Reconstruction theorems for semigroups of functions which contain all transpositions of a set and for clones with the same property

TL;DR

This paper proves that the structure of certain semigroups and clones containing all transpositions can be uniquely reconstructed from their algebraic properties, highlighting their algebraic rigidity.

Contribution

It establishes that semigroups and clones containing all transpositions are uniquely determined by their composition structure.

Findings

Reconstruction of semigroup actions from algebraic structure

Analogous reconstruction for clones

Structural rigidity of transposition-containing sets

Abstract

We prove that if S is a set of functions from a set A to itself, S is closed under composition, and S contains all transpositions of A, then the action of S on Acan be recovered from the semigroup consisting of S together with its compositionoperation. We also prove the analogous statement for clones.