# User:Richard Pinch/sandbox-10

## Contents

# Fermat prime

A prime number of the form $F_k = 2^{2^k}+1$ for a natural number $k$. They are named after 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.

#### References

- 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.

#### 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

# Isoptic

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.

#### 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 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* is $\beta$.

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

#### 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

# 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 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 if and only if it is right separated.

# 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.

# 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

**How to Cite This Entry:**

Richard Pinch/sandbox-10.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-10&oldid=42624