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.