The Bergman-Shelah Preorder on Transformation Semigroups

This is the peer-reviewed version of the following article: Mesyan, Z., Mitchell, J. D., Morayne, M. and Péresse, Y. H. (2012), Mathematical Logic Quarterly, Vol. 58: 424–433, 'The Bergman-Shelah preorder on transformation semigroups', which has been published in final form at doi:10.1002/malq.201200002. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. Copyright © 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. http://www.interscience.wiley.com/Let \nat^\nat be the semigroup of all mappings on the natural numbers \nat, and let $U$ and $V$ be subsets of \nat^\nat. We write $U\preccurlyeq V$ if there exists a countable subset $C$ of \nat^\nat such that $U$ is contained in the subsemigroup generated by $V$ and $C$. We give several results about the structure of the preorder $\preccurlyeq$. In particular, we show that a certain statement about this preorder is equivalent to the Continuum Hypothesis. The preorder $\preccurlyeq$ is analogous to one introduced by Bergman and Shelah on subgroups of the symmetric group on \nat. The results in this paper suggest that the preorder on subsemigroups of \nat^\nat is much more complicated than that on subgroups of the symmetric group.Peer reviewe

Generating self-map monoids of infinite sets

Let I be a countably infinite set, S = Sym(I) the group of permutations of I, and E = End(I) the monoid of self-maps of I. Given two subgroups G, G' of S, let us write G \approx_S G' if there exists a finite subset U of S such that the groups generated by G \cup U and G' \cup U are equal. Bergman and Shelah showed that the subgroups which are closed in the function topology on S fall into exactly four equivalence classes with respect to \approx_S. Letting \approx denote the obvious analog of \approx_S for submonoids of E, we prove an analogous result for a certain class of submonoids of E, from which the theorem for groups can be recovered. Along the way, we show that given two subgroups G, G' of S which are closed in the function topology on S, we have G \approx_S G' if and only if G \approx G' (as submonoids of E), and that cl_S (G) \approx cl_E (G) for every subgroup G of S (where cl_S (G) denotes the closure of G in the function topology in S and cl_E (G) its closure in the function topology in E).Comment: 26 pages. In the second version several of the arguments have been simplified, references to related literature have been added, and a few minor errors have been correcte

Automatic continuity, unique Polish topologies, and Zariski topologies on monoids and clones

In this paper we explore the extent to which the algebraic structure of a monoid $M$ determines the topologies on $M$ that are compatible with its multiplication. Specifically we study the notions of automatic continuity; minimal Hausdorff or Polish semigroup topologies; and we formulate a notion of the Zariski topology for monoids. If $M$ is a topological monoid such that every homomorphism from $M$ to a second countable topological monoid $N$ is continuous, then we say that $M$ has \emph{automatic continuity}. We show that many well-known monoids have automatic continuity with respect to a natural semigroup topology, namely: the full transformation monoid $\mathbb{N}^\mathbb{N}$; the full binary relation monoid $B_{\mathbb{N}}$; the partial transformation monoid $P_{\mathbb{N}}$; the symmetric inverse monoid $I_{\mathbb{N}}$; the monoid Inj$(\mathbb{N})$ consisting of the injective functions on $\mathbb{N}$; and the monoid $C(2^{\mathbb{N}})$ of continuous functions on the Cantor set. We show that the pointwise topology on $\mathbb{N}^\mathbb{N}$, and its analogue on $P_{\mathbb{N}}$, are the unique Polish semigroup topologies on these monoids. The compact-open topology is the unique Polish semigroup topology on $C(2^\mathbb{N})$ and $C([0, 1]^\mathbb{N})$. There are at least 3 Polish semigroup topologies on $I_{\mathbb{N}}$, but a unique Polish inverse semigroup topology. There are no Polish semigroup topologies $B_{\mathbb{N}}$ nor on the partitions monoids. At the other extreme, Inj$(\mathbb{N})$ and the monoid Surj$(\mathbb{N})$ of all surjective functions on $\mathbb{N}$ each have infinitely many distinct Polish semigroup topologies. We prove that the Zariski topologies on $\mathbb{N}^\mathbb{N}$, $P_{\mathbb{N}}$, and Inj$(\mathbb{N})$ coincide with the pointwise topology; and we characterise the Zariski topology on $B_{\mathbb{N}}$. In Section 7: clones.Comment: 51 pages (Section 7 about clones was added in version 4

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems