Set Of Finite Subsets Research Articles

The cardinality of the partitions of a set in the absence of the Axiom of Choice

Abstract In the Zermelo–Fraenkel set theory (ZF), $|\textrm {fin}(A)|<2^{|A|}\leq |\textrm {Part}(A)|$ for any infinite set $A$, where $\textrm {fin}(A)$ is the set of finite subsets of $A$, $2^{|A|}$ is the cardinality of the power set of $A$ and $\textrm {Part}(A)$ is the set of partitions of $A$. In this paper, we show in ZF that $|\textrm {fin}(A)|<|\textrm {Part}_{\textrm {fin}}(A)|$ for any set $A$ with $|A|\geq 5$, where $\textrm {Part}_{\textrm {fin}}(A)$ is the set of partitions of $A$ whose members are finite. We also show that, without the Axiom of Choice, any relationship between $|\textrm {Part}_{\textrm {fin}}(A)|$ and $2^{|A|}$ for an arbitrary infinite set $A$ cannot be concluded.

Relations between cardinalities of the finite sequences and the finite subsets of a set

AbstractWe write and for the cardinalities of the set of finite sequences and the set of finite subsets, respectively, of a set which is of cardinality . With the axiom of choice (), for every infinite cardinal but, without , any relationship between and for an arbitrary infinite cardinal cannot be proved. In this paper, we give conditions that make and comparable for an infinite cardinal . Among our results, we show that, if we assume the axiom of choice for sets of finite sets, then for every Dedekind‐infinite cardinal and the condition that is Dedekind‐infinite cannot be weakened to weakly Dedekind‐infinite.

The Length of Chains in Modular Algebraic Lattices

We show that, for a large class of countable order types α, a modular algebraic lattice L contains no chain of type α if and only if K(L), the join-semilattice of compact elements of L, contains neither a chain of type α nor a join-subsemilattice isomorphic to [ω] < ω , the set of finite subsets of ω ordered by inclusion. We give a description of the indivisible members of this class. It includes the order types ω * of the chain of negative integers and η of the chain of rational numbers.

On the Finite and Non-finite Generation of Finitary Power Semigroups

The finitary power semigroup of a semigroup S, denoted Pf(S), is the set of finite subsets of S with respect to the usual set multiplication. Semigroups with finitely generated finitary power semigroups are characterised in terms of three other properties. From this statement there are drawn several corollaries. It follows that Pf(S) is not finitely generated if S is infinite and in any of the following classes: commutative; Bruck-Reilly extensions; inverse semigroups that contain an infinite group; completely zero-simple; completely regular.

The Induced Partial Order on the set of Finite Subsets of a Partially Ordered Set

SYNOPTIC ABSTRACTLet X be a partially ordered set and X be the set of all finite subsets of X. We introduce a partial order on X, which is canonically induced by the partial order on X. We study the interrelations between these two orders and find out a few properties shared by both.

On a certain class of group presentations

In [8] and [9] S. J. Pride has initiated a study of group presentations in which each defining relator involves exactly two members of the generating set. The methods there involve the use of graphs and so-called edge groups – the building blocks of such presentations. In this paper we replace ‘graph’ by ‘set of finite subsets of a given set’, and ‘edge group’ by ‘face group’ in order to study a larger class of presentations. This way we are able to extend to this larger class a Freiheitssatz and a result on diagrammatic asphericity which appear in the references cited above.

Polynomial rings and weak second-order logic

This article is a rewriting of my Ph.D. Thesis, supervised by Professor G. Sabbagh, and incorporates a suggestion from Professor B. Poizat. My main result can be crudely summarized (but see below for detailed statements) by the equality: first-order theory of F[Xi]i∈I = weak second-order theory of F.§I.1. Conventions. The letter F will always denote a commutative field, and I a nonempty set. A field or a ring (A; +, ·) will often be written A for short. We shall use symbols which are definable in all our models, and in the structure of natural numbers (N; +, ·):— the constant 0, defined by the formula Z(x): ∀y (x + y = y);— the constant 1, defined by the formula U(x): ∀y (x · y = y);— the operation ∹ x − y = z ↔ x = y + z;— the relation of division: x ∣ y ↔ ∃ z(x · z = y).A domain is a commutative ring with unity and without any zero divisor.By “… → …” we mean “… is definable in …, uniformly in any model M of L”.All our constructions will be uniform, unless otherwise mentioned.§I.2. Weak second-order models and languages. First of all, we have to define the models Pf(M), Sf(M), Sf′(M) and HF(M) associated to a model M = {A; ℐ) of a first-order language L [CK, pp. 18–20]. Let L1 be the extension of L obtained by adjunction of a second list of variables (denoted by capital letters), and of a membership symbol ∈. Pf(M) is the model (A, Pf(A); ℐ, ∈) of L1, (where Pf(A) is the set of finite subsets of A. Let L2 be the extension of L obtained by adjunction of a second list of variables, a membership symbol ∈, and a concatenation symbol ◠.