Clubs: A Student-Presented Mathematics Club Program—Infinities of Numbers

1981 ◽  
Vol 74 (9) ◽  
pp. 711-712
Author(s):  
Paul M. Nemecek
Keyword(s):  
Cardinal Number ◽  
Mathematical Discussion ◽  
One To One ◽  
Infinite Set

There is widespread use of the words infinite or infinitely many by students; yet there seldom occurs a sound mathematical discussion of the topic. The proof of the existence of infinity, accomplished in the last hundred years, allows an important opportunity to discuss the concepts of number, cardinal number, infinite set, and one-to-one correspondence.

Proceed with care, because some infinities are larger than others

2020 ◽  
pp. 281-292
Author(s):  
Susan D'Agostino
Keyword(s):  
Number Line ◽  
Mathematics Students ◽  
Natural Numbers ◽  
Real Numbers ◽  
Countable Infinity ◽  
One To One ◽  
Infinite Set

“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.

A Note on z-Ideals and z°-Ideals of the Formal Power Series Rings and Polynomial Rings in an Infinite Set of Indeterminates

2020 ◽  
Vol 27 (03) ◽  
pp. 495-508
Author(s):  
Ahmed Maatallah ◽  
Ali Benhissi
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Cardinal Number ◽  
Prime Ideals ◽  
Infinite Cardinal ◽  
Series Ring ◽  
Formal Power Series Ring ◽  
Power Series Rings ◽  
Formal Power ◽  
Infinite Set

Let A be a ring. In this paper we generalize some results introduced by Aliabad and Mohamadian. We give a relation between the z-ideals of A and those of the formal power series rings in an infinite set of indeterminates over A. Consider A[[XΛ]]3 and its subrings A[[XΛ]]1, A[[XΛ]]2, and A[[XΛ]]α, where α is an infinite cardinal number. In fact, a z-ideal of the rings defined above is of the form I + (XΛ)i, where i = 1, 2, 3 or an infinite cardinal number and I is a z-ideal of A. In addition, we prove that the same condition given by Aliabad and Mohamadian can be used to get a relation between the minimal prime ideals of the ring of the formal power series in an infinite set of indeterminates and those of the ring of coefficients. As a natural result, we get a relation between the z°-ideals of the formal power series ring in an infinite set of indeterminates and those of the ring of coefficients.

scholarly journals The ideal structure of nilpotent-generated transformation semigroups

1999 ◽  
Vol 60 (2) ◽  
pp. 303-318 ◽  
Author(s):  
M. Paula O. Marques-Smith ◽  
R.P. Sullivan
Keyword(s):  
Green’S Relations ◽  
Ideal Structure ◽  
Transformation Semigroups ◽  
Green's Relations ◽  
Algebraic Properties ◽  
Rees Factor ◽  
Free Semigroups ◽  
One To One ◽  
Infinite Set ◽  
The Ideal

In 1987 Sullivan determined the elements of the semigroup N(X) generated by all nilpotent partial transformations of an infinite set X; and later in 1997 he studied subsemigroups of N(X) defined by restricting the index of the nilpotents and the cardinality of the set. Here, we describe the ideals and Green's relations on such semigroups, like Reynolds and Sullivan did in 1985 for the semigroup generated by all idempotent total transformations of X. We then use this information to describe the congruences on certain Rees factor semigroups and to construct families of congruence-free semigroups with interesting algebraic properties. We also study analogous questions for X finite and for one-to-one partial transformations.

scholarly journals The semigroup of one-to-one transformations with finite defects

1989 ◽  
Vol 31 (2) ◽  
pp. 243-249 ◽  
Author(s):  
Inessa Levi ◽  
Boris M. Schein
Keyword(s):  
Symmetric Group ◽  
Disjoint Union ◽  
One To One ◽  
Infinite Set ◽  
Number Of Authors

Let be the semigroup of all total one-to-one transformations of an infinite set X. For an ƒ ∈ let the defect of ƒ def ƒ, be the cardinality of X – R(ƒ), where R(ƒ) = ƒ(X) is the range of ƒ. Then is a disjoint union of the symmetric group x on X, the semigroup S of all transformations in with finite non-zero defects and the semigroup Ā of all transformations in S with infinite defects, such that S U Ā and Ā are ideals of . The properties of x and Ā have been investigated by a number of authors (for the latter it was done via Baer-Levi semigroups, see [2], [3], [5], [6], [7], [8], [9], [10] and note that Ā decomposes into a union of Baer–Levi semigroups). Our aim here is to study the semigroup S. It is not difficult to see that S is left cancellative (we compose functions ƒ, g in S as ƒg(x) = ƒ(g(x)), for x ∈ X) and idempotent-free. All automorphisms of S are inner [4], that is of the form ƒ → hƒhfh-1 ƒ ∈ S, h ∈ x.

scholarly journals Normal semigroups of one-to-one transformations

1991 ◽  
Vol 34 (1) ◽  
pp. 65-76 ◽  
Author(s):  
Inessa Levi
Keyword(s):  
Transformation Semigroup ◽  
One To One ◽  
Infinite Set

Let X be an infinite set and S be a transformation semigroup on X invariant under conjugations by permutations of X. Such S is termed x-normal. In the paper, we describe elements of a x-normal semigroup S of one-to-one transformations.

scholarly journals Group closures of one-to-one transformations

2001 ◽  
Vol 64 (2) ◽  
pp. 177-188 ◽  
Author(s):  
Inessa Levi
Keyword(s):  
Normal Subgroup ◽  
Symmetric Group ◽  
Permutation Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Alternating Group ◽  
One To One ◽  
Necessary And Sufficient ◽  
Infinite Set

For a semigroup S of transformations of an infinite set X let Gs be the group of all the permutations of X that preserve S under conjugation. Fix a permutation group H on X and a transformation f of X, and let 〈f: H〉 = 〈{hfh−1: h ∈ H}〉 be the H-closure of f. We find necessary and sufficient conditions on a one-to-one transformation f and a normal subgroup H of the symmetric group on X to satisfy G〈f:H〉 = H. We also show that if S is a semigroup of one-to-one transformations of X and GS contains the alternating group on X then Aut(S) = Inn(S) ≅ GS.

scholarly journals Injective endomorphisms of 𝒢x-normal semigroups: finite defects

1994 ◽  
Vol 57 (2) ◽  
pp. 261-280
Author(s):  
Inessa Levi
Keyword(s):  
One To One ◽  
Infinite Set ◽  
Semigroup Of Transformations

AbstractA semigroup of transformations of an infinite set X is called 𝒢x-normal if S is invariant under conjugations by permutations of X. In this paper we describe injective endomorphisms of 𝒢x-normal semigroups of total one-to-one transformations f such that the range of f has a finite non-empty complement in X.

scholarly journals On properties of group closures of one-to-one transformations

2005 ◽  
Vol 79 (2) ◽  
pp. 213-229
Author(s):  
Inessa Levi
Keyword(s):  
Congruence Lattice ◽  
Sufficient Conditions ◽  
Distinct Group ◽  
Normal Subgroups ◽  
Free Semigroups ◽  
One To One ◽  
A Chain ◽  
Necessary And Sufficient ◽  
Infinite Set ◽  
Left Simple

AbstractFor a permutation group H on an infinite set X and a transformation f of X, let 〈f: H〉 = 〈{hfh-1:h є; H}〉 be a group closure of f. We find necessary and sufficient conditions for distinct normal subgroups of the symmetric group on X and a one-to-one transformation f of X to generate distinct group closures of f. Amongst these group closures we characterize those that are left simple, left cancellative, idempotent-free semigroups, whose congruence lattice forms a chain and whose congruences are preserved under automorphisms.

Cantor’s theorem

2018 ◽  
Author(s):  
Mary Tiles
Keyword(s):  
Set Theory ◽  
Cardinal Number ◽  
Infinite Cardinality ◽  
Cantor’S Theorem ◽  
Infinite Set

Cantor’s theorem states that the cardinal number (‘size’) of the set of subsets of any set is greater than the cardinal number of the set itself. So once the existence of one infinite set has been proved, sets of ever increasing infinite cardinality can be generated. The philosophical interest of this result lies (1) in the foundational role it played in Cantor’s work, prior to the axiomatization of set theory, (2) in the similarity between its proof and arguments which lead to the set-theoretic paradoxes, and (3) in controversy between intuitionist and classical mathematicians concerning what exactly its proof proves.

The effects of instruction on the stage placement of children in Piaget's seriation experiments

1964 ◽  
Vol 11 (1) ◽  
pp. 4-9 ◽  
Author(s):  
Arthur F. Coxford
Keyword(s):  
Cardinal Number ◽  
Ordinal Number ◽  
Jean Piaget ◽  
Basic Concepts ◽  
One To One

In his book, The Child's Conceplion of Number, Jean Piaget 1 stated that the concept of number has three basic aspects: cardinal number, ordinal number, and unit. He has given criteria for determining when a child understands each of the basic concepts. A child understands cardinal number when he is able' to construct a one-to-one correspondence between two sets of objects and to conserve this corrspondcnce when it is no longer perceptually obvious.

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