Abstracts of Droste's Publications
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The normal subsemigroups of the monoid of injective maps *
(with R. Göbel)
, submitted
Abstract.
We consider the monoid Inj(M) of injective self-maps of a set M and want to determine its normal subsemigroups by numerical invariants. This was established by Mesyan in 2012 if M is countable. Here we obtain an explicit description of all normal subsemigroups of Inj(M) for any uncountable set M.

Automorphism groups of totally ordered sets: a retrospective survey
(with V. V. Bludov, A. M. W. Glass)
, Mathematica Slovaca 61 (2011), 373-388.
Abstract.
In 1963, W. Charles Holland proved that every lattice-ordered group can be embedded in the lattice-ordered group of all order- preserving permutations of a totally ordered set. In this article we examine the context and proof of this result and survey some of the many consequences of the ideas involved in this important theorem.

On extension of coverings
(with I. Rivin)
, Bull. London Math. Soc. 42 (2010), 1044-1054.
Abstract.
We address the question of when a covering of the boundary of a surface F can be extended to a covering of the surface (equivalently: when is th ere a branched cover with a prescribed monodromy ). If such an extension is possible , when can the total space be taken to be connected? When can the extension be taken to be regular? We give necessary and sucient conditions for both finite an d infinite covers (infinite covers are our main focus). In order to prove our resu lts, we show group-theoretic results of independent interests, such as the following extension (and simplification) of the theorem of Ore [11]: every element of the infinite symmetric group is the commutator of two elements which, together, act transitively.

On full groups of measure preserving and ergodic transformations with uncountable cofinalities
(with C.Holland, G.Ulbrich)
Bulletin London Math. Soc. 40 (2008), 463-472.
Abstract.
The group of all measure preserving permutations of the unit interval and the full group of an ergodic transformation of the unit interval are shown to have uncountable cofinality and the Bergman property. Here, a group G is said to have the Bergman property, if for any generating subset E of G, already some bounded power of EE-1 ∪ {1} covers G. This property arose in a recent interesting paper of Bergman where it was derived for the infinite symmetric groups. We give a general sufficient criterion for groups G to have the Bergman property. We show that the criterion applies to a range of further groups, including sufficiently transitive groups of, respectively, measure preserving, non-singular, or ergodic transformations of the reals or of homeomorphisms of the rationals, the irrationals, or Cantor's set.

Uncountable cofinalities of automorphism groups of linear and partial orders
(with J. K. Truss)
, Algebra Universalis 62 (2009), 75-90.
Abstract.
We demonstrate the uncountable cofinality of the automorphism groups of various linear and partial orders. We also relate this to the 'Bergman' property, and discuss cases where this may fail even though the cofinality is uncountable.

Stabilizers of direct composition series
(with R.Göbel)
Algebra Universalis 62 (2009), 209-237
Abstract.
Let R be a domain, V a left R-module, and L a composition series of direct summands of V . Our main results show that if U is a stabilizer group of L containing the McLaingroup associated with L, then U determines the chain (L,⊆) uniquely up to isomorphism or anti-isomorphism.

Construction of some uncountable 2-arc-transitive bipartite graphs
(with R.Gray, J.K.Truss)
Order 25 (2008), 349-357.
Abstract.
We give various constructions of uncountable arc-transitive bipartite graphs employing techniques from partial orders, starting with the cyclefree case, but generalizing to cases where this may be violated.

Absolute graphs with prescribed endomorphism monoids
(with R.Göbel, S.Pokutta)
Semigroup Forum 76 (2008), 256-267, Full version: pdf
Abstract.
We consider endomorphism monoids of graphs. It is well-known that any monoid can be represented as the endomorphism monoid M of some graph Γ with countably many colors. We give a new proof of this theorem such that the isomorphism between the endomorphism monoid End(Γ) and M is absolute, i.e. End(Γ) ≅ M holds in any generic extension of the given universe of set theory. This is true if and only if |M| , |Γ| are smaller than the first Erdös cardinal (which is known to be strongly inaccessible). We will encode Shelah's absolutely rigid family of trees [15] into Γ. The main result will be used to construct fields with prescribed absolute endomorphism monoids, see [8].

Bifinite Chu Spaces
(with G.-Q. Zhang)
in: 2nd Conf. on Algebra and Coalgebra in Computer Science (CALCO), Lecture Notes in Comp. Science vol. 4624, Springer, 2007, pp. 179-193.
Abstract.
This paper studies colimits of sequences of finite Chu spaces and their ramifications. We consider three bases categories of Chu spaces: the generic Chu spaces (C), the extensional Ch spaces (E), and the biextensional Chu spaces (B). The main results are: (1) a characterization of monics in each of the three categories; (2) existence (or the lack thereof) of colimits and a characterization of finite objects in each of the corresponding categories using monomorphisms/injections (denoted as iC, iE, and iB, respectively); (3) a formulation of bifinite Chu spaces with respect of iC; (4) the existence of universal, homogeneous Chu spaces in this category. Unanticipated results driving this development include the fact that: (a) in C, a morphism (f,g) is monic iff f is injective (but g is not necessarily surjective); (b) while colimits always exist in iE, it is not the case for iC and iB; (c) not all finite Chu spaces (considered set-theoretically) are finite objects in their categories. This study opens up opportunities for further investigations into recursively defined Chu spaces, as well as constructive models of linear logic.

Normal Subgroups of BuAut(Ω)
(with W.Ch. Holland),
Applied Categorical Structures 15 (2007), 153-162
Abstract.
Aut(Ω) denotes the group of all order preserving permutations of the totally ordered set Ω, and if e≤u ∈ Aut(Ω), then BuAut(Ω) denotes the subgroup of all those permutations bounded pointwise by a power of u. It is known that if Aut(Ω) is highly transitive, then Aut(Ω) has just five normal subgroups. We show that if Aut(Ω) is highly transitive and u has just one interval of support, then BuAut(Ω) has 220 normal subgroups, and there is a certain ideal Z of the lattice of subsets of P(Z), the power set of the integers, such that the lattice of normal subgroups of every such Aut(Ω) is isomorphic to Z.

Uncountable cofinalities of permutation groups
(with R. Göbel),
J. London Math. Soc. (2) 71 (2005), 335-344.
Abstract.
We will discover a sufficient criterion for certain permutation groups $G$ to have uncountable strong cofinality, i.e. they can not be expressed as the union of a countable, ascending chain $(H_i)_{i\in\o}$ of proper subsets $H_i$ such that $H_iH_i\subseteq H_{i+1}$ and $H_i=H_i^{-1}$. This is a strong form of uncountable cofinality for $G$, where each $H_i$ is a subgroup of $G$. This basic tool comes from a nice, recent article by G. M. Bergman on generating systems of the infinite symmetric groups, which we discuss in the introduction. Our main result is a theorem which can be applied to various classical groups including the symmetric groups and homeomorphism groups of Cantor's discontinuum, the rationals, and the irrationals, respectively. They all have uncountable strong cofinality. So the result also unifies various known results about cofinalities. Our favored example is the group \bsym of all bounded permutations of the rationals $\Q$ which has uncountable cofinality but countable strong cofinality. It is discussed in detail.

Generating automorphism groups of chains.
(with W. Ch. Holland),  Forum Mathematicum 17 (2005), 699-710.
Abstract.
Let (/Omega, /leq) be any doubly homogeneous chain and Aut(/Omega) its group of order-automorphisms. We show that if J is a set of generators of Aut(/Omega), then there is a positive integer n such that every element of Aut(/Omega) is a product of at most n members of J/cup J^{-1}. Also, Aut(/Omega) cannot be written as the union of a countable chain of proper subgroups of Aut(/Omega).

On random relational structures.
(with D. Kuske), Journal of Combinatorial Theory - Series A, 102 (2003), 241 - 254.
Abstract.
Erdös & Rényi gave a probabilistic construction of the countable universal homogeneous graph. We extend their result to more general structures of first order predicate calculus. Our main result shows that if a class of countable relational structures C contains an infinite $\omega$-categorical universal homogeneous structure U, then U can be constructed probabilistically.

Outer automorphism groups of ordered permutation groups.
(with S. Shelah), Forum Mathematicum 14 (2002), 605 - 621.
Abstract.
An infinite linearly ordered set (S,<)is called doubly homogeneous if its automorphism group A(S) acts 2-transitively on it. We show that any group G arises as outer automorphism group $G \cong Out(A(S))$ of the automorphism group A(S), for some doubly homogeneous chain (S,<).

All groups are outer automorphism groups of simple groups.
(with M. Giraudet and R. Göbel), Journal London Math. Soc. 64 (2001), 565 - 575.
Abstract.
We will show that each group is the outer automorphism group of a simple group. Surprisingly, the proof is mainly based on the theory of ordered or relational structures and their symmetry groups. By a recent result of Droste and Shelah (2000) any group is the outer automorphism group Out(Aut T) of the automorphism group Aut T of a doubly homogeneous chain (T,<). But Aut T is never simple. However, following recent investigations on automorphism groups of circles, we are able to turn (T,<) into a circle C such that
$Out(Aut T) \cong Out(Aut C)$. The unavoidable normal subgroups in Aut T evaporate in Aut C, which is now simple, and our result follows.

On the homeomorphism groups of Cantor's discontinuum and the spaces of rational and of irrational numbers.
(with  R. Göbel), Bull. London Math. Soc. 34 (2002), 474 - 478.
Abstract.
We will show that the homeomorphism groups of Cantor's discontinuum, the rationals, and the irrationals, respectively have uncountable cofinality. It is well known that the homeomorphism group of Cantor's
discontinuum is isomorphic to the automorphism group AutB of the countable, atomless boolean algebra B. So also AutB has uncountable cofinality, which answers a question of Droste and Macpherson. The cofinality of a group G is the cardinality of the length of a shortest chain of proper subgroups terminating at G.

(with J.K. Truss), Proc. Amer. Math. Soc., 129 (2001), 1601 - 1608.
Abstract.
A chain (linearly ordered set) is rigid if it has no non-trivial automorphisms. The construction of dense rigid chains was carried out by Dushnik and Miller for subsets of R, and there is a rather different construction of dense rigid chains of cardinality $\kappa$, an uncountable regular cardinal, using stationary sets as 'codes', which was adapted by Droste to show the existence of rigid measurable spaces. Here we examine the possibility that nevertheless there could be many order-embeddings of the chain, in the sense that the whole chain can be embedded into any interval. In the case of subsets of R, an argument involving Baire category is used to modify the original one. For uncountable regular cardinals, a more complicated version of the corresponding argument is used, in which the stationary sets are replaced by sequences of stationary sets, and the chain is built up using a tree. The construction is also adapted to the case of singular cardinals.

Complementary closed relational clones are not always Krasner clones.
(with D. Kuske, R. McKenzie and R. Pöschel), Algebra Universalis 45 (2001), 155 - 160.
Abstract.
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.

The automorphism group of the universal distributive lattice.
(with D. Macpherson), Algebra Universalis 43 (2000), 295 - 306.
Abstract.
We describe the normal subgroup lattice of the automorphism groups of the countable universal homogeneous distributive lattice and of the countable atomless generalized Boolean lattice. Also, we show that subgroups of these automorphism groups of index less than continuum lie between the pointwise and the setwise stabilizer of a finite set.

On homogeneous semilattices and their automorphism groups.
(with D. Kuske and J.K. Truss), Order 16 (1999), 31 - 56.
Abstract.
We show that there are just countably many countable homogeneous semilattices and give an explicit description of them. For the countable universal homogeneous semilattice we show that ist automorphism group has a largest proper nontrivial normal subgroup.

Simple automorphism groups of cycle-free partial orders.
(with J.K. Truss and R. Warren), Forum Mathematicum 11 (1997), 279 - 294.
Abstract.
The purpose of this paper is to show that the automorphism groups of many of the 'cycle-free' partial orders studied are simple. This contrasts strongly with the situation for trees, of which they form a natural generalization. It was shown that the automorphism group of any sufficiently transitive tree has at least $2^{2^{\aleph_0}}$ normal subgroups. All the infinite cycle-free partial orders studied have simple automorphism groups. The finite chain case is more involved; where the ordering on chains of the Dedekind-MacNeille completion can be expressed as a lexicographic product by a non-trivial discrete (transitive) ordering (respected by the group), the automorphism group is not simple. For both finite and infinite chain cases the simple automorphism groups split into two classes: those where there is a bound (<12) on the number of conjugates required to express one non-identity element in terms of another, and those in which there is no such bound.

Set-homogeneous graphs and embeddings of total orders.
(with M. Giraudet and D. Macpherson), Order 14 (1997), 9 - 20.
Abstract.
We construct uncountable graphs in which any two isomorphic subgraphs of size at most 3 can be carried one to the other by an automorphism of the graph, but in which some isomorphism between 2-element subsets does not extend to an automorphism. The corresponding phenomenon does not occur in the countable case. The construction uses a suitable construction of infinite homogeneous coloured chains.

Set-homogeneous graphs.
(with M. Giraudet, D. Macpherson and N. Sauer), J. Combinatorial Theory Ser. B, 62 (1994), 63 - 95.
Abstract.
We investigate set-homogeneity (a weakening of Fraissé's notion of homogeneity), give an example of a set-homogeneous graph which is not homogeneous, and characterize it by its symmetry properties. A variant of Fraissé's amalgamation theorem is also given.

Representations of free lattice-ordered groups.
Order 10 (1993), 375 - 381.
Abstract.
We show for any uncountable cardinal $\eta$ that the free group $G_\eta$ of rank $\eta$ has a linear right ordering on which the natural action of free lattice-ordered group $F_\eta$ of rank $\eta$ is faithful and pathologically 2-transitive. As a consequence, we obtain results on the root system of prime subgroups of $F_\eta$. This generalizes previous results of McCleary which required the generalized continuum hypothesis and $\eta$ to be regular.

On k-homogeneous posets and graphs.
(with D. Macpherson), J. Combinatorial Theory Ser. A56 (1991), 1 - 15.
Abstract.
A relational structure A is called k-homogeneous if each isomorphism between two k-element substructures of A extends to an automorphism of A. We show that if a countable poset $(\Omega,\leq)$ is 1- or 4-homogeneous, then it is k-homogeneous for each $k\in N$. There are infinitely many examples of $\aleph_0$-categorical universal countable posets $(\Omega,\leq)$ showing that here the number 4 may not be replaced by 2 or 3. We also show that for every $k\in N$ there are continuously many countable $\aleph_0$-categorical universal graphs which are k-homogeneous but not (k+1)-homogeneous; this answers a question of R. Fraissé.

The root system of prime subgroups of a free lattice-ordered group (without G.C.H.).
(with S.H. McCleary), Order 6 (1989), 305 - 309.
Abstract.
For the free lattice-ordered group $F_\eta$ of rank $\eta$ the root system $P_\eta$ of prime subgroups has been described in considerable detail by McCleary, for $\eta\leq\omega_0$ and (with G.C.H.) for regular $\eta>\omega_0$. Here, almost all of that description is obtained without G.C.H., and most of it without regularity.

Embeddings into simple lattice-ordered groups with different first order theories.
(with M. Giraudet), Forum Mathematicum 1 (1989), 315 - 321.
Abstract.
We show that any lattice-ordered group G can be embedded into continuously many simple divisible lattice-ordered groups $H_i$ which are pairwise elementarily inequivalent both as groups and as lattices. Moreover, the l-groups $H_i$ can be chosen to have a representation as doubly transitive groups of order-preserving permutations of chains and such that their group structure and their lattice-plus-identity structure are mutually definable in each other withoutadditional parameters.

Automorphism groups of infinite semilinear orders (II).
(with W.C. Holland and D. Macpherson), Proc. London Math. Soc. 58 (1989), 479 - 494.
Abstract.
This paper and its predecessor examine certain infinite semilinear orders ('trees') and their automorphism groups. Here we classify weakly 2-transitive trees up to $L_\infty_\omega$-equivalence, and countable weakly 2-transitive trees up to isomorphism. Various results are obtained about the automorphism groups, concerning torsion, divisibility, and subgroups of small index. The automorphism group of some related treelike struktures and their normal subgroup lattices are also examined.

Automorphism groups of infinite semilinear orders (I).
(with W.C. Holland and D. Macpherson), Proc. London Math. Soc. 58 (1989), 454 - 478.
Abstract.
Results are obtained concerning normal subgroups of the automorphism groups of certain infinite trees. These structures are mostly $\aleph_0$-categorical, and are trees in a poset-theoretic but not graph-theoretic sense. It is shown that the automorphism group has a smallest non-trivial normal subgroup, a largest proper normal subgroup, and at least $2^2^\omega$ normal subgroups between these two. We also obtain and use some results on groups of automorphisms of chains.

The existence of rigid measurable spaces.
Topology and its Applications 31 (1989), 187 - 195.
Abstract.
For each uncountable cardinal k we construct $2^k$ dense unbounded chains $(S_i,\leq)$ of cardinality k which as topological spaces (endowed with the order-topology) have, in particular, the following properties: They are each 0-dimensional and mono-rigid, i.e. the only embedding of $S_i$ into itself is the identity, and they are pairwise nonembeddable into each other. If $k\geq\aleph_2$, the sets $S_i$ can be chosen such that, in addition, each $G_\delta$-set is open and hence the measurable spaces $(S_i,B_i)$, where $B_i$ is the $\delta$-algebra of all clopen subsets of $S_i$, are mono-rigid and pairwise nonembeddable into each other.

Super-rigid families of strongly Blackwell spaces.
Proc. Amer. Math. Soc. 103(1988), 803 - 808.
Abstract.
We construct a complete subfield F of P(R), isomorphic to P(R), of pairwise non-Borel-isomorphic rigid strong Blackwell subsets of R such that there are only 'very few' measurable functions between any two members of F. As a consequence, we obtain large chains and antichains of non-isomorphic rigid strong Blackwell subsets of R. Also, there is a collection of continuously many dense subsets of R such that any two of them differ only by two elements, but none of them is a continuous image of any other.

Normal subgroups and elementary theories of lattice-ordered groups.
Order 5 (1988), 261 - 273.
Abstract.
We show that any lattice-ordered group (l-group) G can be l-embedded into continuously many l-groups $H_i$ which are pairwise elementarily inequivalent both as groups and as lattices with constant e. Our groups $H_i$ can be distinguished by group-theoretical first-order properties which are induced by lattice-theoretically 'nice' properties of their normal subgroup lattices. Moreover. they can be taken to be 2-transitive automorphism groups $A(S_i)$ of infinite linearly ordered sets $(S_i,\leq)$ such that each group $A(S_i)$ has only inner automorphisms. We also show that any countable l-group G can be l-embedded into a countable l-group H whose normal subgroup lattice is isomorphic to the lattice of all ideals of the countable dense Boolean algebra B.

Squares of conjugacy classes in the infinite symmetric groups.
Trans. Amer. Math. Soc. 303 (1987), 503 - 515.
Abstract.
Using combinatorial methods, we will examine squares of conjugacy classes in the symmetric groups $S_\nu$ of all permutations of an infinite set of cardinality $\aleph_\nu$. For arbitrary permutations $p\in S_\nu$, we will characterize when each element $s\in S_\nu$ with finite support can be written as a product of two conjugates of p, and if p has infinitely many fixed points, we determine when all elements of $S_\nu$ are products of two conjugates of p. Classical group-theoretical theorems are obtained from similar results.

Completeness properties of certain normal subgroup lattices.
European J. Combinatorics 8 (1987), 129 - 137.
Abstract.
We examine the normal subgroup lattice of 2-transitive automorphism groups $A(\Omega)$ of infinite linearly ordered sets $(\Omega,\leq)$. Using combinatorial methods, we prove that in each of these lattices the partially ordered subset of all those elements which are finitely generated as normal subgroups is a lattice in which infima and suprema of subsets of cardinality $\leq\aleph$, always exist; two infinite distributive identities are also shown to hold. Similar methods are used to give a completeness result for reduced products of partially ordered sets.

On the universality of systems of words in permutation groups.
(with S. Shelah), Pacific J. Math. 127 (1987), 321 - 328.
Abstract.
In the classes of infinite symmetric groups, their normal subgroups, and their factor groups, we determine those groups which are equivalent in the sense that they may not be distinguished by the solvability of a system of finitely many equations in variables and parameters.

Partially ordered sets with transitive automorphism groups.
Proc. London Math. Soc. 54 (1987), 514 - 543.
Abstract.
In this paper, we study the structure of partially ordered sets $(\Omega,\leq)$ under suitable transitivity assumptions on their group $A(\Omega)$ of all order-automorphisms of $(\Omega,\leq)$. Let us call $A(\Omega)$ k-transitive (k-homogeneous) if whenever A,B are two isomorphic subsets of $\Omega$ each with k elements, then some (any) isomorphism from $(A,\leq)$ onto $(B,\leq)$ extends to an automorphism of $\Omega$, respectively. We show that if $k\geq 4 (k=3), there are precisely k (5) non-isomorphic countable partially ordered sets$(\Omega,\leq)$not containing the pentagon such that$(A(\Omega)$is k-transitive but not k-homogeneous; if k=2, there are a unique countable, and many different uncountable sets$(\Omega,\leq)$of this type. We also give necessary and sufficient conditions for two partially ordered sets$(\Omega,\leq)$not containing the pentagon and with k-transitive automorphism group$(k\geq 2)$to be$L_\infty_\omega$-equivalent. Complete embeddings of linear orderings and embeddings of lattice-ordered groups. Israel J. Math. 56 (1986), 315 - 334. Abstract. An infinite linearly ordered set$(S,\leq)$is called doubly homogeneous, if its automorphism group Aut$(\Omega,\leq)$acts 2-transitively on it. We study embeddings of linearly ordered sets into Dedekind-completions of doubly homogeneous chains which preserve all suprema and infima, and obtain necessary and sufficient conditions for the existence of such embeddings. As one of several consequences, for each lattice-ordered group G and each regular uncountable cardinal$k\geq|G|$there are$2^k$non-isomorphic simple divisible lattice-ordered groups H of cardinality k all containing G as an l-subgroup. On the universality of words for the alternating groups. Proc. Amer. Math. Soc. 96 (1986), 18 - 22. Abstract. We prove the following theorem on the finite alternating groups$A_n$: For each pair (p,q) of nonzero integers there exists an integer N(p,q) such that, for each$n\geq N$, any even permutation$a\in A_n$can be written in the form$a=b^p\cdot c^q$for some suitable elements$b,c\in A_n$. A similar result is shown to be true for the finite symmetric groups$S_n$provided that p or q is odd. The normal subgroup lattice of 2-transitive automorphism groups of linearly ordered sets. Order 2 (1985), 291 - 319. Abstract. Using combinatorial and model-theoretic means, we examine the structure of normal subgroup lattices$N(A(\Omega))$of 2-transitive automorphism groups$A(\Omega)$of infinite linearly ordered sets$(\Omega,\leq)$. Certain natural sublattices of$N(A(\Omega))$are shown to be Stone algebras, and several first order properties of their dense and dually dense elements are characterized within the Dedekind-completion$(\bar\Omega,\leq)$of$(\Omega,\leq)$. As a consequence,$A(\Omega)$has either precisely 5 or at least$2^2^{\aleph_1}$(even maximal) normal subgroups, and various other group- and lattice-theoretic results follow. A construction of all normal subgroup lattices of 2-transitive automorphism groups of linearly ordered sets. (with S. Shelah), Israel J. Math. 51 (1985), 223 - 261. Abstract. We give a complete classification and construction of all normal subgroup lattices of 2-transitive automorphism groups$A(\Omega)$of linearly ordered sets$(\Omega,\leq)$. We also show that in each of these normal subgroup lattices the partially ordered subset of all those elements which are finitely generated as normal subgroups forms a lattice which is closed under even countably-infinite intersections, and we derive several further group-theoretical consequences from our classification. Normal subgroups of doubly transitive automorphism groups of chains. (with R.N. Ball), Trans. Amer. Math. Soc. 290 (1985), 647 - 664. Abstract. We characterize the structure of the normal subgroup lattice of 2-transitive automorphism groups$A(\Omega)$of infinite chains$(\Omega,\leq)$by the structure of the Dedekind completion$(\bar\Omega,\leq)$of the chain$(\Omega,\leq)$. As a consequence we obtain various group-theoretical results on the normal subgroups of$A(\Omega)$, including that any proper subnormal subgroup of$A(\Omega)$is indeed normal and contained in a maximal proper normal subgroup of$A(\Omega)$, and that$A(\Omega)$has precisely 5 normal subgroups if and only if the coterminality of the chain$(\Omega,\leq)$is countable. Cubes of conjugacy classes covering the infinite symmetric group. Trans. Amer. Math. Soc. 288 (1985), 381 - 393. Abstract. Using combinatorial methods, we prove the following theorem on the group S of all permutations of a countably-infinite set: Whenever$p\in S$has infinite support without being a fixed-point-free involution, then any$s\in S$is a product of three conjugates of p. Furthermore, we present uncountably many new conjugacy classes C of S satisfying that any$s\in S$is a product of two elements of C. Similar results are shown for permutations of uncountable sets. Products of conjugacy classes of the infinite symmetric group. Discrete Math. 47 (1983), 35 - 48. Abstract. Using combinatorial methods, we will examine products of conjugacy classes in the symmetric group$S_0$of all permutations of a countably infinite set. If$p\in S_0$has at least one infinite orbit in the underlying set and if$s\in S_0$, we give a characterization of when s is a product of two conjugates of p. From this, we derive that if four permutations$p_i\in S_0 (i=1,2,3,4)$are given which all have infinite support, then any permutation of$S_0$is a product of four elements conjugate to$p_1, p_2, p_3$and$p_4$, respectively. Similar results for permutations of uncountable sets are shown and classical group-theoretical results are obtained from these theorems. Products of conjugate permutations. (with R. Göbel), Pacific J. Math. 94 (1981), 47 - 60. Abstract. Using combinatorial methods, we will prove the following theorem on the permutation group$S_0$of a countable set: If a permutation$p\in S_0$contains at least one infinite cycle then any permutation of$S_0$is a product of three permutations each conjugate to p. Similar results for permutations of uncountable sets are shown and classical group-theoretical results are derived from this. Structure of partially ordered sets with transitive automorphism groups. Memoirs Amer. Math. Soc. 334 (1985), reprinted and updated 2003, in preparation. Abstract. In this paper, we study the structure of infinite partially ordered sets$(\Omega,\leq)$under suitable transitivity assumptions on their group$A(\Omega)=Aut(\Omega\leq)$of all order-automorphisms of$(\Omega,\leq)$. Let$k\in N$. We call$A(\Omega)$k-transitive (k-homogeneous) if whenever$A,B\subseteq \Omega$are two subsets of$\Omega$each with k elements and$\varphi: (A,\leq) \to (B,\leq)$is an isomorphism, then there exists an automorphism$\alpha\in A(\Omega)$which maps A onto B (which extends$\varphi$), respectively.$A(\Omega)$is$\omega$-transitive ($\omega$-homogeneous), if$A(\Omega)$is k-transitive (k-homogeneous) for each$k\in N$. We show that under the assumption that$A(\Omega)$is k-transitive or k-homogeneous for some$2\leq k\in N$various sufficiently complicated structures$(\Omega,\leq)$exist, and we give a classification and characterization of these structures. As one of many consequences we obtain that for each$k\geq 2$, k-transitivity of$A(\Omega)$is indeed weaker than k-homogeneity, but, surprisingly, for any partially ordered set$(\Omega,\leq)$,$A(\Omega)$is$\omega$-transitive iff$A(\Omega)$is$\omega\$-homogeneous.