Monoid intervals in lattices of clones

Suppose A is a finite set. For every clone C over A, the family C⁽¹⁾ of all unary functions in C is a monoid of transformations of the set A. We study how the lattice of clones is partitioned into intervals, where two clones belong to the same partition iff they have the same monoids of unary functions. The problem of Szendrei concerning the power of such intervals is investigated. We give new examples of intervals which are continual, one-element, and finite but not one-element. Moreover, it is proved that every lattice that is not more than a direct product of countably many finite chains is isomorphic to some interval in the lattice of clones, establishing, in passing, the number of E-minimal algebras on a finite set. Translated fromAlgebra i Logika, Vol. 34, No. 3, pp. 288-310, May-June, 1995.     

Maximal partial clones with no finite basis

Let A be a nonsingleton finite set. We give a criterion for recognizing not finitely generated strong partial clones and show that none of the |A| strong maximal partial clones of Słupecki type over A has a finite basis. Received November 14, 1997; accepted in final form August 18, 1998.     

Polynomial clones on groups of order <Emphasis Type="Italic">pq</Emphasis>

For two distinct primes p, q, we describe those clones on a set of size pq that contain a given group operation and all constant operations. We show that each such clone is determined by congruences and commutator relations. Thus we obtain that there is only a finite number of such clones on a fixed set. Supported by the Austrian Science Fund (FWF), grant P15691.     

Properties of the quasilinear clones containing creative functions

We study the problem of characterizing clones on a three-element set by hyperidentities. We prove that there exists a hyperidentity separating any clone of quasilinear functions defined on the set {0, 1, 2} each of them is either a selector or such that all its values belong to {0, 1} from any noncreative clone constituted by such functions incomparable with the initial clone.     

Isomorphism and local isomorphism of clones of spaces

We construct metric spaces X and Y such that their clones cloX and cloY (i.e. the categories of all continuous maps on their respective finite powers) are not isomorphic while any initial segment of cloX is isomorphic to the corresponding segment of cloY.     

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.     

Equivalence of operations with respect to discriminator clones

For each clone C on a set A there is an associated equivalence relation, called C-equivalence, on the set of all operations on A, which relates two operations iff each one is a substitution instance of the other using operations from C. In this paper we prove that if C is a discriminator clone on a finite set, then there are only finitely many C-equivalence classes. Moreover, we show that the smallest discriminator clone is minimal with respect to this finiteness property. For discriminator clones of Boolean functions we explicitly describe the associated equivalence relations.     

Oligomorphic clones

A permutation group on a countably infinite domain is called oligomorphic if it has finitely many orbits of finitary tuples. We define a clone on a countable domain to be oligomorphic if its set of permutations forms an oligomorphic permutation group. There is a close relationship to ω-categorical structures, i.e., countably infinite structures with a first-order theory that has only one countable model, up to isomorphism. Every locally closed oligomorphic permutation group is the automorphism group of an ω-categorical structure, and conversely, the canonical structure of an oligomorphic permutation group is an ω-categorical structure that contains all first-order definable relations. There is a similar Galois connection between locally closed oligomorphic clones and ω-categorical structures containing all primitive positive definable relations. In this article we generalise some fundamental theorems of universal algebra from clones over a finite domain to oligomorphic clones. First, we define minimal oligomorphic clones, and present equivalent characterisations of minimality, and then generalise Rosenberg’s five types classification to minimal oligomorphic clones. We also present a generalisation of the theorem of Baker and Pixley to oligomorphic clones. Presented by A. Szendrei. Received July 12, 2005; accepted in final form August 29, 2006.     

The generation of clones with majority operations

A theorem by Baker and Pixley implies that any clone on a finite set is finitely generated if it contains a near-unanimity operation. This raises the question of what arity the generating operations must have. In this paper, we solve the last open bits of this problem for the majority case by showing that 5 and 8 are the smallest integers k such that every clone with a majority operation on a 3 and 4-element set, respectively, is generated by its k-ary part.     

Primitive positive clones of groupoids

We prove that if two finite groupoids with unity have the same ternary compatible relations, then they have the same primitive positive clones. Received October 23, 2007; accepted in final form February 22, 2008.     

Amalgams of clones

The structure of a clone of functions respecting an amalgam is described. Also, for clones R_A and R_B of relations, conditions are specified under which the amalgam of R_A and R_B, restricted to the domains of R_A and R_B, coincides with these.Translated from Algebra i Logika, Vol. 44, No. 1, pp. 97–113, January–February, 2005.     

On Boolean primitive positive clones

Kuznecov introduced the concept of primitive positive clones and proved in 1977 that there are 25 Boolean primitive positive clones in a notoriously unavailable article. This paper presents a new proof of his result, relating it to Post's lattice and exhibiting finite bases for those clones.     

Clone classification of dually discriminator algebras with finite support

Dually discriminator algebras are considered up to clones generated by the algebra operations. In terms of binary relations, all clones of the operators on a finite set that contain the Pixley dual discriminator are efficiently described. As a consequence, a similar clone classification of quasi-primal algebras with finite support is determined. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 359–366, March, 1997. Translated by A. I. Shtern     

Meet-irreducible submaximal clones determined by nontrivial equivalence relations

Let θ and ρ be a nontrivial equivalence relation and a binary relation on a finite set A, respectively. It is known from Rosenberg’s classification theorem (1965) that the clone Pol θ, which consists of all operations on A that preserve θ, is among the maximal clones on A. In the present paper, we find all binary relations ρ such that the clone Pol ρ is a meet-irreducible maximal subclone of Pol θ.     

Approximation to a parabolic system modeling the thermoelastic contacts of two rods

In this article, we study a sequence of finite difference approximate solutions to a parabolic system, which models two dissimilar rods that may come into contact as a result of thermoelastic expansion. We construct the approximate solutions based on a set of finite difference schemes to the system, and we will prove that the approximate solutions converge strongly to the exact solutions. Moreover, we obtain and prove rigorously the error bound, which measures the difference between the exact solutions and approximate solutions in a reasonable norm. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 1–25, 1998     

Complementary closed relational clones are not always Krasner clones

In ZFC, it is shown that every relational clone on a set A closed under complementation is a Krasner clone if and only if A is at most countable. This is achieved by solving an equivalent problem on locally invertible monoids: A partially ordered set is constructed whose endomorphism monoid is not contained in the local closure of its automorphism group. Received September 5, 1998; accepted in final form December 10, 1998.     

Separation of clones of cooperations by cohyperidentities

An n-ary cooperation is a mapping from a nonempty set A to the nth copower of A. A clone of cooperations is a set of cooperations which is closed under superposition and contains all injections. Coalgebras are pairs consisting of a set and a set of cooperations defined on this set. We define terms for coalgebras, coidentities and cohyperidentities. These concepts will be applied to give a new solution of the completeness problem for clones of cooperations defined on a two-element set and to separate clones of cooperations by coidentities.