Details
| Original language | English |
|---|---|
| Pages (from-to) | 572-586 |
| Number of pages | 15 |
| Journal | Mathematical logic quarterly |
| Volume | 55 |
| Issue number | 6 |
| Publication status | Published - 17 Nov 2009 |
Abstract
We compare diverse degrees of compactness and finiteness in Boolean algebras with each other and investigate the influence of weak choice principles. Our arguments rely on a discussion of infinitary distributive laws and generalized prime elements in Boolean algebras. In ZF set theory without choice, a Boolean algebra is Dedekind finite if and only if it satisfies the ascending chain condition. The Denumerable Subset Axiom (DS) implies finiteness of Boolean algebras with compact top, whereas the converse fails in ZF. Moreover, we derive from DS the atomicity of continuous Boolean algebras. Some of the results extend to more general structures like pseudocomplemented semilattices.
Keywords
- ℳ-distributive, ℳ-prime, Atomistic lattice, Boolean algebra, Chain condition, Compact, Continuous lattice, Dedekind finite, Pseudocomplemented
ASJC Scopus subject areas
- Mathematics(all)
- Logic
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In: Mathematical logic quarterly, Vol. 55, No. 6, 17.11.2009, p. 572-586.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Finiteness conditions and distributive laws for Boolean algebras
AU - Erné, Marcel
PY - 2009/11/17
Y1 - 2009/11/17
N2 - We compare diverse degrees of compactness and finiteness in Boolean algebras with each other and investigate the influence of weak choice principles. Our arguments rely on a discussion of infinitary distributive laws and generalized prime elements in Boolean algebras. In ZF set theory without choice, a Boolean algebra is Dedekind finite if and only if it satisfies the ascending chain condition. The Denumerable Subset Axiom (DS) implies finiteness of Boolean algebras with compact top, whereas the converse fails in ZF. Moreover, we derive from DS the atomicity of continuous Boolean algebras. Some of the results extend to more general structures like pseudocomplemented semilattices.
AB - We compare diverse degrees of compactness and finiteness in Boolean algebras with each other and investigate the influence of weak choice principles. Our arguments rely on a discussion of infinitary distributive laws and generalized prime elements in Boolean algebras. In ZF set theory without choice, a Boolean algebra is Dedekind finite if and only if it satisfies the ascending chain condition. The Denumerable Subset Axiom (DS) implies finiteness of Boolean algebras with compact top, whereas the converse fails in ZF. Moreover, we derive from DS the atomicity of continuous Boolean algebras. Some of the results extend to more general structures like pseudocomplemented semilattices.
KW - ℳ-distributive
KW - ℳ-prime
KW - Atomistic lattice
KW - Boolean algebra
KW - Chain condition
KW - Compact
KW - Continuous lattice
KW - Dedekind finite
KW - Pseudocomplemented
UR - http://www.scopus.com/inward/record.url?scp=76749097194&partnerID=8YFLogxK
U2 - 10.1002/malq.200810034
DO - 10.1002/malq.200810034
M3 - Article
AN - SCOPUS:76749097194
VL - 55
SP - 572
EP - 586
JO - Mathematical logic quarterly
JF - Mathematical logic quarterly
SN - 0942-5616
IS - 6
ER -