Foundations of mathematics. 2. Set theory (continued)

2.1. Tuples, amilies Foundations o mathematics 2. Set theory (continued) Sylvain Poirier settheory.net A tuple (or n-tuple, or any integer n) is an interpretation o a list o n variables. It is thus a meta-unction rom a inite meta-set, to the universe. Tuples o a given kind (list o variables with their types) can be added to any theory as a new type o objects, whose variables are meant as abbreviations o packs o n variables (copies o the list with the same old types) x = (x 0,, x n 1 ). In practice, the domain o considered n-tuples will be the (meta-)set V n o n digits rom 0 to n 1. Set theory can represent its own n-tuples as unctions, iguring V n as a set o objects all named by constants. A 2-tuple is called an oriented pair, a 3-tuple is a triple, a 4-tuple is a quadruple... The n-tuple deiner is not a binder but an n-ary operator, placing its n arguments in a parenthesis and separated by commas: (,, ). The evaluator appears (curried by ixing the meta-argument) as a list o n unctors called projections: or each i V n, the i-th projection π i gives the value π i (x) = x i o each tuple x = (x 0,, x n 1 ) at i (value o the i-th variable inside x). They are subject to the ollowing axioms (where the irst sums up the next ones) : or any x 0,, x n 1 and any n-tuple x, x = (x 0,, x n 1 ) (π 0 (x) = x 0 π n 1 (x) = x n 1 ) x i = π i ((x 0,, x n 1 )) x = (π 0 (x),, π n 1 (x)) or each i V n Oriented pairs suice to build (copies o) n-tuples or each n > 2, in the sense that we can deine operators to play the roles o deiner and projections, satisying the same axioms. For example, triples t = (x, y, z) can be deined as t = ((x, y), z)) and evaluated by x = π 0 (π 0 (t)), y = π 1 (π 0 (t)), z = π 1 (t). Conditional connective The 3-ary connective I A then B else C, is written as ollows (applying (C, B) to A V 2 ) : (A B C) ( C A B) ((A B) ( A C)) ( A C B) (C, B)(A) ((A B) ( A C)) ((C A) (A B)) (A B C) Any n+1-ary connective K amounts to two n-ary ones: K(A) (A K(1) K(0)). Thus, A (A 0 1); (A B) (A B 1); (A B) (A 1 B); (A B) (A B B). Families A amily is a unction intuitively seen as a generalized tuple: its domain (a set) is seen as simple, ixed, outside the main studied system, as i it was a set o meta-objects. A amily o... is a amily whose image is a set o.... Families use the ormalism o unctions disguised in the style o tuples (whose tools cannot apply due to the initeness o symbols). So the notation u i that looks like a meta-variable symbol o variable, evaluates u at i (abbreviating u(i) or π i (u)). A amily deined by a term t, is written (t(i)) instead o (I i t(i)) or (t(0),, t(n 1)). The argument i is called index, and the amily is said indexed by its domain I. A amily indexed by the set N o integers is called a sequence. Structures and binding symbols Each n-ary structure can be interpreted as a unary structure whose class o deiniteness is a class o n-tuples, just like a binder can be seen as a unary structure over a class o unctions or subsets rom the given set. Indeed, binders are the generalization o structures when tuples are replaced by amilies. In particular, quantiiers and are the respective generalizations o chains o conjunctions and disjunctions: (B 0 B n 1 ) ( i V n, B i ). We can see the equality condition between ordered pairs, (x, y) = (z, t) (x = z y = t), as similar to the one or unctions. Let R a unary predicate deinite on E, and C a boolean. We have distributivities (C x E, R(x)) ( x E, C R(x)) (C x E, R(x)) ( x E, C R(x)) (C x E, R(x)) ( x E, C R(x)) (( x E, R(x)) C) ( x E, R(x) C) ( x E, C) (C E ) C (C E = ) ( x E, C) 1

Extensional deinitions o sets The unctor Im deines the binder {T (x) x E} = {T (x)} x E = Im(E x T (x)). As this notation looks similar to the set builder, we can combine both : {T (x) x E R(x)} = {T (x) x {y E R(y)}} Applying Im to tuples (a, b, ), deines the operator symbols o exensional deinition o sets (listing their elements), written {a, b, }. For example V n = {0,, n 1}. Those o arity 0,1,2 are the empty set, the singleton {a} and the pairing {a, b} already presented in 1.10. Images o tuples are inite sets (initeness will be ormally deined in text 5, with Galois connections). 2.2. Boolean operators on amilies o sets Union o a amily o sets For any list o unary predicates A i with index i I, all deinite in (at least) a common class C, applying them to the same variable x with range C reduces them to Booleans whose value depends in x i.e. Boolean variables with parameter x (in the sense o 1.5). This way, any Boolean operation between these variables (a connector or quantiier Q) deines a meta-operation between unary predicates, with result the unary predicate R deined by C x, (R(x) Qi, A i (x)). When C is a set E, this operates between subsets o E (through and the set builder). For example, the quantiier Q = deines the union o a amily o sets: x F i i I, x F i This class is a set independent o E, as it can also be deined rom the union o a set o sets (1.11): F i = {F i i I} ( x F i, B(x)) i I, x F i, B(x) x A B (x A x B) A A B = B A All extensional deinition operators (except ) are deinable rom pairing and binary union. Intersection Now any ixed amily o sets (F i ) can be seen as a amily o subsets o some common set, such as their union U = F i, or any other set E such that U E. Then or any operation (quantiier) Q between Booleans indexed by I, the predicate R(x) deined as (Qi, x F i ) takes value (Qi, 0) or all x / U. Thus Q needs to satisy (Qi, 0) (i.e. to be alse when all entries are alse) or the class R to be a set, then expressible as {x E R(x)}, or as {x U R(x)}. This condition was always satisied or the quantiier (that deines the union) including on the empty amily ( = ), but or (that deines the intersection) it requires a non-empty amily o sets (i.e. (F i ) with I ): j I, F i = {x F j i I, x F i } x F i i I, x F i x A B (x A x B) A = A B B A B = A B A B = B A = {x A x B} A Two sets A and B are called disjoint when A B =, which is equivalent to x A, x / B. Union and intersection have the same associativity and distributivity properties as and : A B C = (A B) C = A (B C) = {A, B, C} ( A i ) C = i C) (A ( A i ) C = i C) (A (A B) C = (A C) (B C) (A B) C = (A C) (B C) Other operators The dierence is deined by A\B = {x A x / B} so that x A\B x A x / B. Finally the connector gives the symmetric dierence: A B = (A B)\(A B). When (Qi, 0) is true, we must choose a set E to deine operations between subsets o E : Negation deines E F = E\F, called the complement o F in E : x E, x F x E F. The intersection o the empty amily gives E: or a amily o subsets F i o E, F i = {F i } = {x E i I, x F i } = E E F i 2

2.3. Products, graphs and composition Finite product For two sets E and F, the product E F is the set o (x, y) where x E and y F. Similarly, the product o n sets E 0 E n 1 is the set o n-tuples (x 0,, x n 1 ) where i V n, x i E i. An n-ary operation is a unction with domain a product o n sets. A relation (or example between E and F ) can be expressed as a set o tuples (G E F ). The domains E and F can be speciied by taking the triple (E, F, G). A set o oriented pairs (such as G) is called a graph. For any binder L and any graph G, the ormula Lx G, S(x 0, x 1 ) that binds x = (x 0, x 1 ) on a binary structure S with domain G, can be seen as binding 2 variables x 0, x 1 on S(x 0, x 1 ), and thus be denoted with an oriented pair o variables: L(y, z) G, S(y, z). The existence o the product (in all arity) is justiied by the set generation principle: ( (x, y) E F, R(x, y)) ( x E, y F, R(x, y)) ( y F, x E, R(x, y)) ( (x, y) E F, R(x, y)) ( x E, y F, R(x, y)) ( y F, x E, R(x, y)) The quantiier (x, y) E E will be abbreviated as x, y E, and the same or. When F = V 2, ( x E, A(x) B(x)) (( x E, A(x)) ( x E, B(x)) ( x E, A(x) B(x)) (( x E, A(x)) ( x E, B(x)) ( x E, C A(x)) ((C (E )) x E, A(x)) ( x E, C A(x)) ((C (E = )) x E, A(x)) Sum or disjoint union The transpose o an ordered pair is t (x, y) = (y, x); that o a graph R is t R = {(y, x) (x, y) R}. Graphs can be expressed in curried orms R and R : y R(x) (x, y) R x R(y) = t R(y), justiied by deining the unctor R as R(x) = {y (z, y) R (z = x)}. Inversely, any amily o sets (E i ) I I deines a graph called their sum (or disjoint union) E i: (i I x E i ) (i, x) E i = {i} E i = {(i, x) x E i } I E i ( x E i, A(x)) ( i I, y E i, A(i, y)) E 0 E n 1 = i V n E i E F = x E F E = = E (E E F F ) E F E F ( i I, E i E i) E i E i Composition, restriction, graph o a unction For any set E, the unction identity on E is deined by Id E = (E x x). For any unctions, g with Im Dom g (namely, : E F and g : F G), their composite is g = ((Dom ) x g((x))) : E G The same with h : G H, h g = (h g) = h (g ) = (E x h(g((x)))) and so on. The restriction o to A Dom is A = (A x (x)) = Id A. The graph o a unction is deined by Gr = {(x, (x)) x Dom } = x Dom {(x)} (x, y) Gr (x Dom y = (x)) We can deine domains, images and restrictions or graphs, letting those or unctions be particular cases (i.e. Dom = Dom(Gr ), Im = Im(Gr ) and Gr( A ) = (Gr ) A ): Dom R = {x (x, y) R} = Im t R x, x Dom R R(x) R E F (Dom R E Im R F ) R A = R (A Im R) = {(x, y) R x A} = x A R(x) 3

Then we have R = E i (Dom R I i I, R(i) = E i ) Im R = E i. For any unctions, g, any graph R, and E = Dom, Gr R x E, (x) R(x) R Gr ( (x, y) R, x E y = (x)) (Dom R E (x, y) R, y = (x)) Gr Gr g ((E Dom g) = g E ) Direct image, inverse image The direct image o a set A by a graph R is R (A) = Im R A = x A R(x). Dom R A R A = R R (A) = Im R R ( A i ) = R (A i ) R ( A i ) R (A i ) A B R (A) R (B) The direct image o A Dom by a unction is [A] = (A) = (Gr ) (A) = Im( A ) = {(x) x A} Im For any : E F and B F, the inverse image o B by, written (B), is deined by (B) = ( t Gr ) (B) = {x E (x) B} = (y) y B (y) = ( Gr)(y) = ({y}) = {x E (x) = y} ( F B) = E (B) For any amily (B i ) o subsets o F, ( B i) = (B i ) where intersections are respectively interpreted as subsets o F and E. 2.4. Uniqueness quantiiers, unctional graphs For all sets F E, all unary predicate A deinite on E, and all x E, x F {x} F ( y E, x = y y F ) ( y E, x = y y F ) x F (( y F, A(y)) A(x) y F, A(y)) F {x} ( y F, x = y) (( y F, A(y)) A(x) ( y F, A(y))) F = {x} (x F y F, x = y) ( y E, y F x = y) Here are 3 new quantiiers: 2 (plurality),! (uniqueness), and! (existence and uniqueness), whose results when applied to R in E only depend on F = {x E, R(x)} (like and unlike ) : ( x E, R(x)) (F ) ( x F, 1) ( x E, {x} F ) ( 2x E, R(x)) ( 2 : F ) ( x, y F, x y) ( x, y E, R(x) R(y) x y) (!x E, R(x)) (!: F ) ( 2 : F ) ( x, y F, x = y) x F, F {x} (!x E, R(x)) (! : F ) ( x F, F {x}) ( x E, F = {x}) F {x} y F, F {y} (!: F ) (! : F ) (F!: F ) F (( y F, A(y)) ( y F, A(y))) (!: F ) (( y F, A(y)) ( y F, A(y))) F = {x} (( y F, A(y)) A(x) y F, A(y)) A unction is said constant when!: Im. The constancy o a tuple is the chain o equalities: x = y = z!:{x, y, z} ((x = y) (y = z)) x = z 4

Translating operators into predicates In a generic theory, any unctor symbol T can be replaced by a binary predicate symbol R (where x R y (y = T (x))) with the axiom x,!y, x R y, replacing any ormula A(T (x)) (where x is a term) by ( y, x R y A(y)), or by ( y, x R y A(y)) (while terms cannot be translated). This way, any predicate R such that x,!y, x R y implicitly deines an operator symbol T. We can extend this to other arities by replacing x by a tuple. But the use o open quantiiers in this construction makes it unacceptable in our set theory. Instead, let us introduce a new operator ɛ on the class (Set(E)! : E) o singletons, giving their element according to the axiom ( x)ɛ{x} = x, or equivalently (Set E! : E) ɛe E. Then or every unary predicate A and every singleton E, A(ɛE) ( x E, A(x)) ( x E, A(x)). Conditional operator Like the conditional connector, it chooses between two objects x, y depending on the boolean B: (B x y) = (y, x) B = ɛ{z {x, y} B z = x z = y} so that or any predicate A we have A(B x y) (B A(x) A(y)). Combined with structures, it is the natural means to deine any other structuring para-operator (like the one translating booleans into objects), making useless their direct registration to the language o a theory. Functional graphs A graph R is said unctional i x Dom R,!: R(x), or equivalently x, y R, x 0 = y 0 x 1 = y 1. This is the condition or it to be the graph o a unction. Namely, R = Gr(ɛ R) where 2.5. The powerset axiom ɛ R = ((Dom R) x ɛ( R(x))) Let us extend set theory by 3 new symbols (powerset, exponentiation, product) together with axioms, that will declare given classes C to be sets K. Such extensions are similar to those given by the set generation principle (1.11), except that they no more satisy its condition o application. In the traditional ZF ormalization o set theory only accepting as primitive stucture, such a declaration is done purely by the axiom (or theorem) ( parameters), K, x, x K C(x). Then, K can be used to represent the ollowing abbreviations: x K means x, C(x) ; the equality X = K means ( x, x X C(x)), and any other A(K) means X, (X = K) A(X). But these ormulas use open quantiiers, orbidden in our ramework. Cases justiied by the set generation principle could still be processed by replacing C by ormulas, while the above X could be interpreted by the axiom with existential elimination (1.9). But this no more works in other cases : without open quantiiers, even a given set K that would happen to be identical to a class C, would still not be recognizable as such, so that the very claim that C coincides with a set remains practically meaningless. Thus instead o this, the eective ormalization requires to name K by a new primitive symbol o set theory, with argument the tuple y o parameters o C, with the axiom ( y ), Set(K(y)) x, x K(y) C y (x) Powerset. Set E, Set(P(E)) F, F P(E) (Set(F ) F E). We shall also shorten P into in binding symbols: ( A E, ) ( A P(E), ). Cantor Theorem. ( Set E, Fnc ) Dom = E P(E) Im. Proo: F = {x E x / (x)} ( x E, x F x (x)) ( x E, F (x)) F / Im. (The Russell paradox may be seen as a particular case) Exponentiation. Set E, F, Set(F E ), ( F E : E F ). Product o a amily o sets. This binder is the generalization o inite product operators : x, x E i (Fnc(x) Dom x = I i I, x i E i ). For all i I we call i-th projection, the unction π i rom j I E j to E i evaluating every amily x at i : π i (x) = x i. This is the unction evaluator seen as curried in the unusual order. 5

These operators are equivalent in the sense that they are deinable rom each other: P(E) = {{x E (x) = 1} V 2 E } F E = F = {ɛ R R E F x E,! : R(x)} x E E i x E,! : R(x)} E i = {x ( E i ) I i I, x i E i } = {ɛ R R Even some cases are expressible rom previous tools: F {a} = {{a} x y y F } F = { } P({a}) = {, {a}} { E i = (i I x i y i ) J (x, y) E i } E i J i J ( i I,E i = ) E i = ( i I,! : E i ) E i = {(ɛe i ) } I F F then P(F ) P(F ), F E F E, and ( i I, E i E i) 2.6. Injectivity and inversion E i E i. A unction : E F is injective (or : an injection) i t Gr is a unctional graph: Inj ( y F,!: (y)) ( x, x E, (x) = (x ) x = x ) ( x, x E, x x (x) (x )) Then its inverse is deined by 1 = ɛ = (Im y ɛ (y)) so that Gr( 1 ) = t Gr. A unction : E F is said bijective (or a bijection) rom E onto F, and we write : E F, i it is injective and surjective: y F,! : (y), in which case 1 : F E. A permutation (or transormation) o a set E is a bijection : E E. Proposition. Let : E F and g : F G. 1. (Inj Inj g) Inj(g ) 2. Inj(g ) Inj 3. Im(g ) = g[im ] Im g. 4. Im(g ) = G Im g = G. 5. Im = F Im(g ) = Im g, so that (( : E F ) (g : F G)) (g : E G) Proos: 1. (Inj Inj g) x, y E, g((x)) = g((y)) (x) = (y) x = y. 2. x, y E, (x) = (y) g((x)) = (g((y)) x = y. 3. z G, z Im(g ) ( x E, g((x)) = z) ( y Im, g(y) = z) z g[im ]. 3. 4. and 3. 5. are obvious. Proposition. For any sets E, F, G and any F E, Im = F Inj(G F g g ) (Im = F!: G) (Inj G ) {g g G F } = G E (Inj!: G) (E F G ) {g E g G F } = G E (Inj E ) g E F, g = Id E Proos : Im = F g, h G F, (( x E, g((x)) = h((x))) ( y F, g(y) = h(y) g = h)). z, z G, (y z) = (y (y Im z z )) thus Inj(g g ) (z = z Im = F ). z G, h G E, Inj (F y (y Im h 1 (y) z)) = h. z, z G, x E, i g G F is such that y E, g((y)) = (y = x z z ), then y E, (y) = (x) g((y)) = g((x)) = z (y = x z = z ). The last ormulas are particular cases o the second one. Let : E F and F : P(F ) P(E) deined by. Then (by G = V 2 and B Im, [ (B)] = B), Inj Im F = P(E) Im = F Inj F Im = {[A] A E} = P(Im ) 6

Proposition. Let F = Dom g. Then Inj g Inj(F E g ) (Inj g E = ). Proos:, F E, (Inj g x E, g((x)) = g( (x))) ( x E, (x) = (x)). Then rom the middle ormula, y, z F, g(y) = g(z) g (E x y) = g (E x z) ( x E, y = z) (y = z E = ). Proposition. Let : E F and Dom g = F. Then g = Id E ( x E, y F, (x) = y g(y) = x) (Gr t Gr g) ( y F, (y) {g(y)}) ( x E, (x) g (x)) (Inj g Im = 1 ) E Im g (g : F E g = Id E ) (Im = F Inj g g = Id F g = 1 ). Proo o the last ormula: (Inj g g g = g Id F ) g = Id F. Proposition. 1) I, g are injective and Im = Dom g, then (g ) 1 = 1 g 1. 2) I, h : E F and g : F E then (g = Id E h g = Id F ) ((g : F E) = h = g 1 ). Proos: 1) x Dom, y Im g, (g )(x) = y (x) = g 1 (y) x = ( 1 g 1 )(y). Other method: (g ) 1 = (g ) 1 g g 1 = (g ) 1 g 1 g 1 = 1 g 1. 2) We deduce = h rom = h g = h, or rom Gr t Gr g Gr h. The rest is obvious. 2.7. Properties o binary relations on a set ; ordered sets A binary relation on E is a graph R E E. Denoting x R y (x, y) R and omitting the domain E o quantiiers, it will be said relexive x, x R x irrelexive x, (x R x) symmetric R t R R = t R R = R antisymmetric x, y, (x R y y R x) x = y transitive x, y, z, (x R y y R z) x R z For any transitive binary relation R we denote x R y R z ((x R y) (y R z)) x R z. Example. Let A E E and R = A Gr. Then (Id E A) R is relexive (, g A, g A) R is transitive ( A, : E E 1 A) R is symmetric Preorder. A preorder R on a set E is a relexive and transitive binary relation on E. Equivalently, x, y E, x R y R(x) R(y) Proo: x, y E, x R(x) R(y) x R y ; transitivity says x R y R(x) R(y). Note : t R is then also a preorder, i.e. x R y R(y) R(x). Ordered set. An order is an antisymmetric preorder. A preordered set is a set E together with a preorder R (thus an ordered pair (E, R)). An ordered set is a set with an order, usually written as. For x, y in an ordered set E, the ormula x y can be read x is less than y, or y is greater than x. The elements x and y are said incomparable when (x y y x). (This implies x y). Any subset F o a set E with an order (resp. a preorder) R E E, is also ordered (resp. preordered) by its restriction R (F F ) (which is an order, resp. preorder, on F ). Strict order. It is a binary relation both transitive and irrelexive; and thus also antisymmetric. Strict orders < bijectively correspond to orders by x < y (x y x y). The inverse correspondence is deined by x y (x < y x = y). Total order. A total order on a set E is an order R on E such that R t R = E E. Namely, it is an order where no two elements are incomparable. Equivalently, it is an order related with its strict order < by x, y E, x < y y x. Still equivalently, it is a transitive relation such that x, y E, x y (y x x = y). A strict order associated with a total order, called a strict total order, is any transitive relation < in E such that x, y E, x < y (y < x x = y), or equivalently x, y E, x = y (y < x x < y)). 7

Monotone, antitone, strictly monotone unctions Between ordered sets E and F, a unction : E F will be said : monotone i x, y E, x y (x) (y) antitone i x, y E, x y (y) (x) strictly monotone i x, y E, x y (x) (y) strictly antitone i x, y E, x y (y) (x). Any composite o a chain o monotone or antitone unctions, is monotone i the number o antitone unctions in the chain is even, or antitone i it is odd. Any strictly monotone or strictly antitone unction is injective. I F E and g E F are both monotone (resp. both antitone) and g = Id E, then is strictly monotone (resp. strictly antitone). In each ordered set E, a unction E E is said extensive i x E, x (x), i.e. Id E. The composite o two extensive unctions is extensive. Order on sets o unctions For all sets E, F where F is ordered, the set F E (and thus any subset o F E ) is ordered by g ( x E, (x) g(x)) Then,, g F E, h E G, g h g h, i.e. h is always monotone. The particular case F = V 2 is that P(E) (and thus any set o sets) is naturally ordered by, and that h is monotone rom P(E) to P(G). I F and G are ordered and u G F is monotone (resp. antitone) then F E u G E is monotone (resp. antitone). 2.8. Canonical bijections For all objects x, y we shall say that x determines y i there is an invariant unctor T such that T (x) = y. Then the role o y as ree variable can be played by the term T (x), thus by the use o x. This is a preorder on the universe, but not a predicate o set theory as it involves a meta-concept. Instead, it is meant to abbreviate the use o T. Similarly, a unction : E F will be said canonical i it is deined as E x T (x) or some invariant unctor T. A bijection will be said bicanonical i both and 1 are canonical. When a bijection : E F is canonical (resp. bicanonical), we write : E = F (resp. : E F ); or, using its deining unctor, T : E = F which means that (E x T (x)) is injective with image F. We shall write E = F (resp. E F ) to mean the existence o a canonical (resp. bicanonical) bijection, that is kept implicit. Canonical bijections can ail to be bicanonical especially when their deining unctor is not injective: V 2 E = P(E), {x} E = {x} and E {x} = E, whereas {x} E E {x} and E {0} E. This is a preorder on the class o sets, preserved by constructions: or example i E = E and F = F then E F = E F and F E = F E using the direct image o the graph (while we may not have F E = F E when E does not determine E). It will oten look like a property o numbers as the existence o a bijection between inite sets implies the equality o their numbers o elements. The transposition o oriented pairs (E F F E) extends to graphs (P(E F ) P(F E)) and to operations: G E F G F E where G E F is transposed by t (x, y) = (y, x). Sums o sets, sums o unctions I S = E i then : P(E i) = P(S), whose inverse S R R I = (I i R(i)) depends on I. In particular or two sets E and F we have (P(F )) E = P(E F ). The sum over I o unctions i where i I, E i = Dom i is deined by i = (S (i, x) i (x)) = i (Dom = S i I, i = (i) = j i ) where j i = (E i x (i, x)) Thus the canonical bijections (bicanonical i I = Dom S, thus i E in the case I E) F Ei = F S F (i,x) = F c x E i (F E ) I = F I E ( i ) (F E ) I c S (E F ) G E F G Gr i P(I (E F )) P((I E) F ) Gr i 8

Product o unctions or recurrying Transposing a relation R exchanges its curried orms R and R, by a bijection (P(F )) E (P(E)) F with parameter F. Similarly we have a bijection (F E ) I (F I ) E, canonical i I (to let (F E ) I determine E), deined by a binder called the product o the unctions i F E or i I: ( i ) F E i = ( F i ) E i = (E x ( i (x)) ), = i ( : E Im i i I, i = π i ) Dom = Dom g = E g = (E x ((x), g(x))) I E F E (I F ) E F i ) E. φ I E x E F φ(x) ( 2.9. Equivalence relations and partitions Indexed partitions A amily o sets (A i ) is called pairwise disjoint when any pair o them is disjoint : i, j I, i j A i A j = Equivalently, ( (i, x), (j, y) k I A k, x = y i = j), thus, Gr = t A i with Dom = Im A i = A i Im = {i I A i } Inversely, any F E deines a amily F = ( (y)) y F P(E) F o pairwise disjoint sets : y, z F, (y) (z) x (y) (z), y = (x) = z An indexed partition o a set E is a amily o nonempty, pairwise disjoint subsets o E, whose union is E. It is always injective : i, j I, A i = A j A i A j = A i i = j. Equivalence relation associated with a unction An equivalence relation is a symmetric preorder. Any : E F deines an equivalence relation on E by = {(x, y) E (x) = (y)} = ( ) Its composite g = h with any h G F satisies g, with = g Inj h Im = (h Im ) 1 g In particular, coincides with the equality relation Gr Id E on E when is injective. As =, the injectivity o the indexed partition Im (that we will abusively denote as ) gives the characteristic identity o equivalence relations : x y (x) = (y). When F = Im, this map G F h h {g G E g } is not only injective (by 2.6), but bijective. Indeed, with H = Im( g), ( g (y, z), (y, z ) H, y = y z = z ) Dom H = F ( h G F, g = h H Gr h). For any unctions, g such that Dom = Dom g, the unction with graph Im( g) is g called the quotient g/ : Im Im g, and is the only unction h such that Dom h = Im g = h. Inversion comes as a particular case: Inj 1 = Id Dom /. Remark. i R is relexive and x, y, z, (x R y z R y) z R x then R is an equivalence relation. Proo : symmetry is veriied as: x, y, (x R y y R y) y R x. Then comes transitivity. 9

Partition, canonical surjection A partition o E is a set o nonempty, pairwise disjoint sets whose union is E, thus the image o any indexed partition o E (where is any unction with domain E): P = Im = Im( ). For any binary relation R on E, i P = Im R E P(E) then ( x, y E, x R(y) R(x) = R(y)) ( x E, A P, x A R(x) = A) Id P = R Thus i R is an equivalence relation then P is a partition. Conversely or any partition P o E, the indexed partition Id P o E is = g or a unique g = R E P E. Then, P = Dom g = Im g thus R is an equivalence relation (deinable by x R y ( A P, x A y A)). The partition Im R associated with an equivalence relation R on E is called the quotient o E by R and denoted E/R; and the unction R is called canonical surjection rom E to E/R. For all x E, the element R(x), only element de E/R containing x, is called the class o x by R. Order quotient o a preorder Any preordered set (E, R) is relected through R by the ordered set (Im R, ), with R(x) = R(y) ( R(x) R(y) R(y) R(x)) (x R y y R x) so that R is injective i and only i R is an order; and the relation in Im R is a copy o the order quotient o the preorder R on the set E/(R t R). On each (ordered) set E, only one order will usually be considered, denoted E, or abusively. This may be justiied by deining ordered sets as sets o sets, ordered by, ignoring their elements. 2.10. Axiom o choice Let us review dierent equivalent orms o this claim, that is called an axiom as it cannot be deduced rom other axioms o set theory but it eels true and can conveniently be added as an axiom (that does not increase the risk o contractions but leads to some interesting theorems) ; but we will actually not need to do so in the rest o this work. Axiom o choice (AC). It says Set X, AC X, where AC X names the ollowing equivalent claims (1) Any product over X o nonempty sets is nonempty : ( x X, A x ) x X A x. (2) Set E, R X E, ( x X, y E, xry) ( E X, x X, xr(x)). Or in short : or any graph R, X = Dom R (Im R) X, Gr R. (3) Fnc g, Im g = X (Dom g) X, g = Id X. Proo o equivalence : (2) (1) by R = x X A x ; (1) (3) by A x = g (x) ; (3) (2) Im π 0 R = Dom R = X (h ) R X, h = Id X Gr R. (We also have (1) (2) by A x = R(x), and (2) (3) by R = t Gr g) Theorem. Each o the ollowing claims is equivalent to the axiom o choice: (4) For any set E o sets, / E ( A E A). (5) For any partition P o a set E, K E, A P,! : K A (6) For any sets E, F, G and any g : F G, {g F E } = G E. Proos: (1) (4) is obvious ; (4) (5) (x A P A K = Im x) (K E A P, x A A B P, x B A B A = B) (5) (3) Let P = Im g. Then = (X x ɛ(k g (x))) = g 1 K g = Id X. (AC E 2) (6) h G E, ( x E, y F, g(y) = h(x)) ( F E, x E, g((x)) = h(x)) (AC G 3) (6) : i F G, g i = Id G h G E, i h F E g i h = h. (6) (3) : E = G Id E {g F E }. Remarks: (4) (1) is also easy : / {A i i I} = E, then A E A ((A i)) A i. (6) has a converse : (Dom g = F E {g F E } = G E ) Im g = G. For explicitly inite X such as X = V n, a direct proo o AC X can be written with one variable per element o X. A general proo o the more abstract (X is inite AC X ) will be seen in text 5. 10

2.11. Notion o Galois connection The set o ixed points o a unction is written Fix = {x Dom (x) = x} Im. Then, Im Fix g ((Im Dom g) (g = )) Im = Fix ((Im Dom ) ( = )) : such a unction is called idempotent. Deinition. For any ordered sets E, F, and F = F with the transposed order, the sets o antitone Galois connections between E and F, and monotone Galois connections rom E to F, are deined by Gal(E, F ) = {(, ) F E E F x E, y F, x E (y) y F (x)} = t Gal(F, E) Gal + (E, F ) = {(u, v) F E E F x E, y F, x E v(y) u(x) F y} = Gal(E, F ) Fundamental example. Any relation R X Y deines a (, ) Gal(P(X), P(Y )) by A X, (A) = {y Y x A, x R y} = R(x) x A B Y, (B) = {x X y B, x R y} = {x X B R(x)} ( ) = Y ( ) = X Proo : A X, B F, A (B) ( x A, y B, x R y) B (A). This will later be shown to be a bijection : Gal(P(X), P(Y )) = P(X Y ). Lemma. F E,! E F, (, ) Gal(E, F ). Proo: E F, (, ) Gal(E, F ) E = F, but E is injective. Properties. For all (, ) Gal(E, F ), the closures Cl = E E and Cl = F F satisy 1) Cl and Cl are extensive. 2) and are antitone 3) Cl and Cl are monotone 4) =, and similarly = 5) Im = Im Cl = Fix Cl, called the set o closed elements o E 6) Cl Cl = Cl 7) ( strictly antitone) Inj Cl = Id E Im = E 8) x, x E, (x) (x ) (Im (x) (x )). 9) Denoting K = Im, K = Id K thus K is strictly antitone and K 1 = Im. Proos: 1) (x) (x) x ( (x)). 1) 2) : x, y E, x y ( (y)) (y) (x). 1) 2) 4) : Id E Cl Cl Cl =. 4) 5) : Cl = Im Cl Im. Then, Cl = Im Fix Cl Im Cl. Deductions 2) 3) and 4) 6) are obvious. 7) (Inj Cl = ) Cl = Id E (Im = E Cl = ) Cl = Id E strictly antitone. 8) (x) (x ) ( y F, x (y) x (y)). 9) K = Fix( ) Id K = Id K. Other proo : ( K, ) Gal(K, F ) with surjective. Finally, in Im K = Id K the roles o and are symmetrical. Remark. Properties 1) and 2) conversely imply that (, ) Gal(E, F ). Indeed, x E, y F, x (y) y ( (y)) (x). The properties o monotone Galois connections are deduced by reversing the order in F. instance i (u, v) Gal + (E, F ) then u and v are monotone, v u is extensive and u v Id F. Closure. A closure o an ordered set E is an E E such that, equivalently: 1) There exists a set F and an (u, v) Gal(E, F ) or Gal + (E, F ) such that v u = 2) is monotone, idempotent and extensive 3) x E, y Im, x y (x) y, i.e. (, Id K ) Gal + (E, K) where K = Im. Proo : 3) 1) 2); or 2) 3), x E, y K, x (x) y x y (x) (y) = y. Notes: The 2) 3) is a particular case o the above remark. Indeed, = Id K Id K (extensivity o the closure in K ). K E, K E, (, Id K ) Gal + (E, K) Im = K according to 7) with Id K injective. 11 For