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=Fermat prime=
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=Cantor–Bendixson characteristics=
A [[prime number]] of the form $F_k = 2^{2^k}+1$ for a natural number $k$.  They are named after [[Fermat, Pierre de|Pierre Fermat]] who observed that $F_0,F_1,F_2,F_3,F_4$ are prime and that this sequence "might be indefinitely extended".  To date (2017), no other prime of this form has been found, and it is known, for example, that $F_k$ is composite for $k=5,\ldots,32$.  Lucas has given an efficient test for the primality of $F_k$.  The Fermat primes are precisely those odd primes $p$ for which a ruler-and-compass construction of the regular $p$-gon is possible: see [[Geometric constructions]] and [[Cyclotomic polynomials]].
 
  
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[[Ordinal number]] invariants of a [[Boolean algebra]].
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Let $B$ be a Boolean algebra, and $I(B)$ the [[ideal]] generated by the [[atom]]s.  We have $I(B) = B$ if and only if $B$ is finite.  We recursively define ideals $I_\alpha$ for [[ordinal number]]s $\alpha$, together with homomorphisms $\pi_\alpha$ and algebras $B_\alpha$ with $\pi_\alpha : B \rightarrow B_\alpha$ with kernel $I_\alpha$, as follows: $I_0(B) =\{0\}$; if $\alpha = \beta+1$ then $I_\alpha = \pi_\beta^{-1}(B_\beta)$ and if $\alpha$ is a limit ordinal then $I_\alpha = \cup_{\beta<\alpha} I_\beta$.  There is a least $\alpha$ such that $I_\alpha = I_\gamma$ for all $\gamma > \alpha$.
  
====References====
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If $B$ is a [[superatomic Boolean algebra]] then each $A_\alpha$ is atomic and the sequence $I_\alpha$ stabilises at $\alpha$ with $\alpha=\beta+1$ where $\beta$ is the least ordinal such that $A_\beta$ is finiteThe Cantor–Bendixson characteristics of $A$ are the quantities $(\beta,n)$ where $n$ is the number of atoms in $A_\beta$.  The ''Cantor–Bendixson height'' or ''rank'' is $\beta$.
* Richard K. Guy, ''Unsolved Problems in Number Theory'' 3rd ed. Springer (2004) ISBN 0-387-20860-7 {{ZBL|1058.11001}}
 
* G.H. Hardy; E.M. Wright. "An Introduction to the Theory of Numbers", Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press (2008) [1938] ISBN 978-0-19-921986-5 {{ZBL|1159.11001}}
 
* Michal Krizek, Florian Luca, Lawrence Somer, "17 Lectures on Fermat Numbers: From Number Theory to Geometry" Springer (2001) ISBN 0-387-21850-5 {{ZBL|1010.11002}}
 
 
 
 
 
=Involution semigroup=
 
A [[semigroup]] $(S,{\cdot})$ with an involution $*$, having the properties $(x\cdot y)^* = y^* \cdot x^*$ and $x^{{*}{*}} = x$.
 
 
 
A ''projection'' in an involution semigroup is an element $e$ such that $e\cdot e = e = e^*$.  There is a partial order on projections given by $e \le f$ if $e\cdot f = e$.   
 
 
 
====References====
 
* Ivan Rival (ed.),"Algorithms and Order", Kluwer (1989) ISBN 940107691X {{ZBL|0709.68004}}
 
 
 
=Foulis semigroup=
 
''Baer $*$-semigroup''
 
 
 
A [[Baer semigroup]] with [[Involution semigroup|involution]].
 
 
 
====References====
 
* T.S. Blyth, "Lattices and Ordered Algebraic Structures" Springer (2005) ISBN 1852339055  {{ZBL|1073.06001}}
 
* Ivan Rival (ed.),"Algorithms and Order", Kluwer (1989) ISBN 940107691X {{ZBL|0709.68004}}
 
  
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For countable superatomic Boolean algebras, the Cantor–Bendixson characteristics determine the isomorphism class.
  
=Isoptic=
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For a [[topological space]] $X$, we may analogously define a Cantor–Bendixson rank as follows.  Let $X_0 = X$ and for ordinal $alpha$ let $X_{\alpha+1}$ be the [[derived set]] of $X_\alpha$.  If $\lambda$ is a limit ordinal, let $X_\lambda = \cap_{\alpha<\lambda} X_\alpha$.  The sequence $(X_\alpha)$ is descending and the smallest $\alpha$ such that $X_{\alpha+1} = X_\alpha$ is the Cantor–Bendixson rank of $X$.
The locus of intersections of tangents to a given curve meeting at a fixed angle; when the fixed angle is a right angle, the locus is an '''orthoptic'''.
 
  
The isoptic of a [[parabola]] is a hyperbola; the isoptic of an [[epicycloid]] is an [[epitrochoid]]; the isoptic of a [[hypocycloid]] is a [[hypotrochoid]]; the isoptic of a [[sinusoidal spiral]] is again a sinusoidal spiral; and the isoptic of a [[cycloid]] is again a cycloid.
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An analogous definition can be made for any partial ordered set $(A,{<})$ equipped with a map $f:A\rightarrow A$ such that $f(x) \le x$.  
  
 
====References====
 
====References====
* J.D. Lawrence,   "A catalog of special plane curves" , Dover (1972) ISBN 0-486-60288-5 {{ZBL|0257.50002}}
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* Winfried Just, Martin Weese, "Discovering Modern Set Theory. II: Set-Theoretic Tools for Every Mathematician",  American Mathematical Society (1997) ISBN 0-8218-7208-7 {{ZBL|0887.03036}}
  
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=Superatomic Boolean algebra=
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A [[Boolean algebra]] for which every homomorphic image is atomic.  Equivalently, the [[Stone space]] is [[scattered space|scattered]]: has no dense-in-itself subset.
  
=Nephroid=
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Countable superatomic Boolean algebras are determined up to isomorphism by their [[Cantor–Bendixson characteristics]].
An [[epicycloid]] with parameter $m=2$; an algebraic plane curve with equation
 
$$
 
x= 3r \cos\theta-r\cos\left[3\theta\right] \,,
 
$$
 
$$
 
y= 3r \sin\theta-r\sin\left[3\theta\right] \ .
 
$$
 
  
The nephroid is the [[catacaustic]] of the [[cardioid]] with respect to a cusp, and of a circle with respect to a point at infinity; the [[evolute]] of a nephroid is another nephroid.
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====References====
 +
* Winfried Just, Martin Weese, "Discovering Modern Set Theory. II: Set-Theoretic Tools for Every Mathematician",  American Mathematical Society (1997) ISBN 0-8218-7208-7  {{ZBL|0887.03036}}
 +
* J. Roitman,  "Superatomic Boolean algebras"  J.D. Monk (ed.)  R. Bonnet (ed.) , ''Handbook of Boolean algebras'' , '''1–3''' , North-Holland  (1989)  pp. Chapt. 19; pp. 719–740 {{ZBL|0671.06001}}
  
The '''nephroid of Freeth''' is the [[strophoid]] of a circle with respect to its centre and a point on the circumference.  It has equation
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=S-space=
$$
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A [[topological space]] which is regular Hausdorff hereditarily separable but not hereditarily Lindelöf.  Dually, an '''L-space''' is regular Hausdorff hereditarily Lindelöf but not hereditarily separable.  The question of the existence of S-spaces and L-spaces is connected to the [[Suslin problem]].  A Suslin line is an L-space, and an S-space may be constructed from a Suslin line.  It is know that non-existence of an S-space is consistent with ZFC.
r = a(1 + 2\sin(\theta/2)) \ .
 
$$
 
 
 
====References====
 
* J.D. Lawrence,  "A catalog of special plane curves" , Dover  (1972) ISBN 0-486-60288-5  {{ZBL|0257.50002}}
 

Latest revision as of 12:35, 18 January 2021

Cantor–Bendixson characteristics

Ordinal number invariants of a Boolean algebra. Let $B$ be a Boolean algebra, and $I(B)$ the ideal generated by the atoms. We have $I(B) = B$ if and only if $B$ is finite. We recursively define ideals $I_\alpha$ for ordinal numbers $\alpha$, together with homomorphisms $\pi_\alpha$ and algebras $B_\alpha$ with $\pi_\alpha : B \rightarrow B_\alpha$ with kernel $I_\alpha$, as follows: $I_0(B) =\{0\}$; if $\alpha = \beta+1$ then $I_\alpha = \pi_\beta^{-1}(B_\beta)$ and if $\alpha$ is a limit ordinal then $I_\alpha = \cup_{\beta<\alpha} I_\beta$. There is a least $\alpha$ such that $I_\alpha = I_\gamma$ for all $\gamma > \alpha$.

If $B$ is a superatomic Boolean algebra then each $A_\alpha$ is atomic and the sequence $I_\alpha$ stabilises at $\alpha$ with $\alpha=\beta+1$ where $\beta$ is the least ordinal such that $A_\beta$ is finite. The Cantor–Bendixson characteristics of $A$ are the quantities $(\beta,n)$ where $n$ is the number of atoms in $A_\beta$. The Cantor–Bendixson height or rank is $\beta$.

For countable superatomic Boolean algebras, the Cantor–Bendixson characteristics determine the isomorphism class.

For a topological space $X$, we may analogously define a Cantor–Bendixson rank as follows. Let $X_0 = X$ and for ordinal $alpha$ let $X_{\alpha+1}$ be the derived set of $X_\alpha$. If $\lambda$ is a limit ordinal, let $X_\lambda = \cap_{\alpha<\lambda} X_\alpha$. The sequence $(X_\alpha)$ is descending and the smallest $\alpha$ such that $X_{\alpha+1} = X_\alpha$ is the Cantor–Bendixson rank of $X$.

An analogous definition can be made for any partial ordered set $(A,{<})$ equipped with a map $f:A\rightarrow A$ such that $f(x) \le x$.

References

  • Winfried Just, Martin Weese, "Discovering Modern Set Theory. II: Set-Theoretic Tools for Every Mathematician", American Mathematical Society (1997) ISBN 0-8218-7208-7 Zbl 0887.03036

Superatomic Boolean algebra

A Boolean algebra for which every homomorphic image is atomic. Equivalently, the Stone space is scattered: has no dense-in-itself subset.

Countable superatomic Boolean algebras are determined up to isomorphism by their Cantor–Bendixson characteristics.

References

  • Winfried Just, Martin Weese, "Discovering Modern Set Theory. II: Set-Theoretic Tools for Every Mathematician", American Mathematical Society (1997) ISBN 0-8218-7208-7 Zbl 0887.03036
  • J. Roitman, "Superatomic Boolean algebras" J.D. Monk (ed.) R. Bonnet (ed.) , Handbook of Boolean algebras , 1–3 , North-Holland (1989) pp. Chapt. 19; pp. 719–740 Zbl 0671.06001

S-space

A topological space which is regular Hausdorff hereditarily separable but not hereditarily Lindelöf. Dually, an L-space is regular Hausdorff hereditarily Lindelöf but not hereditarily separable. The question of the existence of S-spaces and L-spaces is connected to the Suslin problem. A Suslin line is an L-space, and an S-space may be constructed from a Suslin line. It is know that non-existence of an S-space is consistent with ZFC.

How to Cite This Entry:
Richard Pinch/sandbox-10. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-10&oldid=42505