How can we determine associativity of a binary structure from its Cayley table?

Question

Suppose $S$ is a finite set with a binary operation $*$ given by a Cayley table. While the commutativity of $*$ can be determined on the basis of the symmetry of the table across the upper-left to lower-right diagonal, is there any way we can, by inspecting the table alone, decide if $*$ is associative or not?

score 3 · Accepted Answer · answered Oct 02 '13 at 01:58

Since the table is a two by two and associativity involves three elements, it is not as easy to check from the Cayley table. You could brute force it by checking every product of three elements. There is Light's associativity test: http://en.wikipedia.org/wiki/Light%27s_associativity_test

score 1 · Answer 2 · answered Oct 02 '13 at 01:57

1

Wikipedia says

It is not generally possible to determine whether or not an operation is associative simply by glancing at its Cayley table. [...] However, Light's associativity test can determine associativity with less effort than brute force.

answered Oct 02 '13 at 01:57

lhf

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How can we determine associativity of a binary structure from its Cayley table?

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