| Arrow \(a\) | source \(s(a) \in V\) | target \(t(a) \in V\) |
|---|---|---|
| \(a\) | 1 | ? |
| \(b\) | 1 | 3 |
| \(c\) | ? | ? |
| d | ? | ? |
| \(e\) | ? | ? |
columns: Arrow \(a\) | source \(s(a) \in V\) | target \(t(a) \in V\)
| \(\wedge\) | false | true | * | 0 | 1 |
|---|---|---|---|---|---|
| false | false | false | 0 | 0 | 0 |
| true | false | true | 1 | 0 | 1 |
columns: \(\wedge\) | false | true | * | 0 | 1
| v | false | true |
|---|---|---|
| false | false | true |
| true | true | true |
columns: v | false | true
| max | 0 | 1 |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 1 |
columns: max | 0 | 1
| min | no | maybe | yes |
|---|---|---|---|
| no | ? | ? | ? |
| maybe | ? | ? | ? |
| yes | ? | ? | ? |
columns: min | no | maybe | yes
| \(\leq\). | \(p\) | \(q\) | \(r\) | \(s\) | \(t\) |
|---|---|---|---|---|---|
| \(p\) | true | true | true | true | true |
| \(q\) | false | true | false | true | true |
| \(r\) | false | false | true | true | true |
| \(s\) | false | false | false | true | true |
| \(t\) | false | false | false | false | true |
columns: \(\leq\). | \(p\) | \(q\) | \(r\) | \(s\) | \(t\)
| \(d(\nearrow)\) | \(X\) | \(y\) | z |
|---|---|---|---|
| \(x\) | 0 | 4 | 3 |
| y | 3 | 0 | 6 |
| z | 7 | 4 | 0 |
columns: \(d(\nearrow)\) | \(X\) | \(y\) | z
| \(d(\nearrow)\) | \(A\) | \(B\) | \(C\) | \(D\) |
|---|---|---|---|---|
| \(A\) | 0 | ? | ? | ? |
| B | 2 | ? | 5 | ? |
| C | ? | ? | ? | ? |
| D | ? | ? | ? | ? |
columns: \(d(\nearrow)\) | \(A\) | \(B\) | \(C\) | \(D\)
| ↗ | \(A\) | \(B\) | C | D | |
|---|---|---|---|---|---|
| \(A\) | 0 | ? | ? | ? | |
| \(M_{X}=\) | B | 2 | 0 | \(\infty\) | ? |
| C | ? | ? | ? | ? | |
| D | ? | ? | ? | ? |
columns: | ↗ | \(A\) | \(B\) | C | D
| X | \(A\) | \(B\) | \(C\) |
|---|---|---|---|
| \(A\) | 0 | 2 | 5 |
| B | \(\infty\) | 0 | 3 |
| C | \(\infty\) | \(\infty\) | 0 |
columns: X | \(A\) | \(B\) | \(C\)
| y | \(p\) | \(q\) |
|---|---|---|
| \(p\) | 0 | 5 |
| \(q\) | 8 | 0 |
columns: y | \(p\) | \(q\)
| \(x \times y\) | \((A, p)\) | ( \(B, p\) ) | ( \(C, p\) ) | \((A, q)\) | \((B, q)\) | \((C, q)\) |
|---|---|---|---|---|---|---|
| ( \(A, p\) ) | 0 | 2 | 5 | 5 | 7 | 10 |
| \((B, p)\) | \(\infty\) | 0 | 3 | \(\infty\) | 5 | 8 |
| \((C, p)\) | \(\infty\) | \(\infty\) | 0 | \(\infty\) | \(\infty\) | 5 |
| \((A, q)\) | 8 | 10 | 13 | 0 | 2 | 5 |
| \((B, q)\) | \(\infty\) | 8 | 11 | \(\infty\) | 0 | 3 |
| \((C, q)\) | \(\infty\) | \(\infty\) | 8 | \(\infty\) | \(\infty\) | 0 |
columns: \(x \times y\) | \((A, p)\) | ( \(B, p\) ) | ( \(C, p\) ) | \((A, q)\) | \((B, q)\) | \((C, q)\)
| \(M_{Y}\) | \(x\) | \(y\) | \(z\) |
|---|---|---|---|
| \(x\) | 0 | 4 | 3 |
| \(y\) | 3 | 0 | \(\infty\) |
| \(z\) | \(\infty\) | 4 | 0 |
columns: \(M_{Y}\) | \(x\) | \(y\) | \(z\)
| \(d_{Y}\) | \(x\) | \(y\) | \(z\) |
|---|---|---|---|
| \(x\) | 0 | 4 | 3 |
| \(y\) | 3 | 0 | 6 |
| \(z\) | 7 | 4 | 0 |
columns: \(d_{Y}\) | \(x\) | \(y\) | \(z\)
| \(M_{Y}^{2}=\) | ↗ | \(X\) | \(y\) | \(z\) |
|---|---|---|---|---|
| \(x\) | 0 | 4 | 3 | |
| y | 3 | 0 | 6 | |
| z | 7 | 4 | 0 |
columns: \(M_{Y}^{2}=\) | ↗ | \(X\) | \(y\) | \(z\)
| \(M_{Y}^{3}=\) | ↗ | \(X\) | \(y\) | z |
|---|---|---|---|---|
| \(x\) | 0 | 4 | 3 | |
| y | 3 | 0 | 6 | |
| z | 7 | 4 | 0 |
columns: \(M_{Y}^{3}=\) | ↗ | \(X\) | \(y\) | z
| Employee | FName | WorksIn | Mngr |
|---|---|---|---|
| 1 | Alan | 101 | 2 |
| 2 | Ruth | 101 | 2 |
| 3 | Kris | 102 | 3 |
columns: Employee | FName | WorksIn | Mngr
| Department | DName | Secr |
|---|---|---|
| 101 | Sales | 1 |
| 102 | IT | 3 |
columns: Department | DName | Secr
| Planet of Sol | Prime number | Flying pig |
|---|---|---|
| Mercury | 2 | |
| Venus | 3 | |
| Earth | 5 | |
| Mars | 7 | |
| Jupiter | 11 | |
| Saturn | 13 | |
| Uranus | 17 | |
| Neptune | ⋮ |
columns: Planet of Sol | Prime number | Flying pig
| Beatle | Played |
|---|---|
| George | lead guitar |
| John | rhythm guitar |
| Paul | bass guitar |
| Ringo | drums |
columns: Beatle | Played
| Beatle | Played |
|---|---|
| George | lead guitar |
| John | rhythm guitar |
| Paul | bass guitar |
| Ringo | drums |
columns: Beatle | Played
| Rock-and-roll instrument |
|---|
| bass guitar |
| drums |
| keyboard |
| lead guitar |
| rhythm guitar |
columns: Rock-and-roll instrument
| \(\mathbb{N}_{\geq 2}\) | Smallest prime factor |
|---|---|
| 2 | 2 |
| 3 | 3 |
| 4 | 2 |
| ⋮ | ⋮ |
| 49 | 7 |
| 50 | 2 |
| 51 | 3 |
| ⋮ | ⋮ |
columns: \(\mathbb{N}_{\geq 2}\) | Smallest prime factor
| Arrow | source | target | Vertex |
|---|---|---|---|
| \(a\) | 1 | 2 | 1 |
| \(b\) | 1 | 3 | 2 |
| c | 1 | 3 | 3 |
| \(d\) | 2 | 2 | 4 |
| \(e\) | 2 | 3 |
columns: Arrow | source | target | Vertex
| State | next |
|---|---|
| 1 | 4 |
| 2 | 4 |
| 3 | 5 |
| 4 | 5 |
| 5 | 5 |
| 6 | 7 |
| 7 | 6 |
columns: State | next
| Arrow | source | target | Vertex |
|---|---|---|---|
| 1 | 1 | 4 | 1 |
| 2 | 2 | 4 | 2 |
| 3 | 3 | 5 | 3 |
| 4 | 4 | 5 | 4 |
| 5 | 5 | 5 | 5 |
| 6 | 6 | 7 | 6 |
| 7 | 7 | 6 | 7 |
columns: Arrow | source | target | Vertex
| Migration functor | pronounced | reminiscent of | database idea |
|---|---|---|---|
| \(\Delta\) | delta | duplicate or destroy | duplicate or destroy tables or columns |
| \(\Sigma\) | sigma | sum | union (sum up) data |
| \(\Pi\) | pi | product | pair \({ }^{9}\) and query data |
columns: Migration functor | pronounced | reminiscent of | database idea
| sent_by | received_by | Address | |
|---|---|---|---|
| Em_1 | Bob | Grace | Bob |
| Em_2 | Grace | Pat | Doug |
| Em_3 | Bob | Emmy | Emmy |
| Em_4 | Sue | Doug | Grace |
| Em_5 | Doug | Sue | Pat |
| Em_6 | Bob | Bob | Sue |
columns: Email | sent_by | received_by | Address
| Bool-category | preorder |
|---|---|
| Bool-functor | monotone map |
| Bool-profunctor | feasibility relation |
columns: Bool-category | preorder
| \(c\) | \(d\) | \(c \Rightarrow d\) |
|---|---|---|
| true | true | true |
| true | false | false |
| false | true | true |
| false | false | true |
columns: \(c\) | \(d\) | \(c \Rightarrow d\)
| \(\Phi\) | \(a\) | \(b\) | \(c\) | \(d\) | \(e\) |
|---|---|---|---|---|---|
| \(N\) | ? | ? | ? | ? | true |
| E | true | ? | ? | ? | ? |
| W | ? | ? | ? | false | ? |
| \(S\) | ? | ? | ? | ? | ? |
columns: \(\Phi\) | \(a\) | \(b\) | \(c\) | \(d\) | \(e\)
| \(\Phi\) | \(x\) | \(y\) | \(z\) |
|---|---|---|---|
| \(A\) | ? | ? | 20 |
| B | 11 | ? | ? |
| C | ? | 17 | ? |
| D | ? | ? | ? |
columns: \(\Phi\) | \(x\) | \(y\) | \(z\)
| \(M_{X}\) | \(A\) | \(B\) | C | D |
|---|---|---|---|---|
| \(A\) | 0 | \(\infty\) | 3 | \(\infty\) |
| B | 2 | 0 | \(\infty\) | 5 |
| C | \(\infty\) | 3 | 0 | \(\infty\) |
| D | \(\infty\) | \(\infty\) | 4 | 0 |
columns: \(M_{X}\) | \(A\) | \(B\) | C | D
| \(M_{\Phi}\) | \(x\) | \(y\) | z |
|---|---|---|---|
| \(A\) | \(\infty\) | \(\infty\) | \(\infty\) |
| \(B\) | 11 | \(\infty\) | \(\infty\) |
| C | \(\infty\) | \(\infty\) | \(\infty\) |
| D | \(\infty\) | 9 | \(\infty\) |
columns: \(M_{\Phi}\) | \(x\) | \(y\) | z
| \(M_{Y}\) | \(x\) | \(y\) | \(z\) |
|---|---|---|---|
| \(x\) | 0 | 4 | 3 |
| y | 3 | 0 | \(\infty\) |
| z | \(\infty\) | 4 | 0 |
columns: \(M_{Y}\) | \(x\) | \(y\) | \(z\)
| \(\Phi\) | \(a\) | \(b\) | \(c\) | \(d\) | \(e\) |
|---|---|---|---|---|---|
| \(N\) | true | false | true | false | false |
| \(E\) | true | false | true | false | true |
| W | true | true | true | true | false |
| \(S\) | true | true | true | true | true |
columns: \(\Phi\) | \(a\) | \(b\) | \(c\) | \(d\) | \(e\)
| \(\Psi\) | \(x\) | \(y\) |
|---|---|---|
| \(a\) | false | true |
| \(b\) | true | true |
| \(c\) | false | true |
| d | true | true |
| \(e\) | false | false |
columns: \(\Psi\) | \(x\) | \(y\)
| \(\Phi{ }_{9}^{\circ} \Psi\) | \(x\) | \(y\) |
|---|---|---|
| \(N\) | false | true |
| E | false | true |
| W | true | true |
| S | true | true |
columns: \(\Phi{ }_{9}^{\circ} \Psi\) | \(x\) | \(y\)
| \(\Phi{ }_{9}^{\circ} \Psi\) | \(p\) | \(q\) | \(r\) | \(s\) |
|---|---|---|---|---|
| \(A\) | ? | 24 | ? | ? |
| B | ? | ? | ? | ? |
| C | ? | ? | ? | ? |
| D | ? | ? | 9 | ? |
columns: \(\Phi{ }_{9}^{\circ} \Psi\) | \(p\) | \(q\) | \(r\) | \(s\)
| X | \(A\) | \(B\) | \(\Phi\) | \(\chi\) | \(y\) | \(y\) | \(\chi\) | \(y\) |
|---|---|---|---|---|---|---|---|---|
| \(A\) | 0 | 2 | \(A\) | 5 | 8 | \(x\) | 0 | 3 |
| B | \(\infty\) | 0 | B | \(\infty\) | \(\infty\) | \(y\) | 4 | 0 |
columns: X | \(A\) | \(B\) | \(\Phi\) | \(\chi\) | \(y\) | \(y\) | \(\chi\) | \(y\)
| \(\operatorname{Col}(\Phi)\) | \(A\) | \(B\) | \(x\) | \(y\) |
|---|---|---|---|---|
| \(A\) | 0 | 2 | 5 | 8 |
| \(B\) | \(\infty\) | 0 | \(\infty\) | \(\infty\) |
| \(x\) | 0 | 0 | 0 | 3 |
| \(y\) | 0 | 0 | 4 | 0 |
columns: \(\operatorname{Col}(\Phi)\) | \(A\) | \(B\) | \(x\) | \(y\)
| \(a\) | \(b\) | |||
|---|---|---|---|---|
| \(x\) | 15 | 3 | \(\left(\begin{array}{c}15 \\ 0\end{array}\right.\) | 21 ) |
| \(y\) | 0 | 21 |
columns: | \(a\) | \(b\) | |
| Generator | Icon | Matrix | Arity |
|---|---|---|---|
| amplify by \(a \in R\) | - \(a\) - | (a) | \(1 \rightarrow 1\) |
| add | D- | \(\binom{1}{1}\) | \(2 \rightarrow 1\) |
| zero | ○- | () | \(0 \rightarrow 1\) |
| copy | - C | \(\left(\begin{array}{ll}1 & 1\end{array}\right)\) | \(1 \rightarrow 2\) |
| discard | → | () | \(1 \rightarrow 0\) |
columns: Generator | Icon | Matrix | Arity
| \(P\) | \(Q\) | \(\ulcorner(\) true, true \()\urcorner(P, Q)\) |
|---|---|---|
| true | true | true |
| true | false | false |
| false | true | false |
| false | false | false |
columns: \(P\) | \(Q\) | \(\ulcorner(\) true, true \()\urcorner(P, Q)\)
| \(P\) | \(Q\) | \(P \vee Q\) |
|---|---|---|
| true | true | true |
| true | false | true |
| false | true | true |
| false | false | false |
columns: \(P\) | \(Q\) | \(P \vee Q\)
| \(P\) | \(Q\) | \(P \wedge Q\) | \(P=(P \wedge Q)\) |
|---|---|---|---|
| true | true | ? | ? |
| true | false | ? | ? |
| false | true | ? | ? |
| false | false | ? | ? |
columns: \(P\) | \(Q\) | \(P \wedge Q\) | \(P=(P \wedge Q)\)
| Arrow \(a\) | Source \(s(a) \in V\) | Target \(t(a) \in V\) |
|---|---|---|
| \(a\) | 1 | 2 |
| \(b\) | 1 | 3 |
| \(c\) | 1 | 3 |
| d | 2 | 2 |
| \(e\) | 2 | 3 |
columns: Arrow \(a\) | Source \(s(a) \in V\) | Target \(t(a) \in V\)
| \(p\) | \(q\) | \(f(p)\) | \(g(q)\) | \(f(p) \leq ? \quad q\) | \(p \leq ? g(q)\) | Same? |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 1 | yes | yes | yes! |
| 1 | 2 | 1 | 2 | no | no | yes! |
| 1 | 4 | 1 | 4 | yes | yes | yes! |
| 2 | 1 | 2 | 1 | no | no | yes! |
| 2 | 2 | 2 | 2 | yes | yes | yes! |
| 2 | 4 | 2 | 4 | yes | yes | yes! |
| 3.9 | 1 | 4 | 1 | no | no | yes! |
| 3.9 | 2 | 4 | 2 | no | no | yes! |
| 3.9 | 4 | 4 | 4 | yes | yes | yes! |
| 4 | 1 | 4 | 1 | no | no | yes! |
| 4 | 2 | 4 | 2 | no | no | yes! |
| 4 | 4 | 4 | 4 | yes | yes | yes! |
columns: \(p\) | \(q\) | \(f(p)\) | \(g(q)\) | \(f(p) \leq ? \quad q\) | \(p \leq ? g(q)\) | Same?
| min | no | maybe | yes |
|---|---|---|---|
| no | no | no | no |
| maybe | no | maybe | maybe |
| yes | no | maybe | yes |
columns: min | no | maybe | yes
| \(d(\nearrow)\) | \(A\) | \(B\) | \(C\) | \(D\) |
|---|---|---|---|---|
| A | 0 | 6 | 3 | 11 |
| B | 2 | 0 | 5 | 5 |
| C | 5 | 3 | 0 | 8 |
| D | 11 | 9 | 6 | 0 |
columns: \(d(\nearrow)\) | \(A\) | \(B\) | \(C\) | \(D\)
| ↗ | \(A\) | \(B\) | C | \(D\) | |
|---|---|---|---|---|---|
| \(A\) | 0 | \(\infty\) | 3 | \(\infty\) | |
| \(M_{X}=\) | B | 2 | 0 | \(\infty\) | 5 |
| C | \(\infty\) | 3 | 0 | \(\infty\) | |
| D | \(\infty\) | \(\infty\) | 6 | 0 |
columns: | ↗ | \(A\) | \(B\) | C | \(D\)
| X(↗) | \(A\) | \(B\) | \(C\) | \(D\) |
|---|---|---|---|---|
| A | \(M\) | \{boat\} | \(\varnothing\) | \{boat\} |
| B | \(\varnothing\) | \(M\) | \(\varnothing\) | \{boat\} |
| C | \{foot, boat\} | \{boat\} | \(M\) | \(M\) |
| D | \{foot\} | \(\varnothing\) | \{foot, car\} | \(M\) |
columns: X(↗) | \(A\) | \(B\) | \(C\) | \(D\)
| \(M(\nearrow)\) | \(A\) | \(B\) | \(C\) |
|---|---|---|---|
| \(A\) | \(\infty\) | 10 | 10 |
| B | 6 | \(\infty\) | 6 |
| C | 10 | 10 | \(\infty\) |
columns: \(M(\nearrow)\) | \(A\) | \(B\) | \(C\)
| ⇒ | false | true |
|---|---|---|
| false | true | true |
| true | false | true |
columns: ⇒ | false | true
| ↗ | \(v_{1}\) | \(f_{1}\) | \(f_{1}{ }^{\circ} f_{2}\) | \(v_{2}\) | \(f_{2}\) | \(v_{3}\) |
|---|---|---|---|---|---|---|
| \(v_{1}\) | \(v_{1}\) | \(f_{1}\) | \(f_{1} ; f_{2}\) | ⊠ | ⊠ | ⊠ |
| \(f_{1}\) | ⊠ | ⊠ | ⊠ | \(f_{1}\) | \(f_{1} \stackrel{\circ}{9} f_{2}\) | ⊠ |
| \(f_{1}{ }^{\circ} f_{2}\) | ⊠ | ⊠ | ⊠ | ⊠ | ⊠ | \(f_{1}{ }^{\circ} f_{2}\) |
| \(v_{2}\) | ⊠ | ⊠ | ⊠ | \(v_{2}\) | \(f_{2}\) | ⊠ |
| \(f_{2}\) | ⊠ | ⊠ | ⊠ | ⊠ | ⊠ | \(f_{2}\) |
| \(v_{3}\) | ⊠ | ⊠ | ⊠ | ⊠ | ⊠ | \(v_{3}\) |
columns: ↗ | \(v_{1}\) | \(f_{1}\) | \(f_{1}{ }^{\circ} f_{2}\) | \(v_{2}\) | \(f_{2}\) | \(v_{3}\)
| Arrow | source | target |
|---|---|---|
| Mngr | Employee | Employee |
| WorksIn | Employee | Department |
| Secr | Department | Employee |
| FName | Employee | string |
| DName | Department | string |
columns: Arrow | source | target
| Vertex |
|---|
| Department Employee string |
columns: Vertex
| State | next | |
|---|---|---|
| 1 | 4 | |
|  | 2 | 4 |
| 3 | 5 | |
| 4 | 5 | |
| 5 | 5 | |
| 6 | 7 | |
| 7 | 6 | |
| Gr |
columns: | | State | next
| Arrow | source | target |
|---|---|---|
| 1 | 4 | 1 |
| 2 | 4 | 2 |
| 3 | 5 | 3 |
| 4 | 5 | 4 |
| 5 | 5 | 5 |
| 6 | 7 | 6 |
| 7 | 6 | 7 |
columns: Arrow | source | target
| Vertex |
|---|
| 1 |
| 2 |
| 3 |
| 4 |
| 5 |
| 6 |
| 7 |
columns: Vertex
| sent_by | received_by | Address | |
|---|---|---|---|
| Em_1 | Bob | Grace | Bob |
| Em_2 | Grace | Pat | Doug |
| Em_3 | Bob | Emmy | Emmy |
| Em_4 | Sue | Doug | Grace |
| Em_5 | Doug | Sue | Pat |
| Em_6 | Bob | Bob | Sue |
columns: Email | sent_by | received_by | Address
| \(\Phi\) | \(a\) | \(b\) | \(c\) | \(d\) | \(e\) |
|---|---|---|---|---|---|
| \(N\) | true | false | true | false | true |
| E | true | true | true | true | true |
| W | true | false | true | false | true |
| \(S\) | true | true | true | true | true |
columns: \(\Phi\) | \(a\) | \(b\) | \(c\) | \(d\) | \(e\)
| \(\Phi: \Psi\) | \(p\) | \(q\) | \(r\) | \(s\) |
|---|---|---|---|---|
| \(A\) | 22 | 24 | 20 | 21 |
| B | 16 | 18 | 14 | 15 |
| C | 19 | 21 | 17 | 18 |
| D | 11 | 13 | 9 | 10 |
columns: \(\Phi: \Psi\) | \(p\) | \(q\) | \(r\) | \(s\)
| r1 | r2 | r3 | c1 | i1 | |
|---|---|---|---|---|---|
| \(s(-)\) | dl | ul | ur | ul | dl |
| \(t(-)\) | ul | ur | dr | ur | dr |
| \(\ell(-)\) | \(1 \Omega\) | \(2 \Omega\) | \(1 \Omega\) | \(3 F\) | \(1 H\) |
columns: | r1 | r2 | r3 | c1 | i1
| 1 | 2 | |
|---|---|---|
| \(f^{\prime}(-)\) | l | d |
| \(g^{\prime}(-)\) | r | r |
columns: | 1 | 2
| r1' | r2' | |
|---|---|---|
| \(s(-)\) | l | r |
| \(t(-)\) | r | d |
| \(\ell(-)\) | \(5 \Omega\) | \(8 \Omega\) |
columns: | r1' | r2'
| r1 | r2 | r3 | c1 | i1 | r1' | r2' | |
|---|---|---|---|---|---|---|---|
| \(s^{\prime \prime}(-)\) | dl | ul | m | ul | dl | m | r |
| \(t^{\prime \prime}(-)\) | ul | m | dr | m | dr | r | m |
| \(\ell^{\prime \prime}(-)\) | \(1 \Omega\) | \(2 \Omega\) | \(1 \Omega\) | \(3 F\) | \(1 H\) | \(5 \Omega\) | \(8 \Omega\) |
columns: | r1 | r2 | r3 | c1 | i1 | r1' | r2'
| \(P\) | \(Q\) | \(P \wedge Q\) | \(P=(P \wedge Q)\) |
|---|---|---|---|
| true | true | true | true |
| true | false | false | false |
| false | true | false | true |
| false | false | false | true |
columns: \(P\) | \(Q\) | \(P \wedge Q\) | \(P=(P \wedge Q)\)