70 pages; Dissertation written at the Vienna University of Technology under the supervision of Martin Goldstern; essentially consists of the author's papers "The clone generated by the median functions", "Clones containing all almost unary functions, "Maximal clones on uncountable sets that include all permutations" which are all available from arXiv
A clone on a set X is a set of finitary functions on X which contains the projections and which is closed under composition. The set of all clones on X forms a complete algebraic lattice Cl(X). We obtain several results on the structure of Cl(X) for infinite X. In the first chapter we prove the combinatorial result that if X is linearly ordered, then the median functions of different arity defined by that order all generate the same clone. The second chapter deals with clones containing the almost unary functions, that is, all functions whose value is determined by one of its variables up to a small set. We show that on X of regular cardinality, the set of such clones is always a countably infinite descending chain. The third chapter generalizes a result due to L. Heindorf from the countable to all uncountable X of regular cardinality, resulting in an explicit list of all clones containing the permutations but not all unary functions of X. Moreover, all maximal submonoids of the full transformation monoid which contain the permutations of X are determined, on all infinite X; this is an extension of a theorem by G. Gavrilov for countable base sets.
A clone on a set X is a set of finitary operations on X which contains all projections and which is moreover closed under functional composition. Ordering all clones on X by inclusion, one obtains a complete algebraic lattice, called the clone lattice. We summarize what we know about the clone lattice on an infinite base set X and formulate what we consider the most important open problems.
The clone lattice Cl(X) over an infinite set X is a complete algebraic lattice with 2^X compact elements. We show that every algebraic lattice with at most 2^X compact elements is a complete sublattice of Cl(X).
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 latte…
We consider clones on countable sets. If such a clone has quasigroup operations, is locally closed and countable, then there is a function $f : \mathbb{N} \to \mathbb{N}$ such that the $n$-ary part of $C$ is equal to the $n$-ary part of $\mathrm{Pol}\,\mathrm{Inv}^{[f(n)]} C$, where $\mathrm{Inv}^{[f(n)]} C$ denotes the set of $f(n)$-ary invariant relations of $C$.
Let X be an infinite set of regular cardinality. We determine all clones on X which contain all almost unary functions. It turns out that independently of the size of X, these clones form a countably infinite descending chain. Moreover, all such clones are finitely generated over the unary functions. In particular, we obtain an explicit description of the only maximal clone in this part of the clone lattice. This is especially interesting if X is countably infinite, in which case it is known that such a description cannot be obtained for the second maximal clone over the unary functions.
We calculate the number of unary clones (submonoids of the full transformation monoid) containing the permutations, on an infinite base set. It turns out that this number is quite large, on some cardinals as large as the whole clone lattice. Moreover we find that, with one exception, even the cardinalities of the intervals between the monoid of all permutations and the maximal submonoids of the full transformation monoid are as large. Whether or not the only exception is of the same cardinality as the other intervals depends on additional axioms of set theory.
We first determine the maximal clones on a set X of infinite regular cardinality which contain all permutations but not all unary functions, extending a result of Heindorf's for countably infinite X. If |X| 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.
On an infinite base set X, every ideal of subsets of X can be associated with the clone of those operations on X which map small sets to small sets. We continue earlier investigations on the position of such clones in the clone lattice.
We show that for an infinite set X, if L is a completely distributive algebraic lattice with not more completely join irreducible elements than the size of the power set of X, then there is a monoidal interval in the clone lattice on X which is isomorphic to 1+L, which is L plus a new smallest element added. Concerning cardinalities of monoidal intervals this result implies that there exist monoidal intervals of all cardinalities of at most the size of the power set of X, as well as monoidal intervals of cardinality 2^k, for all cardinals k which are not greater than the power set of X.
Given an infinite set X and an ideal I of subsets of X, the set of all finitary operations on X which map all (powers of) I-small sets to I-small sets is a clone. In a 2001 article, G. Czedli and L. Heindorf asked whether or not for two particular ideals I and J on a countably infinite set X, the corresponding ideal clones were a covering in the lattice of clones. We give an affirmative answer to this question.
We determine the atoms of the interval of the clone lattice consisting of those clones which contain all permutations, on an infinite base set. This is equivalent to the description of the atoms of the lattice of transformation monoids above the permutations.
We consider finitary relations (also known as crosses) that are definable via finite disjunctions of unary relations, i.e. subsets, taken from a fixed finite parameter set $Γ$. We prove that whenever $Γ$ contains at least one non-empty relation distinct from the full carrier set, there is a countably infinite number of polymorphism clones determined by relations that are disjunctively definable from $Γ$. Finally, we extend our result to finitely related polymorphism clones and countably infinite sets $Γ$.
We investigate the complexity of the lattice of local clones over a countably infinite base set. In particular, we prove that this lattice contains all algebraic lattices with at most countably many compact elements as complete sublattices, but that the class of lattices embeddable into the local clone lattice is strictly larger than that.
We investigate the lattice of clones that are generated by a set of functions that are induced on a finite field $\mathbb{F}$ by monomials. We study the atoms and coatoms of this lattice and investigate whether this lattice contains infinite ascending chains, or infinite descending chains, or infinite antichains. We give a connection between the lattice of these clones and semi-affine algebras. Furthermore, we show that the sublattice of idempotent clones of this lattice is finite and every idempotent monomial clone is principal.
We investigate the structure of the lattice of clones on an infinite set X. We first observe that ultrafilters naturally induce clones; this yields a simple proof of Rosenberg's theorem: "there are 2^2^kappa many maximal (=precomplete) clones on a set of size kappa." The clones we construct here do not contain all unary functions. We then investigate clones that do contain all unary functions. Using a strong negative partition theorem we show that for many cardinals kappa there are 2^2^kappa many such clones on a set of size kappa. Finally, we show that on a weakly compact cardinal there are e…
Let X be a linearly ordered set of arbitrary size (finite or infinite). Natural functions on such a set one can define using the linear order include maximum, minimum and median functions. While it is clear what the clone generated by the maximum or the minimum looks like, this is not obvious for median functions. We show that every clone on X contains either no median function or all median functions, that is, the median functions generate each other.
A clone on a set X is a set of finitary operations on X which contains all the projections and is closed under composition. The set of all clones forms a complete lattice Cl(X) with greatest element O, the set of all finitary operations. For finite sets X the lattice is "dually atomic": every clone other than O is below a coatom of Cl(X). It was open whether Cl(X) is also dually atomic for infinite X. Assuming the continuum hypothesis, we show that there is a clone C on a countable set such that the interval of clones above C is linearly ordered, uncountable, and has no coatoms.
We give three examples of large intervals in the lattice of (local) clones on an infinite set X, by exhibiting clones C_1, C_2, C_3 such that: (1) the interval [C_1, O] in the lattice of local clones is (as a lattice) isomorphic to {0,1,2, ...} under the divisibility relation, (2) the interval [C_2, O] in the lattice of local clones is isomorphic to the congruence lattice of an arbitrary semilattice, (3) the interval [C_3, O] in the lattice of all clones is isomorphic to the lattice of all filters on X. These examples explain the difficulty of obtaining a satisfactory analysis of the clone lat…
$\mathit{C}$-clones are polymorphism sets of so-called clausal relations, a special type of relations on a finite domain, which first appeared in connection with constraint satisfaction problems in [Creignou et al. 2008]. We completely describe the relationship w.r.t. set inclusion between maximal $\mathit{C}$-clones and maximal clones. As a main result we obtain that for every maximal $\mathit{C}$-clone there exists exactly one maximal clone in which it is contained. A precise description of this unique maximal clone, as well as a corresponding completeness criterion for $\mathit{C}$-clones i…
The main aim of this paper is to study aggregation functions on lattices via clone theory approach. Observing that the aggregation functions on lattices just correspond to $0,1$-monotone clones, as the main result we show that for any finite $n$-element lattice $L$ there is a set of at most $2n+2$ aggregation functions on $L$ from which the respective clone is generated. Namely, the set of generating aggregation functions consists only of at most $n$ unary functions, at most $n$ binary functions, and lattice operations $\wedge,\vee$, and all aggregation functions of $L$ are composed of them by…
We give a full description of all sets of functions on the group $(\mathbb{ Z}_p, +)$ of prime order which are closed under the composition with the clone generated by $+$ from both sides. Thereby, we also get a description of all iterative algebras on $\mathbb{Z}_p$ which are closed under the composition with the clone generated by $+$ from both sides. As another consequence, there are infinitely many non finitely generated clones above $\operatorname{Clo}({\mathbb{ Z}_p \times \mathbb{ Z}_p, +})$ for $p > 2$.
The general Galois theory for functions and relational constraints over arbitrary sets described in the authors' previous paper is refined by imposing algebraic conditions on relations.
Function clones are sets of functions on a fixed domain that are closed under composition and contain the projections. They carry a natural algebraic structure, provided by the laws of composition which hold in them, as well as a natural topological structure, provided by the topology of pointwise convergence, under which composition of functions becomes continuous. Inspired by recent results indicating the importance of the topological ego of function clones even for originally algebraic problems, we study questions of the following type: In which situations does the algebraic structure of a …
We prove that every clone of operations on a finite set A, if it contains a Malcev operation, is finitely related -- i.e., identical with the clone of all operations respecting R for some finitary relation R over A. It follows that for a fixed finite set A, the set of all such Malcev clones is countable. This completes the solution of a problem that was first formulated in 1980, or earlier: how many Malcev clones can finite sets support? More generally, we prove that every finite algebra with few subpowers has a finitely related clone of term operations. Hence modulo term equivalence and a ren…
To convert a fractional solution to an instance of a constraint satisfaction problem into a solution, a rounding scheme is needed, which can be described by a collection of symmetric operations with one of each arity. An intriguing possibility, raised in a recent paper by Carvalho and Krokhin, would imply that any clone of operations on a set $D$ which contains symmetric operations of arities $1, 2, \ldots, \lvert D \rvert$ contains symmetric operations of all arities in the clone. If true, then it is possible to check whether any given family of constraint satisfaction problems is solved by i…
One of the most fundamental mathematical contributions of Garrett Birkhoff is the HSP theorem, which implies that a finite algebra B satisfies all equations that hold in a finite algebra A of the same signature if and only if B is a homomorphic image of a subalgebra of a finite power of A. On the other hand, if A is infinite, then in general one needs to take an infinite power in order to obtain a representation of B in terms of A, even if B is finite. We show that by considering the natural topology on the functions of A and B in addition to the equations that hold between them, one can do wi…
A zig-zag order is like a directed path, only with alternating directions. A generating set of minimal size for the semigroup of all full transformations on a finite set preserving the zig-zag order was determined by Fenandes et al. in 2019. This paper deals with generating sets of the semigroup $F_{\mathbb{N}}$ of all full transformations on the set of all natural numbers preserving the zig-zag order. We prove that $F_{\mathbb{N}}$ has no minimal generating sets and present two particular infinite decreasing chains of generating sets of $F_{\mathbb{N}}$.
In the paper we introduce a notion of a key relation, which is similar to the notion of a critical relation introduced by Keith A.Kearnes and Ágnes Szendrei. All clones on finite sets can be defined by only key relations. In addition there is a nice description of all key relations on 2 elements. These are exactly the relations that can be defined as a disjunction of linear equations. In the paper we show that, in general key relations do not have such a nice description. Nevertheless, we obtain a nice characterization of all key relations preserved by a weak near-unanimity function. This char…
