I read this thread here the other day and I also read about Cayley tables that day.
As I understood it every column and every row in the Cayley table of the group will contain each element exactly once.
Then, later, this answer was posted and it suggests to fill the first row with $1$s and $3$s. But this would mean that we can have an element appear more than once in a Cayley table.
Please could someone help me?
The question asked in the linked table specifically asks for the Cayley table for a set which is NOT a group, and the term is obviously used in the sense of operation table. Cayley tables are normally associated with groups and hence an element can appear only once in any row (or column). Groups have (unique) inverses but in the linked question one is asked about a Cayley table for $S=\{1,2,3\}$ where $1 \ast x =2$ has no solution (thus 2 has no inverse). While one may object to the use of the term Cayley table, people have considered generalizations of the concept to semigroups and other algebraic objects. See, for example, this research monograph