelian groups of finite exponent.The variety A_n,0 of abelian groups provides an example of a variety which has no finite base of hyperidentities (cf. [12]) but has a finite base of hybrid identities.