Mati Rubin: Transpositions, semi-transposition and clones

Set Theory and Topology seminar (BGU)

 Time: Tuesday, November 3, 12:15-13:40.

Place: Seminar room -101, Math building 58.
Speaker: Mati Rubin (BGU).
Title: On the reconstruction of the action of a clone from its algebraic structure

OR
Transpositions, semi-transposition and clones.
Yonah Maissel and Matatyahu Rubin
Ben Gurion University, Beer Sheva, Israel
Abstract: 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 $a,b \in A$ such that $f(a) = b$,
and for every $c \in A$: if $c \not= a$, 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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