On the set of all unique elements

[I.V. Serov]

Consider, as a metaphysical example, the set of all unique "days", where 
each day (such as today) has both a unique predecessor (such as 
yesterday for today) and a unique successor (such as tomorrow for today) 
in the set, so that the days are naturally ordered one after another, 
and there is no first day, and all days are elements of the set.

Substitute, if you wish, the word "day" with "thing", "event", "state", 
"thought" or "element".

Is there a known formal mathematical object/structure/logic that seeks 
to describe this set?

I.V. Serov