# Work φ

## Numerical Incidence Properties

### LaTeX

$\begin{matrix} L & \text{is} & (< \! c) \textit{-regular at}~ j & \text{if and only if} & |L_{x\,@\,j}| < c & \text{for all}~ x \in X_j. \\[4pt] L & \text{is} & c \textit{-regular at}~ j & \text{if and only if} & |L_{x\,@\,j}| = c & \text{for all}~ x \in X_j. \\[4pt] L & \text{is} & (> \! c) \textit{-regular at}~ j & \text{if and only if} & |L_{x\,@\,j}| > c & \text{for all}~ x \in X_j. \end{matrix}$

$\begin{matrix} L & \text{is} & (< \! c) \textit{-regular at}~ m & \text{if and only if} & |L_{x\,@\,m}| < c & \text{for all}~ x \in X_m. \\[4pt] L & \text{is} & c \textit{-regular at}~ m & \text{if and only if} & |L_{x\,@\,m}| = c & \text{for all}~ x \in X_m. \\[4pt] L & \text{is} & (> \! c) \textit{-regular at}~ m & \text{if and only if} & |L_{x\,@\,m}| > c & \text{for all}~ x \in X_m. \end{matrix}$

## Dyadic Relations • Total • Tubular

### LaTeX

$\begin{matrix} L & \text{is} & \textit{total} & \text{at}~ S & \text{if and only if} & L & \text{is} & (\ge 1)\text{-regular} & \text{at}~ S. \\[4pt] L & \text{is} & \textit{total} & \text{at}~ T & \text{if and only if} & L & \text{is} & (\ge 1)\text{-regular} & \text{at}~ T. \\[4pt] L & \text{is} & \textit{tubular} & \text{at}~ S & \text{if and only if} & L & \text{is} & (\le 1)\text{-regular} & \text{at}\ S. \\[4pt] L & \text{is} & \textit{tubular} & \text{at}~ T & \text{if and only if} & L & \text{is} & (\le 1)\text{-regular} & \text{at}~ T. \end{matrix}$

## Dyadic Relations • Surjective, Injective, Bijective

### LaTeX

$\begin{matrix} f & \text{is} & \textit{surjective} & \text{if and only if} & f & \text{is} & \text{total} & \text{at}~ T. \\[4pt] f & \text{is} & \textit{injective} & \text{if and only if} & f & \text{is} & \text{tubular} & \text{at}~ T. \\[4pt] f & \text{is} & \textit{bijective} & \text{if and only if} & f & \text{is} & 1\text{-regular} & \text{at}~ T. \end{matrix}$