Imagine a system on which you can perform various operations. You would like to analyze the behavior of the system, to determine what positions can be reached from what other positions. One of the most important mathematical tools for analyzing this is the notion of an ``invariant,'' a property of the system (often numerical) which does not change under various operations. Invariants can be used to show that one configuration cannot be reached from another. But if you wanted to hear about invariants, you came to the wrong place, because I'm talking instead about a closely related topic: monovariants. A monovariant is a property of the system -- most often a real number, especially an integer, though any sort of ordered set provides a meaningful context for monovariants -- that may vary, but only in one direction. In general, this means that with each operation, the monovariant either always increases or always decreases.
That was pretty vague, so here are a few examples of monovariants in everyday life. Perhaps the simplest is your age. As time passes, you only get older (sadly). Another is the physical concept of entropy -- the second law of thermodynamics states that it is a monovariant. If you're trapped in a room full of fine china and you proceed to drop it on the concrete floor, the number of pieces of china in the room will always increase; this is a monovariant. These quantities can be used to study the attainability of certain configurations, starting from other configurations. For example, because the number of pieces increases, you cannot reassemble a plate from a bunch of shards by repeatedly dropping them on the floor. However, monovariants have much more substantial mathematical uses, as we shall see. I classify these broadly into two categories...