There are lots of places where we encounter one-way operations where there is a defined way "forward" but no way to really go "backwards". I can think of a number of examples but the so called "arrow of time" is what really got me thinking about this. Another example I have been thinking about is in cellular automata (such as the game of Life) where for a given state there may be any number of possible "seeds" or maybe none at all so one can not really move "backwards".
I was thinking that surely the abstraction of this idea must have been studied. Structures where we can move "forward" but an inverse is not possible. I can't seem to find a formal discussion of this. Can someone please give me a push in the the "forward" direction here?
I am thinking (roughly) about a set $S$ that has an ordering $(a \leq b)$ and some operation $a * b = c$ such that $a \leq c$ and $b \leq c$ but without a corresponding inverse. As well as a universe $U$ such that $S$ is a subset of $U$ but there exists element $d$ in $U$ but not $S$ such that $d \leq x$ and $x \leq d$ is undefined for all $x$ in $S$ but $x * d = e$ is defined in $U$.
What areas of mathematics deals with such sets?
I am so lost I can't even think of a good tag for the question!