Hajime Machida, Michael Pinsker
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.
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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.
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 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…
Permutation clones that preserve relations
本文研究由关系定义的置换克隆，刻画有限集上其格结构，证明极大借位闭克隆的两类定义，并确定二元集上共有13个此类克隆。
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…
We determine all majority operations on a four-element set that generate a minimal clone.
We characterize minimal clones generated by a majority function containing at most seven ternary operations.
We classify binary minimal clones into seven categories: affine algebras, rectangular bands, $p$-cyclic groupoids, spirals, non-Taylor partial semilattices, melds, and dispersive algebras. Each category has nice enough properties to guarantee that any clone of one of these types contains a minimal clone of the same type.
We present two minimal clones containing 26 and 78 majority operations respectively, more than any other previously known example.
We show that a minimal clone has a nontrivial weakly abelian representation iff it has a nontrivial abelian representation, and that in this case all representations are weakly abelian.
Characterizations of "almost associative" binary operations generating a minimal clone are given for two interpretations of the term "almost associative". One of them uses the associative spectrum, the other one uses the index of nonassociativity to measure how far an operation is from being associative.
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 …
Minimal operations over permutation groups
本文对任意置换群上极小运算的可能类型进行分类，证明非平凡置换群上至多存在四种类型，改进了Bodirsky和Chen关于寡形置换群的结果，并回答了三个先前未解决的问题。
We study bounded width algebras which are minimal in the sense that every proper reduct does not have bounded width. We show that minimal bounded width algebras can be arranged into a pseudovariety with one basic ternary operation. We classify minimal bounded width algebras which have size at most three, and prove a structure theorem for minimal bounded width algebras which have no majority subalgebra, which form a pseudovariety with a commutative binary operation. As a byproduct of our results, we also classify minimal clones which have a Taylor term.
We determine all composition-closed equational classes of Boolean functions. These classes provide a natural generalization of clones and iterative algebras: they are closed under composition, permutation and identification (diagonalization) of variables and under introduction of inessential variables (cylindrification), but they do not necessarily contain projections. Thus the lattice formed by these classes is an extension of the Post lattice. The cardinality of this lattice is continuum, yet it is possible to describe its structure to some extent.
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