Is there a term that designates a "straight mapping" function? Suppose I have two sets $A = \{A_1, A_2, A_3, A_4\}$, and $B = \{1,2,3,4\}$. I can create a bijection mapping between $A$ and $B$, but is there a term that implies that the mapping itself doesn't matter? For example, I can map $A_1 \mapsto 1$, $A_2 \mapsto 2$, and so on. However, I can also do $A_1 \mapsto 2$, $A_2 \mapsto 1$, ... etc. As long as the bijective property is satisfied, this seems valid. Though, I want to know if there is a term for saying whether how the mapping is defined matters or not. I'm self-defining "straight mapping" as a mapping that goes directly from left to right if we pictorially represent $A$ and $B$ (i.e., imagine we draw $A$ and $B$ as circles with each member of the set vertically and we draw an arrow from each element in $A$ to each element in $B$). I'll also define a "cross mapping" as a bijection that is not a straight mapping.
In essence, are there terms for this sort of thing or this property of a function? I haven't taken an abstract algebra class, so I wouldn't be sure one way or the other.
 A: Let me expand on the comments, which give the right approach. When you write $A = \{A_1, A_2, A_3, A_4\}$, you implicitly define a linear order (also called total order) on the set $A$, namely $A_1 < A_2 < A_3 < A_4$. You also implicitly define a linear order when you "pictorially represent $A$ vertically".
What you call a straight mapping is called an order-preserving (or monotone) map in mathematics.
Now your question raises an interesting issue. The cardinality of a finite set is just the number of its elements, but this notion can be extended to infinite sets as well. In particular, two sets $A$ and $B$ have the same cardinality if there exists a bijection from $A$ to $B$. When you want to prove that two sets have the same cardinality, finding a bijection is enough and the choice of the bijection is irrelevant. For instance, one can show that $\Bbb N$ and $\Bbb Z$ have the same cardinality.
What happens now if you consider linearly ordered sets instead of sets?
Two linearly ordered sets $(A,<)$ and $(B,<)$ are said to have the same order type if there is an order preserving bijection from $A$ to $B$.
This makes a huge difference with the definition of cardinals.
For instance, $(\Bbb N, <)$ and $(\Bbb Z, <)$ do not have the same order type, as there is no order-preserving bijection between them.
At the risk of confusing you even more, let me add a last remark. If two linearly ordered sets have the order type, then they have the same cardinal. Moreover, if two finite linearly ordered sets $(A,<)$ and $(B,<)$ have the same cardinal, then they have the same the order type. In other words, if there is a bijection from $A$ to $B$, then there is always an order-preserving bijection between them (a non-trivial fact). But this is not true if $A$ and $B$ are infinite, like $(\Bbb N, <)$ and $(\Bbb Z, <)$.
