Let us investigate the powers of $<$:
- ${<^1} = \{(0,1);(0,2);(0,3);...;(1;2);...\}$
- ${<^2} = \{(0,2);(0,3);(0,4);...;(1;3);...\}$
- ${<^3} = \{(0,3);(0,4);(0,5);...;(1;4);...\}$
- ...
- ${<^N} = \{(0,N);(0,N+1);(0,N+2);...;(1;N+1);...\}$
- ${<^0} = \{(0,0);(0,1);(0,2);...;(1;1);...\} = {\le}$
- $<^{-1} = \{(0,-1);(0,0);(0,1);...;(1;0);...\}\ne {>}$
But why?
You got it wrong. So, you're looking at a relation $<\,\, \in \mathbb{N}_0 \times \mathbb{N}_0$. By definition,
$$<^n \,\,:= \{(a_0, a_n) \mid \exists a_0, a_1, \ldots, a_n \in \mathbb{N}_0 : a_0 < a_1 < \ldots < a_n \}$$
for positive $n$, but
$$<^0 \,\,:= \{(a, a) \mid a \in \mathbb{N_0}\} = \,\,\,=$$
and
$$<^{-1} \,\,:= \{(b, a) \mid (a, b) \in \,\,<\} = \{(b, a) \mid a < b\} = \{(b, a) \mid b > a\} =\,\,>.$$
Note that these are entirely different definitions than positive powers of relations. So you can't just extrapolate for negative numbers.