I have an algebraic structure $(S,\cdot)$ and $a,b,c,d \in S$ where $a,b,c,d$ are not necessarily four distinct elements. This is part of a larger problem that I am working on and based on what I know, the following is the operation table or "Cayley table" (quotes because this isn't necessarily a group):
$$\begin{array}{c|cccc}
\cdot & a & b & c & d \\\hline
a & a & & c & \\
b & & b & & d \\
c & & c & & a \\
d & d & & b & \\
\end{array}
$$
The table is made such that $d \cdot c = b$.
The minimum assumption I want to make is that one of associativity, because I can't even manipulate equations without it.
My primary question is: Is it possible to fill in any more spaces on this binary operation table with the information given and assuming associativity only?
If not, what (minimum number of) assumptions would I have to make to fill up the table completely?
Best Answer
Computer search finds these three completions and no others:
$$\begin{array}{c|cccc} \cdot & a & b & c & d \\\hline a & a & c & c & a \\ b & d & b & b & d \\ c & a & c & c & a \\ d & d & b & b & d \\ \end{array} $$
$$\begin{array}{c|cccc} \cdot & a & b & c & d \\\hline a & a & a & c & c \\ b & b & b & d & d \\ c & c & c & a & a \\ d & d & d & b & b \\ \end{array} $$
$$\begin{array}{c|cccc} \cdot & a & b & c & d \\\hline a & a & b & c & d \\ b & a & b & c & d \\ c & d & c & b & a \\ d & d & c & b & a \\ \end{array} $$