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{\Large Department of Mathematics, BGU}

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{\Huge Logic, Set Theory and Topology}\\[0.2\baselineskip]

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\textbf{On} \emph{Tuesday, November 24, 2015}
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\textbf{At} \emph{12:15 -- 13:40}
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\textbf{In} \emph{Math -101}

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{\large\scshape Mati Rubin 
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  (Ben-Gurion University of the Negev)
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will talk about
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{\Large\bfseries On the reconstruction of the action of a clone from its algebraic structure\par}
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\textsc{Abstract:}
Yonah Maissel and Matatyahu Rubin
Ben Gurion University, Beer Sheva, Israel
Ralph McKenzie proved that if \$G\$ is a group of permutations of
a set \$A\$ with cardinality different from 6 and 1, then the action of
\$G\$ on \$A\$ can be recovered from the group \$G\$ using first order
formulas.
The analogous problems for semigroups of functions from a
set \$A\$ to itself and for clones on \$A\$ have not been
considered (so it seems).
I shall present four analogues of McKenzie's theorem.
Here is one of them.
Theorem 1: Let \$A\$ be a set whose cardinality is different
from 6 and 1, and let \$S\$ be a semigroup of functions from
\$A\$ to \$A\$ containing all transpositions of \$A\$.
Then the action of \$S\$ on \$A\$ can be recovered from the
algebraic structure of the semigroup \$S\$ using first order formulas.
A function \$f\$ from \$A\$ to \$A\$ is called a semi-transposition,
if there are distinct \$a,b \textbackslash{}in A\$ such that \$f(a) = b\$,
and for every \$c \textbackslash{}in A\$: if \$c \textbackslash{}neq a\$, then \$f(c)=c\$.
Theorem 2: Let \$A\$ be a set whose cardinality different from 1,
and let \$S\$ be a semigroup of functions from \$A\$ to \$A\$
containing all semi-transpositions of \$A\$.
Then the action of \$S\$ on \$A\$ can be recovered from the
algebraic structure of the semigroup \$S\$ using first order formulas.
Theorem 3: The analogues of Theorems 1 and 2 for clones are
also true.
I shall present several open questions both for
semigroups of functions and for clones.








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