Namespaces
Variants
Actions

Difference between revisions of "User:Richard Pinch/sandbox-10"

From Encyclopedia of Mathematics
Jump to: navigation, search
(→‎Fermat prime: move text)
 
(8 intermediate revisions by the same user not shown)
Line 1: Line 1:
=Involution semigroup=
+
=Cantor–Bendixson characteristics=
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====
+
[[Ordinal number]] invariants of a [[Boolean algebra]].
* Ivan Rival (ed.),"Algorithms and Order", Kluwer (1989) ISBN 940107691X {{ZBL|0709.68004}}
+
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 $\gamma > \alpha$.
  
=Foulis semigroup=
+
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 $\beta$.
''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}}
 
  
 +
For countable superatomic Boolean algebras, the Cantor–Bendixson characteristics determine the isomorphism class.
  
=Isoptic=
+
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.
+
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====
 
* J.D. Lawrence,  "A catalog of special plane curves" , Dover  (1972) ISBN 0-486-60288-5  {{ZBL|0257.50002}}
 
 
 
 
 
=Nephroid=
 
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.
 
 
 
The '''nephroid of Freeth''' is the [[strophoid]] of a circle with respect to its centre and a point on the circumference.  It has equation
 
$$
 
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}}
 
 
 
 
 
=Cantor–Bendixson characteristics=
 
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 $\gamma > \alpha$.
 
 
 
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'' is $\beta$.
 
 
 
For countable superatomic Boolean algebras, the Cantor–Bendixson characteristics determine the isomorphism class.
 
  
 
====References====
 
====References====
Line 64: Line 23:
 
* 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}}
 
* 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}}
 
* 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}}
 
=Separated space=
 
A ''separated space'' may refer to
 
* a [[topological space]] satisfying a [[separation axiom]]; in particular a [[Hausdorff space]];
 
* a [[left separated space]] or [[right separated space]].
 
 
=Right separated space=
 
A [[topological space]] $X$ is '''right''' (resp. '''left''') '''separated''' if there is a [[Well-ordered set|well ordering]] ${<}$ on $X$ such that the segments $\{x \in X : x < y\}$ are all open (resp. closed) in the topology of $X$.
 
 
A Hausdorff space is [[scattered space|scattered]] if and only if it is right separated.
 
  
 
=S-space=
 
=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.
 
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.
 
=Myope topology=
 
A topology on the family $\mathcal{K} = \mathcal{K}_X$ of compact subsets of a topological space $X$.  Let $\mathcal{F}$ denote the family of closed sets in $X$ and $\mathcal{G}$ the family of open sets.  A basic open set for the myope topology is a set $U \subset \mathcal{K}$ of the form
 
$$
 
U = \{ A \in K : A \cap F = \emptyset\,\ A \cap G \ne \emptyset \}
 
$$
 
where $F \in \mathcal{F}$ and $G \in \mathcal{G}$.
 
 
====References====
 
* C. van den Berg, J. P. R. Christensen, P. Ressel, "Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions" Graduate Texts in Mathematics '''100''' Soringer (2012) ISBN 146121128X
 

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=51363