Clones on three elements preserving a binary relation (Q873741)

A clone on a set \(X\) is a set of finitary operations on \(X\) which contains all projections and which is closed under composition. For finite \(X\), every clone is of the form \(\text{Pol}(R)\), where \(R\) is a set of finitary relations on \(X\), and \(\text{Pol}(R)\) is the set of all operations which preserve all \(r\in R\). In particular, every clone is an intersection of clones which can be represented as sets of operations preserving a single finitary relation on \(X\), i.e. of clones of the form \(\text{Pol}(\{r\})\). In her Master's thesis (Université de Montréal 1992), the author of the present paper described all clones on a three-element set which are of the form \(\text{Pol}(\{r\})\) for a binary relation \(r\). There are 266 such clones. This paper is essentially a summary of the thesis.