Algebraic structures for compositions of subsets of the unit square, Representation Theory and Congruences

In this thesis, we first define a certain magma. This magma is an attempt to pass
into mathematical form some aspects of human ‘pictorial (and geometric) thinking’
(the elements are mathematical versions of ‘pictures’). We then generalise this
magma in a natural way to ‘higher dimensions’. And then we study aspects of the
representation theory and structure of these magmas. In particular we investigate
associative quotients, by a variety of means. And also sub-magmas that pass closer
to some classical mathematical structures such as braid groups.
Our core definition is for the magma itself: 3.1.20. But while everything depends
on this, it is relatively straightforward. Our main results are 4.3.19, 5.4.22 and 6.2.2.