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A point-free relation-algebraic approach to general topology Gunther Schmidt Fakult at f ur Informatik, Universit at der Bundeswehr M unchen Gunther.Schmidt@unibw.de May 1, 2014 Contents 1. Motivation my early topology 2.

  1. � � � � � ���� � ��� � Direct power — up to isomorphism � � �

  2. � � � � � ���� � ��� � Direct power — up to isomorphism � � � Any relation ε satisfying I syq ( ε , ε ) ✓ , (i.e., in fact syq ( ε , ε ) = ) I syq ( ε , R ) is surjective for every relation R starting in X . is called a interpreted with 2 -relation direct power P ( X ) DirPow x ε : X � ! P ( X ) Member x

  3. { ♠ , ♥ , ♦ , ♣ } { ♠ , ♥ , ♦ } { ♠ , ♥ , ♣ } { ♠ , ♦ , ♣ } { ♥ , ♦ , ♣ } { ♠ , ♥ } { ♠ , ♦ } { ♥ , ♦ } { ♠ , ♣ } { ♥ , ♣ } { ♦ , ♣ } { ♠ } { ♥ } { ♦ } { ♣ } {} ♠ 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 ♥ 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 ε = B C ♦ @ 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 A ♣ 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

  4. { ♠ , ♥ , ♦ , ♣ } { ♠ , ♥ , ♦ } { ♠ , ♥ , ♣ } { ♠ , ♦ , ♣ } { ♥ , ♦ , ♣ } { ♠ , ♥ } { ♠ , ♦ } { ♥ , ♦ } { ♠ , ♣ } { ♥ , ♣ } { ♦ , ♣ } { ♠ } { ♥ } { ♦ } { ♣ } {} ♠ 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 ♥ 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 ε = B C ♦ @ 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 A ♣ 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 { ♠ , ♥ , ♦ , ♣ } { ♠ , ♦ , ♣ } { ♠ , ♥ , ♦ } { ♠ , ♥ , ♣ } { ♥ , ♦ , ♣ } { ♠ , ♦ } { ♥ , ♦ } { ♥ , ♣ } { ♠ , ♣ } { ♠ , ♥ } { ♦ , ♣ } { ♦ } { ♠ } { ♣ } { ♥ } {} ♠ 1 1 1 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 ♥ 0 1 1 1 0 0 1 0 1 0 0 1 0 0 1 1 ε 0 = B C ♦ 1 1 1 0 1 1 1 0 0 0 0 0 1 0 1 0 @ A ♣ 1 0 1 1 0 0 0 0 1 1 1 0 1 0 1 0

  5. { ♠ , ♥ , ♦ , ♣ } { ♠ , ♥ , ♦ } { ♠ , ♥ , ♣ } { ♠ , ♦ , ♣ } { ♥ , ♦ , ♣ } { ♠ , ♥ } { ♠ , ♦ } { ♥ , ♦ } { ♠ , ♣ } { ♥ , ♣ } { ♦ , ♣ } { ♠ } { ♥ } { ♦ } { ♣ } {} ♠ 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 ♥ 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 ε = B C ♦ @ 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 A ♣ 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 { ♠ , ♥ , ♦ , ♣ } { ♠ , ♦ , ♣ } { ♠ , ♥ , ♦ } { ♠ , ♥ , ♣ } { ♥ , ♦ , ♣ } { ♠ , ♦ } { ♥ , ♦ } { ♥ , ♣ } { ♠ , ♣ } { ♠ , ♥ } { ♦ , ♣ } { ♦ } { ♠ } { ♣ } { ♥ } {} ♠ 1 1 1 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 ♥ 0 1 1 1 0 0 1 0 1 0 0 1 0 0 1 1 ε 0 = B C ♦ 1 1 1 0 1 1 1 0 0 0 0 0 1 0 1 0 @ A ♣ 1 0 1 1 0 0 0 0 1 1 1 0 1 0 1 0 P := syq ( ε , ε 0 ) satisfies ε ; syq ( ε , ε 0 ) = ε 0

  6. Membership relations { a,b,c,d } { a,b,d } { b,c,d } { a,b,c } { a,c,d } { a,b } { a,d } { b,d } { a,c } { b,c } { c,d } { a } { b } { d } { c } {} {} 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  0    { a } 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 U = ε ; e e = syq ( ε , U )   { b }  0  0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1     { a,b }   0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0       { a,b,c,d } { c }  0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1  0   { a,b,d }   { a,b,c } { a,c,d } { b,c,d } { a,c } 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0     { b,d } { a,b } { b,c } { a,d } { c,d } { a,c }     { b,c } 0 { b } { d } 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 { a } { c }     {}   { a,b,c } 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1     a  0     0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1  { d }   0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0     b 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 { a,d }  0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1  0           c 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 { b,d } 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1         d 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 { a,b,d } 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1         { c,d } 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0       { a,c,d } 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1  0  ( 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ) = e T       { b,c,d } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 { a,b,c,d } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 Subset U and corresponding point e in the powerset via ε , Ω

  7. Contents 1. Motivation — my early topology 2. Topology 3. Interlude on prerequisites 4. Cryptomorphy of topology concepts 5. Continuity 6. Interlude on structure comparison 7. Interlude on the existential and inverse image 8. Relating continuity with the inverse image

  8. Topology as defined by Felix Hausdor ff — recalled A set X endowed with a system U ( p ) of subsets for every p 2 X is called a topological structure , provided

  9. Topology as defined by Felix Hausdor ff — recalled A set X endowed with a system U ( p ) of subsets for every p 2 X is called a topological structure , provided i) p 2 U for every neighborhood U 2 U ( p )

  10. Topology as defined by Felix Hausdor ff — recalled A set X endowed with a system U ( p ) of subsets for every p 2 X is called a topological structure , provided i) p 2 U for every neighborhood U 2 U ( p ) ii) If U 2 U ( p ) and V ◆ U , then V 2 U ( p )

  11. Topology as defined by Felix Hausdor ff — recalled A set X endowed with a system U ( p ) of subsets for every p 2 X is called a topological structure , provided i) p 2 U for every neighborhood U 2 U ( p ) ii) If U 2 U ( p ) and V ◆ U , then V 2 U ( p ) iii) If U 1 , U 2 2 U ( p ), then U 1 \ U 2 2 U ( p ) and X 2 U ( p )

  12. Topology as defined by Felix Hausdor ff — recalled A set X endowed with a system U ( p ) of subsets for every p 2 X is called a topological structure , provided i) p 2 U for every neighborhood U 2 U ( p ) ii) If U 2 U ( p ) and V ◆ U , then V 2 U ( p ) iii) If U 1 , U 2 2 U ( p ), then U 1 \ U 2 2 U ( p ) and X 2 U ( p ) iv) For every U 2 U ( p ) there exists a V 2 U ( p ) such that U 2 U ( y ) for all y 2 V

  13. Topology — lifted i) p 2 U for every neighborhood U 2 U ( p ) U ✓ ε

  14. Topology — lifted i) p 2 U for every neighborhood U 2 U ( p ) U ✓ ε ii) If U 2 U ( p ) and V ◆ U , then V 2 U ( p ) U ; Ω ✓ U

  15. Topology — lifted i) p 2 U for every neighborhood U 2 U ( p ) U ✓ ε ii) If U 2 U ( p ) and V ◆ U , then V 2 U ( p ) U ; Ω ✓ U iii) If U 1 , U 2 2 U ( p ), then U 1 \ U 2 2 U ( p ) and X 2 U ( p ) ( U � < U ) ; M ✓ U U ; =

  16. Topology — lifted i) p 2 U for every neighborhood U 2 U ( p ) U ✓ ε ii) If U 2 U ( p ) and V ◆ U , then V 2 U ( p ) U ; Ω ✓ U iii) If U 1 , U 2 2 U ( p ), then U 1 \ U 2 2 U ( p ) and X 2 U ( p ) ( U � < U ) ; M ✓ U U ; = iv) For every U 2 U ( p ) there exists a V 2 U ( p ) such that U 2 U ( y ) for all y 2 V U ✓ U ; ε T ; U

  17. The same with ε conceiving U as a relation: ! 2 X ! 2 X ε : X � and U : X � “For every U 2 U ( p ) there exists a V 2 U ( p ) such that U 2 U ( y ) for all y 2 V ”

  18. The same with ε conceiving U as a relation: ! 2 X ! 2 X ε : X � and U : X � “For every U 2 U ( p ) there exists a V 2 U ( p ) such that U 2 U ( y ) for all y 2 V ” � � �� 8 p, U : U 2 U ( p ) ! 9 V : V 2 U ( p ) ^ 8 y : y 2 V ! U 2 U ( y )

  19. The same with ε conceiving U as a relation: ! 2 X ! 2 X ε : X � and U : X � “For every U 2 U ( p ) there exists a V 2 U ( p ) such that U 2 U ( y ) for all y 2 V ” � � �� 8 p, U : U 2 U ( p ) ! 9 V : V 2 U ( p ) ^ 8 y : y 2 V ! U 2 U ( y ) 8 p, U : U pU ! � 9 V : U pV ^ � 8 y : ε yV ! U yU ��

  20. The same with ε conceiving U as a relation: ! 2 X ! 2 X ε : X � and U : X � “For every U 2 U ( p ) there exists a V 2 U ( p ) such that U 2 U ( y ) for all y 2 V ” � � �� 8 p, U : U 2 U ( p ) ! 9 V : V 2 U ( p ) ^ 8 y : y 2 V ! U 2 U ( y ) 8 p, U : U pU ! � 9 V : U pV ^ � 8 y : ε yV ! U yU �� � � 8 p, U : U pU ! 9 V : U pV ^ 9 y : ε yV ^ U yU

  21. The same with ε conceiving U as a relation: ! 2 X ! 2 X ε : X � and U : X � “For every U 2 U ( p ) there exists a V 2 U ( p ) such that U 2 U ( y ) for all y 2 V ” � � �� 8 p, U : U 2 U ( p ) ! 9 V : V 2 U ( p ) ^ 8 y : y 2 V ! U 2 U ( y ) 8 p, U : U pU ! � 9 V : U pV ^ � 8 y : ε yV ! U yU �� � � 8 p, U : U pU ! 9 V : U pV ^ 9 y : ε yV ^ U yU � � 8 p, U : U pU ! 9 V : U pV ^ ε T ; U V U

  22. The same with ε conceiving U as a relation: ! 2 X ! 2 X ε : X � and U : X � “For every U 2 U ( p ) there exists a V 2 U ( p ) such that U 2 U ( y ) for all y 2 V ” � � �� 8 p, U : U 2 U ( p ) ! 9 V : V 2 U ( p ) ^ 8 y : y 2 V ! U 2 U ( y ) 8 p, U : U pU ! � 9 V : U pV ^ � 8 y : ε yV ! U yU �� � � 8 p, U : U pU ! 9 V : U pV ^ 9 y : ε yV ^ U yU � � 8 p, U : U pU ! 9 V : U pV ^ ε T ; U V U � � 8 p, U : U pU ! U ; ε T ; U pU

  23. The same with ε conceiving U as a relation: ! 2 X ! 2 X ε : X � and U : X � “For every U 2 U ( p ) there exists a V 2 U ( p ) such that U 2 U ( y ) for all y 2 V ” � � �� 8 p, U : U 2 U ( p ) ! 9 V : V 2 U ( p ) ^ 8 y : y 2 V ! U 2 U ( y ) 8 p, U : U pU ! � 9 V : U pV ^ � 8 y : ε yV ! U yU �� � � 8 p, U : U pU ! 9 V : U pV ^ 9 y : ε yV ^ U yU � � 8 p, U : U pU ! 9 V : U pV ^ ε T ; U V U � � 8 p, U : U pU ! U ; ε T ; U pU U ✓ U ; ε T ; U

  24. A neighborhood topology and the basis of its open sets ! 2 X will be called a neighborhood A relation U : X � topology if the following properties are satisfied: i) U ; = and U ✓ ε , ii) U ; Ω ✓ U , iii) ( U � < U ) ; M ✓ U , iv) U ✓ U ; ε T ; U .

  25. � � � � A neighborhood topology and the basis of its open sets ! 2 X will be called a neighborhood A relation U : X � topology if the following properties are satisfied: i) U ; = and U ✓ ε , ii) U ; Ω ✓ U , iii) ( U � < U ) ; M ✓ U , iv) U ✓ U ; ε T ; U . { a,b,c,d } { a,b,d } { a,b,c } { a,c,d } { b,c,d } { a,b } { a,d } { b,d } { a,c } { b,c } { c,d } { b } { d } { a } { c } {} a 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 b 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 U = B C c 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 @ A d 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1

  26. Topology given by transition to the open kernel We call a relation K : 2 X � ! 2 X a mapping-to-open-kernel topology , if i) K is a kernel forming, i.e., K ✓ Ω T , Ω ; K ✓ K ; Ω , K ; K ✓ K ,

  27. Topology given by transition to the open kernel We call a relation K : 2 X � ! 2 X a mapping-to-open-kernel topology , if i) K is a kernel forming, i.e., K ✓ Ω T , Ω ; K ✓ K ; Ω , K ; K ✓ K , contracting isotonic idempotent ii) ε ; K T is total, iii) ( K � ⇥ K ) ; M ✓ M ; K ; Ω T , ( K � ⇥ K ) ; M = M ; K . in fact

  28. Topology given by transition to the open kernel We call a relation K : 2 X � ! 2 X a mapping-to-open-kernel topology , if i) K is a kernel forming, i.e., K ✓ Ω T , Ω ; K ✓ K ; Ω , K ; K ✓ K , contracting isotonic idempotent ii) ε ; K T is total, iii) ( K � ⇥ K ) ; M ✓ M ; K ; Ω T , ( K � ⇥ K ) ; M = M ; K . in fact kernel forming commutes with intersection

  29. � � � � A topology in di ff erent forms { a,b,c,d } { a,b,d } { a,b,c } { a,c,d } { b,c,d } { b,d } { a,b } { b,c } { a,d } { c,d } { a,c } { a,b,c,d } { a } { b } { c } { d } { a,b,d } { b,c,d } {} { a,b,c } { a,c,d } { a,b } { a,d } { b,d } { a,c } { b,c } { c,d } a  0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1  { b } { d } { a } { c } {} b 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1   {} 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  1  c 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1     { a } 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 d 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1   { b }   1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 { a,b,c,d }       { a,b,d } { a,b,c } { a,c,d } { b,c,d } { a,b }  0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0   { b,d }   { a,b } { a,c } { b,c } { a,d } { c,d }   { c } 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1   { b } { d } { a } { c }     {}   { a,c } 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1       { b,c } 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0  1  a  0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1        { a,b,c } b 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0  1          c 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 { d } 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0         d 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 { a,d }  0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0  0       { b,d } 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 { a,b,c,d }       { a,b,d } { a,b,c } { a,c,d } { b,c,d } { a,b,d }   0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   { b,d }   { a,b } { a,c } { b,c } { a,d } { c,d }   { c,d } 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0  1  { b } { d } { a } { c }     {}   { a,c,d } 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1       0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 1 a   { b,c,d }  0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0  1   b 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 { a,b,c,d } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1   c 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1   d 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 U ε O := ε ∩ ; O T V = ε ; K ∩ ε K := syq ( U , ε ) indicating O D as diagonal O V ε

  30. Non-topological kernel-forming { a,b,c,d } { a,b,d } { b,c,d } { a,b,c } { a,c,d } { b,d } { a,b } { a,d } { a,c } { b,c } { c,d } { a } { b } { d } { c } {} {} 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   { a } 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   { b } 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   { a,b }  1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0    { c } 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     { a,c } 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     { b,c } 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     { a,b,c } 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 K =     { d } 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   { a,d }  0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0    { b,d } 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0     { a,b,d } 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0     { c,d } 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     { a,c,d } 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0     { b,c,d } 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 { a,b,c,d } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 Kernel-forming that is not a topology, since not intersection-closed

  31. � � � � { 1,2,3,4 } { 1,2,3 } { 1,2,4 } { 1,3,4 } { 2,3,4 } { 1,2 } { 1,3 } { 2,3 } { 1,4 } { 2,4 } { 3,4 } { 1 } { 2 } { 3 } { 4 } {} 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 2 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 B C 3 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 @ A 4 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1

  32. � � � � � 5 4 3 2 1 @ B B B 0 {} 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 { 1 } { 2 } { 1,2 } { 3 } { 1,3 } { 2,3 } { 1,2,3 } { 4 } { 1,4 } { 2,4 } { 1,2,4 } { 3,4 } { 1,3,4 } { 2,3,4 } { 1,2,3,4 } { 5 } { 1,5 } { 2,5 } { 1,2,5 } { 3,5 } { 1,3,5 } { 2,3,5 } { 1,2,3,5 } { 4,5 } { 1,4,5 } { 2,4,5 } { 1,2,4,5 } { 3,4,5 } { 1,3,4,5 } { 2,3,4,5 } { 1,2,3,4,5 } A C C C 1

  33. � � � � { a,b,c,d } { a,b,d } { b,c,d } { a,b,c } { a,c,d } { b,d } { a,b } { b,c } { a,d } { c,d } { a,c } { a } { b } { d } { c } {} a 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 1 b 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 B C c 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 @ A d 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1

  34. Cryptomorphy of diverse topology concepts K := syq ( U , ε ) : 2 X � ! 2 X U 7! U := ε ; K T : X � ! 2 X . K 7! O D 7! U := ε ; O D ; Ω K , U 7! O D := \ ε T ; U = K T ; K

  35. Cryptomorphy of diverse topology concepts K := syq ( U , ε ) : 2 X � ! 2 X U 7! U := ε ; K T : X � ! 2 X . K 7! O D 7! U := ε ; O D ; Ω K , U 7! O D := \ ε T ; U = K T ; K This means the obligation to prove, e.g. U ; = , K ✓ Ω T , U ✓ ε , Ω ; K ✓ K ; Ω , U ; Ω ✓ U , ( ) K ; K ✓ K , ( U � < U ) ; M ✓ U , ε ; K T ; = , ( K � ⇥ K ) ; M = M ; K . U ✓ U ; ε T ; U .

  36. Separation axioms Let a topology on X be given via neighborhoods, open sets, kernel mapping as required. It is T 0 -space (sometimes a Kolmogorov space) if for any two points in X an open set exists that contains one of them but not the other. It is T 1 -space when 8 x, y : x = / y ! 9 U, V 2 O : x 2 U ^ y 2 / U ^ y 2 V ^ x 2 / V . It is T 2 -space, i.e., a topology satisfying the Hausdor ff property, when 8 x, y : x = / y ! 9 U, V 2 O : x 2 U ^ y 2 V ^ ; = U \ V .

  37. Separation axioms Let a topology given in relational form, i.e., by U , O , K , ε O as required. It is called a i) T 0 -space if syq ( U T , U T ) = T . ii) T 1 -space if ✓ U ; U iii) T 2 -space or a Hausdor ff space if ✓ U ; ε T ; ε ; U T .

  38. Contents 1. Motivation — my early topology 2. Topology 3. Interlude on prerequisites 4. Cryptomorphy of topology concepts 5. Continuity 6. Interlude on structure comparison 7. Interlude on the existential and inverse image 8. Relating continuity with the inverse image

  39. � � � � � � � � � � � � Continuity — standard vs. relational definition � � T Let any two neighborhood topologies U , U 0 be given on sets X, X 0 , and a mapping f : X � ! X 0 . For p 2 X and every neighborhood U 0 2 U 0 ( f ( p )), there exists a neighborhood f continuous : ( ) U 2 U ( p ) satisfying f ( U ) ✓ U 0 .

  40. � � � � � � � � � { a,b,c,d } { a,b,d } { b,c,d } { a,b,c } { a,c,d } { a,b } { a,d } { b,d } { a,c } { b,c } { c,d } { b } { d } { a } { c } {} {} 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   { 1 } 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0   { 2 } 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0     { 1,2 } 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0     { 3 } 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 b d   a c { 1,3 }  0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0    1 0 1 0 0 0 1 { 2,3 } 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0     { 1,2,3 } 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0     { 4 } B C 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0   f = 3 B 0 0 1 0 C   { 1,4 } 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0   B C   { 2,4 } 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 1 @ A   { 1,2,4 }   0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0   5 0 0 1 0 { 3,4 }  0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0    { 1,3,4 } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0     { 2,3,4 } 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0     { 1,2,3,4 } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ϑ f =   { 5 }   0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0   { 1,5 }  0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0    { 2,5 }  0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0    { 1,2,5 } 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0     { 3,5 } 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0     { 1,3,5 } 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0     { 2,3,5 } 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0   { 1,2,3,5 }  0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0    { 4,5 }  0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0    { 1,4,5 } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0     { 2,4,5 } 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0     { 1,2,4,5 } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1     { 3,4,5 } 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0   { 1,3,4,5 }   0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0   { 2,3,4,5 } 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0   { 1,2,3,4,5 } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

  41. � � � � � � � � � � b d a c e 1 0 1 0 0 0 0 1 2 0 0 0 1 0 B C f = 3 B 1 0 0 0 0 C B C 4 1 0 0 0 0 @ A 5 0 0 1 0 0

  42. Contents 1. Motivation — my early topology 2. Topology 3. Interlude on prerequisites 4. Cryptomorphy of topology concepts 5. Continuity 6. Interlude on structure comparison 7. Interlude on the existential and inverse image 8. Relating continuity with the inverse image

  43. Structure-preserving mappings Let be given two “structures” of whatever kind abstracted to relations R 1 : X 1 � ! Y 1 and R 2 : X 2 � ! Y 2 . Y Y 1 2 R R 1 2 X X 1 2

  44. � � � � � � Structure-preserving mappings Let be given two “structures” of whatever kind abstracted to relations R 1 : X 1 � ! Y 1 and R 2 : X 2 � ! Y 2 . Ψ 1 2 1 2 Φ 1 2 Given mappings Φ : X 1 � ! X 2 and Ψ : Y 1 � ! Y 2 , we may ask whether these mappings transfer the first structure “su ffi ciently nice” into the second one.

  45. � � � � � � Structure-preserving mappings Let be given two “structures” of whatever kind abstracted to relations R 1 : X 1 � ! Y 1 and R 2 : X 2 � ! Y 2 . Ψ 1 2 1 2 Φ 1 2 Given mappings Φ : X 1 � ! X 2 and Ψ : Y 1 � ! Y 2 , we may ask whether these mappings transfer the first structure “su ffi ciently nice” into the second one. If any two elements x, y are in relation R 1 , then their images Φ ( x ) , Ψ ( y ) shall be in relation R 2 .

  46. � � � � � � Structure-preserving mappings Let be given two “structures” of whatever kind abstracted to relations R 1 : X 1 � ! Y 1 and R 2 : X 2 � ! Y 2 . Ψ 1 2 1 2 Φ 1 2 Given mappings Φ : X 1 � ! X 2 and Ψ : Y 1 � ! Y 2 , we may ask whether these mappings transfer the first structure “su ffi ciently nice” into the second one. If any two elements x, y are in relation R 1 , then their images Φ ( x ) , Ψ ( y ) shall be in relation R 2 . 8 x 2 X 1 : 8 y 2 Y 1 : ( x, y ) 2 R 1 ! ( Φ ( x ) , Ψ ( y )) 2 R 2

  47. � � � � � � Structure-preserving mappings Let be given two “structures” of whatever kind abstracted to relations R 1 : X 1 � ! Y 1 and R 2 : X 2 � ! Y 2 . Ψ 1 2 1 2 Φ 1 2 Given mappings Φ : X 1 � ! X 2 and Ψ : Y 1 � ! Y 2 , we may ask whether these mappings transfer the first structure “su ffi ciently nice” into the second one. If any two elements x, y are in relation R 1 , then their images Φ ( x ) , Ψ ( y ) shall be in relation R 2 . 8 x 2 X 1 : 8 y 2 Y 1 : ( x, y ) 2 R 1 ! ( Φ ( x ) , Ψ ( y )) 2 R 2 R 1 ; Ψ ✓ Φ ; R 2

  48. Homomorphism This concept works for groups, fields and other algebraic structures , but also for relational structures as, e.g., graphs. Φ , Ψ is a homomorphism from R to R 0 , if Φ , Ψ are mappings satisfying R ; Φ ✓ Ψ ; R 0 . Φ , Ψ is an isomorphism between R and R 0 , if Φ , Ψ as well as Φ T , Ψ T are homomorphisms. Theorem If Φ , Ψ are mappings, then R ; Ψ ✓ Φ ; R 0 ( ) R ✓ Φ ; R 0 ; Ψ T ( ) Φ T ; R ✓ R 0 ; Ψ T ( ) Φ T ; R ; Ψ ✓ R 0 If relations Φ , Ψ are not mappings, one cannot fully execute this rolling; there remain di ff erent forms of (bi-)simulations.

  49. � � � � � � � � � � � � Continuity compares structures in a di ff erent way! � � T Let any two neighborhood topologies U , U 0 be given on sets X, X 0 , and a mapping f : X � ! X 0 . For p 2 X and every neighborhood U 0 2 U 0 ( f ( p )), there exists a neighborhood f continuous : ( ) U 2 U ( p ) satisfying f ( U ) ✓ U 0 .

  50. Contents 1. Motivation — my early topology 2. Topology 3. Interlude on prerequisites 4. Cryptomorphy of topology concepts 5. Continuity 6. Interlude on structure comparison 7. Interlude on the existential and inverse image 8. Relating continuity with the inverse image

  51. � � � � � � � � � � � � � � � � � � Existential image of relations � � T

  52. � � � � � � � � � � � � � � � � � � Existential image of relations � � T ϑ := ϑ R := syq ( R T ; ε , ε 0 ) existential image .

  53. � � � � � � � � � � � � � � � � � � Existential image of relations � � T ϑ := ϑ R := syq ( R T ; ε , ε 0 ) existential image . ϑ is (lattice-)continuous wrt. the powerset orderings Ω = ε T ; ε

  54. � � � � � � � � � � � � � � � � � � Existential image of relations � � T ϑ := ϑ R := syq ( R T ; ε , ε 0 ) existential image . ϑ is (lattice-)continuous wrt. the powerset orderings Ω = ε T ; ε X = ϑ Q ; R = ϑ Q i.e. multiplicative ϑ ; ϑ R 2 X ε 0 T ; R T = ϑ R T ; ε T ; ε 0 T ε T ; R = ϑ R i.e. mutual simulation R may be re-obtained from ϑ as R = ε ; ϑ ; ε 0 T

  55. � � � � � � � � � � � � � � � � � � Existential image of relations � � T ϑ := ϑ R := syq ( R T ; ε , ε 0 ) existential image . ϑ is (lattice-)continuous wrt. the powerset orderings Ω = ε T ; ε X = ϑ Q ; R = ϑ Q i.e. multiplicative ϑ ; ϑ R 2 X ε 0 T ; R T = ϑ R T ; ε T ; ε 0 T ε T ; R = ϑ R i.e. mutual simulation R may be re-obtained from ϑ as R = ε ; ϑ ; ε 0 T but there exist many relations W satisfying R = ε ; W ; ε 0 T

  56. � � � � � � � � � � � � � � � � � � { a,b,c,d } Existential image { a,b,d } { b,c,d } { a,b,c } { a,c,d } { b,d } { a,b } { a,c } { b,c } { a,d } { c,d } { a } { b } { c } { d } {} {}  1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  { 1 } 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0   { 2 } 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0     { 1,2 } 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0     { 3 } 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0   { 1,3 }   0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0   { 2,3 }  0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0    � � T { 1,2,3 } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1     { 4 } 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0     { 1,4 } 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0     { 2,4 } 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0   { 1,2,4 }  0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0    { 3,4 } 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0     { 1,3,4 }  0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0    { 2,3,4 } 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0     { 1,2,3,4 } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 b d a c ϑ R =   { 5 }   0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1     { 1,5 }  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1    2 1 0 0 0 { 2,5 }  0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0    { 1,2,5 } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1     R = 3 0 0 1 0   { 3,5 } 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0         { 1,3,5 } 4 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1   { 2,3,5 }   0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0   5 1 0 1 0 { 1,2,3,5 }   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1   { 4,5 }  0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0    { 1,4,5 } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1     { 2,4,5 } 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0     { 1,2,4,5 } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1     { 3,4,5 } 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0   { 1,3,4,5 }   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1   { 2,3,4,5 }  0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0  { 1,2,3,4,5 } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

  57. � � � � � � � � � � � � � � � � � � Inverse image � � T b d a c 1 0 1 0 1   2 1 0 0 0   R = 3 0 0 1 0      4 0 0 0 1  5 1 0 1 0 { } { 1,2,3,4,5 } { 1,2,3,4 } { 1,2,3,5 } { 1,2,4,5 } { 1,3,4,5 } { 2,3,4,5 } { 1,2,3 } { 1,2,4 } { 1,3,4 } { 2,3,4 } { 1,2,5 } { 1,3,5 } { 2,3,5 } { 1,4,5 } { 2,4,5 } { 3,4,5 } { 1,2 } { 1,3 } { 2,3 } { 1,4 } { 2,4 } { 3,4 } { 1,5 } { 2,5 } { 3,5 } { 4,5 } { 1 } { 2 } { 3 } { 4 } { 5 } {} {} 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   { a } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0   { b } 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   { a,b }  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0    { c } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0     { a,c } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0     { b,c } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0     { a,b,c } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 ϑ R T =     { d } 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   { a,d }  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0    { b,d }  0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0    { a,b,d } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0     { c,d } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0     { a,c,d } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1   { b,c,d }   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 { a,b,c,d } 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

  58. Contents 1. Motivation — my early topology 2. Topology 3. Interlude on prerequisites 4. Cryptomorphy of topology concepts 5. Continuity 6. Interlude on structure comparison 7. Interlude on the existential and inverse image 8. Relating continuity with the inverse image

  59. � � � � � � � � � � � � Continuity compares structures in a di ff erent way! � � T Let any two neighborhood topologies U , U 0 be given on sets X, X 0 , and a mapping f : X � ! X 0 . For p 2 X and every neighborhood U 0 2 U 0 ( f ( p )), there exists a neighborhood f continuous : ( ) U 2 U ( p ) satisfying f ( U ) ✓ U 0 .

  60. Lifting the continuity condition For all p 2 X , all V 2 U 0 ( f ( p )), exists a U 2 U ( p ) with f ( U ) ✓ V . 8 p 2 X : 8 V 2 U 0 ( f ( p )) : 9 U 2 U ( p ) : f ( U ) ✓ V 8 p 2 X : 8 v 2 2 X 0 : U 0 � ⇥ ⇤� f ( p ) ,v � ! 9 u : U p,u ^ 8 y : ε yu ! ε 0 f ( y ) ,v � ⇥ ⇤� 8 p : 8 v : ( f ; U 0 ) pv � ! 9 u : U pu ^ 8 y : ε yu ! ( f ; ε 0 ) yv � � 8 p : 8 v : ( f ; U 0 ) pv � ! 9 u : U pu ^ 9 y : ε yu ^ ( f ; ε 0 ) yv � � 8 p : 8 v : ( f ; U 0 ) pv � ! 9 u : U pu ^ ε T ; f ; ε 0 uv � U ; ε T ; f ; ε 0 � 8 p : 8 v : ( f ; U 0 ) pv � ! pv f ; U 0 ✓ U ; ε T ; f ; ε 0 f ; U 0 ✓ U ; ϑ T The last step is proved as follows: f T U ; ε T ; f ; ε 0 ✓ U ; ε T ; f ; ε 0 ; ϑ f T ; ϑ T because ϑ f T is total f T = U ; ε T ; f ; ε 0 ; syq ( f ; ε 0 , ε ) ; ϑ T by definition of ϑ f T f T ✓ U ; ε T ; ε ; ϑ T cancellation f T = U ; ε T ; ε ; ϑ T since ϑ f T is a mapping f T = U ; Ω ; ϑ T f T = U ; ϑ T f T

  61. � � � � � � � � � � � � Continuity — standard vs. relational definition � � T Let any two neighborhood topologies U , U 0 be given on sets X, X 0 , and a mapping f : X � ! X 0 . For p 2 X and every neighborhood U 0 2 U 0 ( f ( p )), there exists a neighborhood f continuous : ( ) U 2 U ( p ) satisfying f ( U ) ✓ U 0 . : ( ) f ; U 0 ; ϑ f T ✓ U f continuous f ; U 0 ✓ U ; ϑ T ( ) f T

  62. Cryptomorphy of continuity concepts Given sets X, X 0 with topologies, we consider a mapping ! X 0 together with its inverse image ϑ f T : 2 X 0 � ! 2 X . f : X � Then we say that the pair ( f, ϑ f T ) is i) K - continuous : ( ) K T 2 ; ϑ f T ✓ ε 2 T ; f T ; ε 1 ; K T 1 : ( ) O D 2 ; ϑ f T ✓ ϑ f T ; O D 1 ii) O D - continuous iii) O V - continuous : ( ) ϑ T f T ; O 0 V ✓ O V iv) ε O - continuous : ( ) f ; ε O 2 ; ϑ f T ✓ ε O 1

  63. Cryptomorphy of continuity concepts Given sets X, X 0 with topologies, we consider a mapping ! X 0 together with its inverse image ϑ f T : 2 X 0 � ! 2 X . f : X � Then we say that the pair ( f, ϑ f T ) is i) K - continuous : ( ) K T 2 ; ϑ f T ✓ ε 2 T ; f T ; ε 1 ; K T 1 : ( ) O D 2 ; ϑ f T ✓ ϑ f T ; O D 1 ii) O D - continuous iii) O V - continuous : ( ) ϑ T f T ; O 0 V ✓ O V iv) ε O - continuous : ( ) f ; ε O 2 ; ϑ f T ✓ ε O 1 Again, there is an obligation to prove f is K - continuous ( ) f is O D - continuous ( ) ( ) f is O V - continuous f is ε O - continuous

  64. Thank you!

  65. Language and system

  66. Systems to support work with relations I RelView : RBDD-Implementierung; auch f¨ ur große Relationen I T ITU R EL eine relationale Sprache, transformierbar, interpretierbar I Ralf : weiland ein guter Formel-Manipulator und Beweis-Assistent I RATH : Exploring (finite) relation algebras with tools written in Haskell

  67. Aims in designing T ITU R EL I Formulate all problems so far tackled with relational methods I Transform relational terms and formulae in order to optimize them I Interpret the relational constructs as boolean matrices, in RelView , in the T ITU R EL substrate, or in Rath I Prove relational formulae with system support in the style of Ralf or Rasiowa-Sikorski I Translate relational formulae into T EX-representation, or to first-order predicate logic, e.g.

  68. Recalling syntax vs. semantics for PL/I: K const. an element K I for K interpretation I ϕ fct. a function table ϕ I for ϕ tokens in supporting set p pred. s subset p I for p

  69. Recalling syntax vs. semantics for PL/I: K const. an element K I for K interpretation I ϕ fct. a function table ϕ I for ϕ tokens in supporting set p pred. s subset p I for p Out of this and the variables V one forms terms and formulae T = V | K | ϕ ( T ) F = p ( T ) | ¬ F | 8 V : F

  70. Recalling syntax vs. semantics for PL/I: K const. an element K I for K interpretation I ϕ fct. a function table ϕ I for ϕ tokens in supporting set p pred. s subset p I for p Out of this and the variables V one forms terms and formulae T = V | K | ϕ ( T ) F = p ( T ) | ¬ F | 8 V : F With a variable valuation v : x 7! v ( x ) terms are evaluated v ⇤ ( x ) := v ( x ) v ⇤ ( k ) := k I v ⇤ ( ϕ ( t )) := ϕ I ( v ⇤ ( t ))

  71. Recalling syntax vs. semantics for PL/I: K const. an element K I for K interpretation I ϕ fct. a function table ϕ I for ϕ tokens in supporting set p pred. s subset p I for p Out of this and the variables V one forms terms and formulae T = V | K | ϕ ( T ) F = p ( T ) | ¬ F | 8 V : F With a variable valuation v : x 7! v ( x ) terms are evaluated v ⇤ ( x ) := v ( x ) v ⇤ ( k ) := k I v ⇤ ( ϕ ( t )) := ϕ I ( v ⇤ ( t )) and formulae interpreted | = I,v p ( t ) : ( ) v ⇤ ( t ) ✓ p I | = I,v ¬ F : ( ) | = / I,v F | = I,v 8 x : F : ( ) For all s holds | = I,v x s F

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