Among the variety of regularities affecting dyadic relations we pay special attention to the 
Let 
![\begin{array}{lll} P ~\text{is total at}~ X & \iff & P ~\text{is}~ (\ge 1)\text{-regular at}~ X. \\[6pt] P ~\text{is total at}~ Y & \iff & P ~\text{is}~ (\ge 1)\text{-regular at}~ Y. \\[6pt] P ~\text{is tubular at}~ X & \iff & P ~\text{is}~ (\le 1)\text{-regular at}~ X. \\[6pt] P ~\text{is tubular at}~ Y & \iff & P ~\text{is}~ (\le 1)\text{-regular at}~ Y. \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++P+%7E%5Ctext%7Bis+total+at%7D%7E+X+%26+%5Ciff+%26+P+%7E%5Ctext%7Bis%7D%7E+%28%5Cge+1%29%5Ctext%7B-regular+at%7D%7E+X.++%5C%5C%5B6pt%5D++P+%7E%5Ctext%7Bis+total+at%7D%7E+Y+%26+%5Ciff+%26+P+%7E%5Ctext%7Bis%7D%7E+%28%5Cge+1%29%5Ctext%7B-regular+at%7D%7E+Y.++%5C%5C%5B6pt%5D++P+%7E%5Ctext%7Bis+tubular+at%7D%7E+X+%26+%5Ciff+%26+P+%7E%5Ctext%7Bis%7D%7E+%28%5Cle+1%29%5Ctext%7B-regular+at%7D%7E+X.++%5C%5C%5B6pt%5D++P+%7E%5Ctext%7Bis+tubular+at%7D%7E+Y+%26+%5Ciff+%26+P+%7E%5Ctext%7Bis%7D%7E+%28%5Cle+1%29%5Ctext%7B-regular+at%7D%7E+Y.++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002 "\begin{array}{lll} P ~\text{is total at}~ X & \iff & P ~\text{is}~ (\ge 1)\text{-regular at}~ X. \\[6pt] P ~\text{is total at}~ Y & \iff & P ~\text{is}~ (\ge 1)\text{-regular at}~ Y. \\[6pt] P ~\text{is tubular at}~ X & \iff & P ~\text{is}~ (\le 1)\text{-regular at}~ X. \\[6pt] P ~\text{is tubular at}~ Y & \iff & P ~\text{is}~ (\le 1)\text{-regular at}~ Y. \end{array}")
We previously examined dyadic relations that separately exemplified each of these regularity conditions. And we introduced a few bits of terminology and special-purpose notations for working with tubular relations:
![\begin{array}{lll} P ~\text{is a pre-function}~ P : X \rightharpoonup Y & \iff & P ~\text{is tubular at}~ X. \\[6pt] P ~\text{is a pre-function}~ P : X \leftharpoonup Y & \iff & P ~\text{is tubular at}~ Y. \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++P+%7E%5Ctext%7Bis+a+pre-function%7D%7E+P+%3A+X+%5Crightharpoonup+Y+%26+%5Ciff+%26+P+%7E%5Ctext%7Bis+tubular+at%7D%7E+X.++%5C%5C%5B6pt%5D++P+%7E%5Ctext%7Bis+a+pre-function%7D%7E+P+%3A+X+%5Cleftharpoonup+Y+%26+%5Ciff+%26+P+%7E%5Ctext%7Bis+tubular+at%7D%7E+Y.++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002 "\begin{array}{lll} P ~\text{is a pre-function}~ P : X \rightharpoonup Y & \iff & P ~\text{is tubular at}~ X. \\[6pt] P ~\text{is a pre-function}~ P : X \leftharpoonup Y & \iff & P ~\text{is tubular at}~ Y. \end{array}")
We arrive by way of this winding stair at the special stamps of dyadic relations 
If 
To say that a relation 
![\begin{array}{lll} P ~\text{is a function}~ P : X \to Y & \iff & P ~\text{is}~ 1\text{-regular at}~ X. \\[6pt] P ~\text{is a function}~ P : X \leftarrow Y & \iff & P ~\text{is}~ 1\text{-regular at}~ Y. \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++P+%7E%5Ctext%7Bis+a+function%7D%7E+P+%3A+X+%5Cto+Y+%26+%5Ciff+%26+P+%7E%5Ctext%7Bis%7D%7E+1%5Ctext%7B-regular+at%7D%7E+X.++%5C%5C%5B6pt%5D++P+%7E%5Ctext%7Bis+a+function%7D%7E+P+%3A+X+%5Cleftarrow+Y+%26+%5Ciff+%26+P+%7E%5Ctext%7Bis%7D%7E+1%5Ctext%7B-regular+at%7D%7E+Y.++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002 "\begin{array}{lll} P ~\text{is a function}~ P : X \to Y & \iff & P ~\text{is}~ 1\text{-regular at}~ X. \\[6pt] P ~\text{is a function}~ P : X \leftarrow Y & \iff & P ~\text{is}~ 1\text{-regular at}~ Y. \end{array}")
For example, let 
 |
(39) |
We observe that 
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