Changes

It always helps to begin by recalling the pertinent definitions.

It always helps to begin by recalling the pertinent definitions.

For a 2-adic relation ''L'' ⊆ ''X'' × ''Y'', we have:

For a 2-adic relation ''L'' ⊆ ''X'' × ''Y'', we have:

: ''L'' is a "function" ''L'' : ''X'' ← ''Y'' if and only if ''L'' is 1-regular at ''Y''.

: ''L'' is a "function" ''L'' : ''X'' ← ''Y'' if and only if ''L'' is 1-regular at ''Y''.

As for the definition of relational composition, it is enough to consider the coefficient of the composite on an arbitrary ordered pair like ''i'':''j''.

As for the definition of relational composition, it is enough to consider the coefficient of the composite on an arbitrary ordered pair like ''i'':''j''.

: (''P'' o ''Q'')_''ij'' = ∑_''k'' (''P''_''ik'' ''Q''_''kj'').

: (''P'' o ''Q'')_''ij'' = ∑_''k'' (''P''_''ik'' ''Q''_''kj'').

So let us begin.

So let us begin.

: ''P'' : ''X'' ← ''Y'', or ''P'' being 1-regular at ''Y'', means that there is exactly one ordered pair ''i'':''k'' in ''P'' for each ''k'' in ''Y''.

: ''P'' : ''X'' ← ''Y'', or ''P'' being 1-regular at ''Y'', means that there is exactly one ordered pair ''i'':''k'' in ''P'' for each ''k'' in ''Y''.

: ''Q'' : ''Y'' ← ''Z'', or ''Q'' being 1-regular at ''Z'', means that there is exactly one ordered pair ''k'':''j'' in ''Q'' for each ''j'' in ''Z''.

: ''Q'' : ''Y'' ← ''Z'', or ''Q'' being 1-regular at ''Z'', means that there is exactly one ordered pair ''k'':''j'' in ''Q'' for each ''j'' in ''Z''.

Thus, there is exactly one ordered pair ''i'':''j'' in ''P'' o ''Q'' for each ''j'' in ''Z'', which means that ''P'' o ''Q'' is 1-regular at ''Z'', and so we have the function ''P'' o ''Q'' : ''X'' ← ''Z''.

Thus, there is exactly one ordered pair ''i'':''j'' in ''P'' o ''Q'' for each ''j'' in ''Z'', which means that ''P'' o ''Q'' is 1-regular at ''Z'', and so we have the function ''P'' o ''Q'' : ''X'' ← ''Z''.