In ordinal arithmetic?, the ordinal sum is a natural addition on ordered sets and so provides a useful tool when manipulating simplicial sets. In ordinal sum you ‘first put one of the two ordinals and then the other’, so that the elements of the second ordinal are all bigger than those in the first one.
so that then
is the singleton set, as usual and
and so on, so that
This counting is off by one compared to the cardinality of these sets.
The monoidal structure on that we are interested in now is, at the level of the sets, just the disjoint union, but we have to consider the order on that union. If we have and , we form the union of the two sets, where we know the order on two elements we keep it, but if we have two elements one, , say, from the and the other, , from we put .
As an example, consider and , where the overlines are just so that we can keep track of where the different elements come from. We form the union of the two sets and the rule says that anything without an overline is less than anything with one. This gives a linear order
We can thus think of the operation as the addition of cardinalities, but must remember that has elements. In terms of the counting ‘off-by-one’, this reads
but remember there is also the order to keep track of.