Maximal subsemigroups of some semigroups of order-preserving mappings on a countably infinite set

AbstractIn this paper, we study the maximal subsemigroups of several semigroups of order-preserving transformations on the natural numbers and the integers, respectively. We determine all maximal subsemigroups of the monoid of all order-preserving injections on the set of natural numbers as well as on the set of integers. Further, we give all maximal subsemigroups of the monoid of all bijections on the integers. For the monoid of all order-preserving transformations on the natural numbers, we classify also all its maximal subsemigroups, containing a particular set of transformations.

Maximal subsemigroups containing a particular semigroup

Mathematica Slovaca ◽

10.2478/s12175-014-0280-0 ◽

2014 ◽

Vol 64 (6) ◽

Author(s):

Jörg Koppitz ◽

Tiwadee Musunthia

Keyword(s):

Natural Numbers ◽

Finitely Generated ◽

Maximal Subsemigroups ◽

AbstractWe characterize maximal subsemigroups of the monoid T(X) of all transformations on the set X = ℕ of natural numbers containing a given subsemigroup W of T(X) such that T(X) is finitely generated over W. This paper gives a contribution to the characterization of maximal subsemigroups on the monoid of all transformations on an infinite set.

Monotone but not positive subsets of the Cantor space

10.1017/s0022481200029790 ◽

1987 ◽

Vol 52 (3) ◽

pp. 817-818 ◽

Author(s):

Randall Dougherty

Keyword(s):

Partial Ordering ◽

Background Information ◽

Countable Union ◽

Abstract Form ◽

Cantor Space ◽

Countable Intersection ◽

Power Set ◽

Countably Infinite ◽

A subset of the Cantor space ω2 is called monotone iff it is closed upward under the partial ordering ≤ defined by x ≤ y iff x(n) ≤ y(n) for all n ∈ ω. A set is -positive (-positive) iff it is monotone and -positive set is a countable union of -positive sets; a -positive set is a countable intersection of -positive sets. (See Cenzer [2] for background information on these concepts.) It is clear that any -positive set is and monotone; the converse holds for n ≤ 2 [2] and was conjectured by Dyck to hold for greater n. In this note, we will disprove this conjecture by giving examples of monotone sets (for n ≥ 3) which are not even -positive.First we note a few isomorphisms. The space (ω2, ≤) is isomorphic to the space (ω2 ≥), so instead of monotone and positive sets we may construct hereditary and negative sets (the analogous notions with “closed upward” replaced by “closed downward”). Also, (ω2, ≤) is isomorphic to ((ω), ⊆), where denotes the power set operator, or to ((S), ⊆) for any countably infinite set S.In order to remove extraneous notation from the proofs, we state the results in an abstract form (whose generality is deceptive).

REVERSE MATHEMATICS, YOUNG DIAGRAMS, AND THE ASCENDING CHAIN CONDITION

10.1017/jsl.2016.17 ◽

2017 ◽

Vol 82 (2) ◽

pp. 576-589 ◽

Cited By ~ 3

Author(s):

KOSTAS HATZIKIRIAKOU ◽

STEPHEN G. SIMPSON

Keyword(s):

Theorem Proving ◽

Chain Condition ◽

Reverse Mathematics ◽

Partial Ordering ◽

Young Diagrams ◽

Equivalence Proof ◽

Image Position ◽

Countably Infinite ◽

Infinite Set ◽

Ascending Chain Condition

AbstractLetSbe the group of finitely supported permutations of a countably infinite set. Let$K[S]$be the group algebra ofSover a fieldKof characteristic 0. According to a theorem of Formanek and Lawrence,$K[S]$satisfies the ascending chain condition for two-sided ideals. We study the reverse mathematics of this theorem, proving its equivalence over$RC{A_0}$(or even over$RCA_0^{\rm{*}}$) to the statement that${\omega ^\omega }$is well ordered. Our equivalence proof proceeds via the statement that the Young diagrams form a well partial ordering.

Proceed with care, because some infinities are larger than others

How to Free Your Inner Mathematician ◽

10.1093/oso/9780198843597.003.0047 ◽

2020 ◽

pp. 281-292

Author(s):

Susan D'Agostino

Keyword(s):

Number Line ◽

Mathematics Students ◽

Natural Numbers ◽

Real Numbers ◽

Countable Infinity ◽

One To One ◽

“Proceed with care, because some infinities are larger than others” explains in detail why the infinite set of real numbers—all of the numbers on the number line—represents a far larger infinity than the infinite set of natural numbers—the counting numbers. Readers learn to distinguish between countable infinity and uncountable infinity by way of a method known as a “one-to-one correspondence.” Mathematics students and enthusiasts are encouraged to proceed with care in both mathematics and life, lest they confuse countable infinity with uncountable infinity, large with unfathomably large, or order with disorder. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

On radicals of infinite matrix rings

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500012694 ◽

1969 ◽

Vol 16 (3) ◽

pp. 195-203 ◽

Author(s):

A. D. Sands

Keyword(s):

Infinite Matrix ◽

Natural Numbers ◽

Index Set ◽

Matrix Rings ◽

Let R be a ring and I an infinite set. We denote by M(R) the ring of row finite matrices over I with entries in R. The set I will be omitted from the notation, as the same index set will be used throughout the paper. For convenience it will be assumed that the set of natural numbers is a subset of I.

How to Program an Infinite Abacus

Canadian Mathematical Bulletin ◽

10.4153/cmb-1961-032-6 ◽

1961 ◽

Vol 4 (3) ◽

pp. 295-302 ◽

Cited By ~ 35

Author(s):

Joachim Lambek

Keyword(s):

Finite Number ◽

Recursive Function ◽

Computable Function ◽

Unlimited Supply ◽

Countably Infinite ◽

This is an expository note to show how an “infinite abacus” (to be defined presently) can be programmed to compute any computable (recursive) function. Our method is probably not new, at any rate, it was suggested by the ingenious technique of Melzak [2] and may be regarded as a modification of the latter.By an infinite abacus we shall understand a countably infinite set of locations (holes, wires etc.) together with an unlimited supply of counters (pebbles, beads etc.). The locations are distinguishable, the counters are not. The confirmed finitist need not worry about these two infinitudes: To compute any given computable function only a finite number of locations will be used, and this number does not depend on the argument (or arguments) of the function.

Baer–Levi semigroups of linear transformations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500003309 ◽

2004 ◽

Vol 134 (3) ◽

pp. 477-499 ◽

Cited By ~ 3

Author(s):

Suzana Mendes-Gonçalves ◽

R. P. Sullivan

Keyword(s):

Vector Space ◽

Dimensional Vector ◽

Linear Transformations ◽

Dimensional Vector Space ◽

Infinite Dimensional ◽

Basic Properties ◽

Maximal Subsemigroups ◽

Linear Version ◽

Given an infinite-dimensional vector space V, we consider the semigroup GS (m, n) consisting of all injective linear α: V → V for which codim ran α = n, where dim V = m ≥ n ≥ ℵ0. This is a linear version of the well-known Baer–Levi semigroup BL (p, q) defined on an infinite set X, where |X| = p ≥ q ≥ ℵ0. We show that, although the basic properties of GS (m, n) are the same as those of BL (p, q), the two semigroups are never isomorphic. We also determine all left ideals of GS (m, n) and some of its maximal subsemigroups; in this, we follow previous work on BL (p, q) by Sutov and Sullivan as well as Levi and Wood.

The extensions of the modal logic K5

10.2307/2273793 ◽

1985 ◽

Vol 50 (1) ◽

pp. 102-109 ◽

Cited By ~ 12

Author(s):

Michael C. Nagle ◽

S. K. Thomason

Keyword(s):

Modal Logic ◽

Classical Logic ◽

Abstract Characterization ◽

Propositional Modal Logic ◽

Normal Logic ◽

Countably Infinite ◽

Our purpose is to delineate the extensions (normal and otherwise) of the propositional modal logic K5. We associate with each logic extending K5 a finitary index, in such a way that properties of the logics (for example, inclusion, normality, and tabularity) become effectively decidable properties of the indices. In addition we obtain explicit finite axiomatizations of all the extensions of K5 and an abstract characterization of the lattice of such extensions.This paper refines and extends the Ph.D. thesis [2] of the first-named author, who wishes to acknowledge his debt to Brian F. Chellas for his considerable efforts in directing the research culminating in [2] and [3]. We also thank W. J. Blok and Gregory Cherlin for observations which greatly simplified the proofs of Theorem 3 and Corollary 10.By a logic we mean a set of formulas in the countably infinite set Var of propositional variables and the connectives ⊥, →, and □ (other connectives being used abbreviatively) which contains all the classical tautologies and is closed under detachment and substitution. A logic is classical if it is also closed under RE (from A↔B infer □A ↔□B) and normal if it is classical and contains □ ⊤ and □ (P → q) → (□p → □q). A logic is quasi-classical if it contains a classical logic and quasi-normal if it contains a normal logic. Thus a quasi-normal logic is normal if and only if it is classical, and if and only if it is closed under RN (from A infer □A).

A note on degrees of subsets1

10.2307/2271101 ◽

1969 ◽

Vol 34 (2) ◽

pp. 256-256 ◽

Author(s):

Robert I. Soare

Keyword(s):

Minimal Degree ◽

Turing Degree ◽

Lower Degree ◽

Natural Numbers ◽

In [2] we constructed an infinite set of natural numbers containing no subset of higher (Turing) degree. Since it is well known that there are nonrecursive sets (e.g. sets of minimal degree) containing no nonrecursive subset of lower degree, it is natural to suppose that these arguments may be combined, but this is false. We prove that every infinite set must contain a nonrecursive subset of either higher or lower degree.

Every analytic set is Ramsey

10.1017/s0022481200092239 ◽

1970 ◽

Vol 35 (1) ◽

pp. 60-64 ◽

Cited By ~ 23

Author(s):

Jack Silver

Keyword(s):

Measurable Cardinal ◽

Infinite Subset ◽

The Other ◽

Natural Numbers ◽

Ramsey Property ◽

Analytic Subset ◽

Borel Set ◽

Analytic Set ◽

Principal Theorem ◽

Countably Infinite

If X is a set, [Χ]ω will denote the set of countably infinite subsets of X. ω is the set of natural numbers. If S is a subset of [ω]ω, we shall say that S is Ramsey if there is some infinite subset X of ω such that either [Χ]ω ⊆ S or [Χ]ω ∩ S = 0. Dana Scott (unpublished) has asked which sets, in terms of logical complexity, are Ramsey.The principal theorem of this paper is: Every Σ11 (i.e., analytic) subset of [ω]ω is Ramsey (for the Σ, Π notations, see Addison [1]). This improves a result of Galvin-Prikry [2] to the effect that every Borel set is Ramsey. Our theorem is essentially optimal because, if the axiom of constructibility is true, then Gödel's Σ21 Π21 well-ordering of the set of reals [3], having the convenient property that the set of ω-sequences of reals enumerating initial segments is also Σ21 ∩ Π21, rather directly gives a Σ21 ∩ Π21 set which is not Ramsey. On the other hand, from the assumption that there is a measurable cardinal we shall derive the conclusion that every Σ21 (i.e., PCA) is Ramsey. Also, we shall explore the connection between Martin's axiom and the Ramsey property.