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{\Large המחלקה למתמטיקה, בן-גוריון}

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{\Huge לוגיקה, תורת הקבוצות וטופולוגיה}\\[0.2\baselineskip]

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\textbf{ב}\emph{יום שלישי, 22 בדצמבר, 2015}
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\textbf{בשעה} \emph{12:15 -- 13:40}
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\textbf{ב}\emph{Math -101}

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ההרצאה

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{\Large\bfseries On the reconstruction of the action of a clone from its algebraic structure\par}
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תינתן על-ידי
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{\large\scshape Mati Rubin 
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  (Ben-Gurion University of the Negev)
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\textbf{תקציר:}
  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)\$ equals \$b\$, and for every \$c\$ 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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