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=Fermat prime=
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]].


====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 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}}