Lesson 15: Relations and algebras Compiled by: Ondřej Kohut (within the Theory of formal systems course)

Relace a algebry2 Contents  Theory of sets (revision)  Relations and mappings (revision) –Relations –Binary relations on a set –Mappings –Partitions, equivalences –Orderings  Algebras –Algebras with one operation –Algebras with two operations –Lattices

Relace a algebry3 Naive theory of sets  Language: Special symbols: –Binary predicates:  (is an element of),  (is a proper subset of),  (is a subset of) –Binary function symbols:  (intersection),  (union)  Cantor – the naive set theory (without axiomatization)  There are many formal axiomatizations, but none of them is complete.  Examples: von Neumann-Bernays-Gödel, Zermelo-Fränkel + axiom of choice

Relace a algebry4 Zermelo-Fränkel set-theory Axiom of extensionalityAxiom of extensionality: Two sets are the same if and only if they have the same elements. Axiom of empty setAxiom of empty set: There is a set with no elements. Axiom of pairingAxiom of pairing: If x, y are sets, then so is {x,y}, a set containing x and y as its only elements. Axiom of unionAxiom of union: Every set has a union. That is, for any set x there is a set y whose elements are precisely the elements of the elements of x.union Axiom of infinityAxiom of infinity: There exists a set x such that {} is in x and whenever y is in x, so is the union y U {y}. Axiom of separationAxiom of separation (or subset axiom): Given any set and any proposition P(x), there is a subset of the original set containing precisely those elements x for which P(x) holds.propositionsubset Axiom of replacementAxiom of replacement: Given any set and any mapping, formally defined as a proposition P(x,y) where P(x,y) and P(x,z) implies y = z, there is a set containing precisely the images of the original set's elements.mapping Axiom of power setAxiom of power set: Every set has a power set. That is, for any set x there exists a set y, such that the elements of y are precisely the subsets of x.power set Axiom of regularityAxiom of regularity (or axiom of foundation): Every non-empty set x contains some element y such that x and y are disjoint sets.disjoint sets Axiom of choiceAxiom of choice: (Zermelo's version) Given a set x of mutually disjoint nonempty sets, there is a set y (a choice set for x) containing exactly one element from each member of x.

Relace a algebry5 Naive theory of sets  Ø – an empty set  Cardinality of a set A: |A| Relations between sets (axioms):  Equality  Inclusion

Relace a algebry6 Naive theory of sets – set theoretical operations  Intersection  Union  Difference  Symetrical difference  Complement with respect to universe U

Relace a algebry7 Naive theory of sets – set theoretical operations  Potential set  Cartesian product  Cartesian power A 1 =A, A 0 ={ Ø } n

Relace a algebry8 Relations  n-ary relation between the sets A 1, A 2,..., A n  Examples : –D = a set of possible days –M = a set of VŠB rooms –Z = a set of VŠB employees A ternary relation meeting (when, where, who):

Relace a algebry9 Binary relations  Inverse relation to r:  Composition of relations

Relace a algebry10 Binary relations A binary relation r on a set A is called:  Reflexive:  x  A: (x,x)  r  Irreflexive:  x  A: (x,x)  r  Symmetric:  x,y  A: (x,y)  r  (y,x)  r  Antisymmetric:  x,y  A: (x,y)  r and (y,x)  r  x=y  Asymmetric:  x,y  A: (x,y)  r  (y,x)  r  Transitive:  x,y,z  A: (x,y)  r and (y,z)  r  (x,z)  r  Cyclic:  x,y,z  A: (x,y)  r and (y,z)  r  (z,x)  r  Linear:  x,y  A: x=y or (x,y)  r or (y,x)  r

Relace a algebry11 Binary relations The important types of binary relations:  Tolerance – reflexive, symmetric  Quasi-ordering – reflexive, transitive  Equivalence – reflexive, symmetric, transitive  Partial ordering – reflexive, antisymmetric, transitive

Relace a algebry12 Binary relations Examples:  Tolerance: – „to be akin to“ on a set of people, – „to have a different age no more than one year“ on the set of people,...  Quasi-ordering: –„if it holds |X|  |Y|, then sets X and Y are in relation“ on a set of sets, – divisibility relation on a set of integers, –„not to be older“ on the set of people,...  Equivalence: – „to be the same age“ on the set of people, – equivalence on a set of natural numbers,...  Ordering: –inclusion relation, –divisibility relation on the set of natural numbers,...

Relace a algebry13 Mappings (functions)  f  A  B is called a mapping from a set A into a set B (partial mapping), iff: (  x  A,  y 1,y 2  B) ( (x,y 1 )  f and (x,y 2 )  f  y 1 = y 2 )  f is called a mapping of a set A into a set B (total mapping, written as: f: A  B), iff: –f is mapping from A to B –(  x  A)(  y  A) ( (x,y)  f)  If f is a mapping and (x,y)  f, then we write: f(x)=y

Relace a algebry14 Mapping (functions)  Examples: u = {(x,y)  Z  Z; x=y 2 },v = {(x,y)  N  N; x=y 2 }, w = {(x,y)  Z  Z; y=x 2 }  r, u – are not mappings  s, v – are partial mappings from A to B, (not total mappings)  t, w – are total mappings r  A  Br  A  Bs  A  Bs  A  Bt  A  Bt  A  B

Relace a algebry15 Mapping (functions) Mapping f: A  B is called  Injection (one to one mapping A into B), iff:  x 1,x 2  A,  y  B: (x 1,y)  f and (x 2,y)  f  x 1 = x 2  Surjection (mapping A onto B), iff:  y  B  x  A: (x,y)  f  Bijection (one to one mapping A onto B) (mutually single-valued), iff it is an injection and surjection.

Relace a algebry16 Mapping (functions)  Examples: j: Z  Z, j(n)=n 2,k: Z  N, k(n)=|n|, l: N  N, l(n)=n+1,m: R  R, j(x)=x 3  f, j – is neither an injection nor a surjection  h, k – are surjections, but not injections  g, l – are injections, but not surjections  I,m – is an injection and a surjection  bijections f : A  Bg : A  Bh : A  Bi : A  B

Relace a algebry17 Partitions and equivalences  Partition on a set A is such a system that: X = { X i ; i  I } –X i  A pro  i  I –X i  X j = Ø pro  i,j  I, i  j –U X = A X i – classes of the partition  Refininment of a partition X = { X i ; i  I } is a system that: Y = { Y j ; j  J }, iff: –  j  J,  i  I so that Y j  X i

Relace a algebry18 Partitions and equivalences  Let r be an equivalence relation on a set A, X is a partition on A, then it holds:  X r = {[x] r ; x  A} – the partition on A (the partition induced by equivalence r, the factor set of the set A according to the equivalence r)  r X = {(x,y); x and y belongs to the same class of the partition X} – equivalence on A (induced by partition X) Examples:  r  Z  Z;r = {(x,y); 3 divides x-y }  X={X 1, X 2, X 3 }  X 1 ={…-6, -3, 0, 3, 6, …}  X 2 ={…-5, -2, 1, 4, 7, …}  X 3 ={…-4, -1, 2, 5, 8, …}

Relace a algebry19 Orderings  If r is an order relation on A, then a couple (A,r) is called an ordered set  Written as: (A,  ) –Examples: (N,  ), (2 M,  )  Cover relation Let (A,  ) be an ordered set, (a,b)  A a –< b („b covers a“), iff: a < b and  c  A: a  c a c  b –Examples: (N,  ), –< ={(n,n+1); n  N}

Relace a algebry20 Orderings  Hasse diagram – graphical picturing –Example: (A,  ), A={a,b,c,d,e} r = {(a,b), (a,c), (a,d), (b,d)}  id A id A ={(a,a): a  A}

Relace a algebry21 Orderings An element a of an ordered set (A,  ) is called:  The least: for  x  A: a  x  The greatest: for  x  A: x  a  Minimal: for  x  A: (x  a  x = a)  Maximal: for  x  A: (a  x  x = a)  Examples: –The least: does not exist –The greatest: does not exist –Minimal: a, e –Maximal: d, c, e

Relace a algebry22 Orderings  A mapping of the ordered sets (A,  ), (B,  ) is called isomorphic, iff the bijection f: A  B exists such that:  x,y  A: x  y, iff f(x)  f(y)  A mapping of the ordered sets (A,  ), (B,  ) is called isotone f: A  B, when it holds:  x,y  A: x  y  f(x)  f(y)  Examples:  f: N  Z, f(x)=kx, k  Z, k  0 is isotone  g: N  Z, g(x)=kx, k  Z, k  0 is not isotone

Relace a algebry23 Orderings  Let (A,  ) be an ordered set, M  A, then  L A (M)={x  A;  m  M: x  m } –A set of lower bounds  U A (M)={x  A;  m  M: m  x } –A set of upper bounds  Inf A (M) – The greatest element of the set L A (M) –Infimum of the set M  Sup A (M) – The least element of the set U A (M) –Supremum of the set M

Relace a algebry24 Lattices - lattice ordered sets  A set (A,  ) is called a lattice (lattice ordered set), iff:  x,y  A  s,i  A : s = sup({x,y}), i = inf({x,y})  Notation: –x  y = sup({x,y}) –x  y = inf({x,y})  If sup(M) a inf(M) exist for every M  A, then (A,  ) is called a complete lattice

Relace a algebry25 Algebras Algebra (abstract algebra) is couple: (A, F A ):  A  Ø – an underlying set of algebra  F A = {f i : A p(fi)  A; i  I} – a set of operations on A  p(f i ) – an arity of operation f i  Examples: –(N, + 2,  2 ) the set of natural numbers with the addition and multiplication operations –(2 M, ,  ) the set of all subsets of a set M with the intersection and union operations –(F, ,  ) The set (F) of the propositional logic formulas with the conjunction and disjunction operations

Relace a algebry26 Algebras with one binary operation Grupoid G=(G,  )   : G  G  G  If a set G is finite, then the grupoid G is called finite  Order of a grupoid = |G| Examples of grupoids:  G 1 =(R,+), G 2 =(R,  ), G 3 =(N,+)...

Relace a algebry27 Algebras with one binary operation  We can express the finite grupoid G=(G,  ) by Cayley table  Example:  G = {a,b,c}  For example: a  b = b, b  a = a, c  c = b...  abc aabc bacc caab

Relace a algebry28 Algebras with one binary operation Let G=(G,  ) is grupoid, G is called:  Commutative, if it holds: –(  a,b  G)(a  b = b  a)  Associative, if it holds: –(  a,b,c  G)((a  b)  c = a  (b  c))  With a neutral element, if it holds: –(  e  G  a  G)(a  e = a = e  a)  With an aggressive element, if it holds: –(  o  G  a  G)(a  o = o = o  a)  With inverse elements, if it holds: –(  a  G  b  G)(a  b = e = b  a)

Relace a algebry29 Algebras with one binary operation Examples:  (R,  ), (N,+) – commutative and associative  (R,  ), a  b = (a+b) / 2 – commutative, not associative  (R,  ), a  b = a b – neither commutative nor associative  (R,  ) – 1 = the neutral element, 0 = the aggressive element

Relace a algebry30 Algebras with one binary operation Let G=(G,  G ) be a grupoid. H  G is called closed (with respect to the operation  G ), if it holds:  (  a,b  H)(a  G b  H) A Grupoid H=(H,  H ) is a subgrupoid of a grupoid G=(G,  G ), if it holds:  Ø  H  G is closed   a,b  H: a  H b = a  G b  Examples:  (N,+ N ) is a subgrupoid of (Z,+ Z )  {0,1,2} is not the base set of a podgrupoid (Z,+ Z )

Relace a algebry31 Algebras with one binary operation  Let G 1 =(G 1,  1 ), G 2 =(G 2,  2 ) be a grupoids.  G 1  G 2 =(G 1  G 2,  ) – direct product G 1 and G 2, where:  (a 1, a 2 )  (b 1, b 2 ) = (a 1  1 b 1, a 2  2 b 2 ) Examples:  G 1 =(Z,+), G 2 =(Z,  ).  G 1  G 2 =(Z  Z,  ),  (a 1, a 2 )  (b 1, b 2 ) = (a 1 + b 1, a 2  b 2 )  (1,2)(3,4) = (1+3, 2  4) = (4,8) and so on.

Relace a algebry32 Algebras with one binary operation  Let G =(G,  G ), H =(H,  H ) be grupoids and h:G  H be a mapping.  h is called homomorphism of grupoid G into grupoid H, if it holds:   a,b  G: h(a  G b) = h(a)  H h(b) The types of homomorphism:  Monomorphism – h is injective  Epimorphism – h is surjective  Isomorphism – h is bijective  Endomorphism – H=G  Automorphism – is bijective and H=G Examples:  (R,+), h(x)= -x, h is automorfismus (R,+) into itself  h(x+y) = -(x+y) = (-x) + (-y) = h(x) + h(y)

Relace a algebry33 Algebras with one binary operation r is called a congruence on a grupoid G=(G,  G ), iff:  r is a binary relation: θ  G  G  r is an equivalence  (a 1, a 2 ), (b 1, b 2 )  r  (a 1  G b 1, a 2  G b 2 )  r A factor grupoid of grupoid G according to the congruence r: G/r=(G/r,  G/r ), [a] r  G/r [b] r = [a  G b] r Examples:  r  Z  Z;r = {(x,y); 3 divides x-y }  r is a congruence on (Z,+)  [0][0][1][1][2][2] [0][0][0][0][1][1][2][2] [1][1][1][1][2][2][0][0] [2][2][2][2][0][0][1][1]

Relace a algebry34 Algebras with one binary operation The types of grupoids:  Semigroup – an associative grupoid  Monoid – a semigroup with the neutral element  Group – a monoid with the inverse elements  Abelian group – a commutative group Examples:  (Z, –) – grupoid, not a semigroup  (N – {0}, +) – semigroup, not a monoid  (N,  ) – monoid, not a group  (Z, +) – Abelian group

Relace a algebry35 Algebras with two binary operation Algebra (A,+,·) is called a Ring, if it holds:  (A,+) is commutative group  (A,·) is monoid  For  a,b,c  A it holds: a·(b+c)=a·b+a·c, (b+c)·a=b·a + c·a  If |A|>1, then (A,+,·) is called a non-trivial ring.  Let 0  A is the neutral element of group (A,+). Then 0 is called the ring zero (A,+,·).  Let 1  A is the neutral element of monoid (A,·). Then 1 is called the ring unit (A,+,·).

Relace a algebry36 Algebras with two binary operation A ring (A,+,·) is called a field, if it holds:  (A - {0},·) is a commutative group Examples:  (Z,+,·) – a ring, not a field  (R,+,·), (C,+,·) – fields

Relace a algebry37 Lattice – algebraic structure  Lattice L = (L, ,  )   : L  L  L,  : L  L  L   x, y, z  L it holds: x  x = xx  x = x idempotention x  y = y  xx  y = y  x commutativity x  (y  z) = (x  y)  zx  (y  z) = (x  y)  z associativity x  (x  y) = xx  (x  y) = x absorption

Relace a algebry38 Lattice – algebraic structure Let (A, ,  ) be a lattice, (B,  ) be a lattice ordered set  Let us define a relation   on A : a   b, iff a  b = b  Let us define the relations   and   on B a   b = sup{a,b}, a   b = inf{a,b}, Then it holds:  (A,  ) is a lattice ordered set, where: sup{a,b}= a  b, inf{a,b}= a  b  (B,  ,   ) is a lattice  (A, ,  ) = (A,  ,   )

Relace a algebry39 Lattice – algebraic structure A lattice (L, ,  ) is called:  Modular, if it holds:  x, y, z  L : x  z  x  (y  z) = (x  y)  z  Distributive, if it holds:  x, y, z  L : x  (y  z) = (x  y)  (x  z) x  (y  z) = (x  y)  (x  z)  Complementary, if it holds: : There is the least element 0  L and the greatest element 1  L  x  L  x’  L : x  x’ = 0, x  x’ = 1 x’ is called a complement of an element x

Relace a algebry40 Lattice – algebraic structure  Each distributive lattice is modular Examples:  M5 (diamond) – a modular lattice which is not distributive  N5 (pentagon) – is not modular M5N5

Relace a algebry41 Lattice – algebraic structure Lattice (L, ,  ) is called Boolean lattice, when it is:  Complementary, distributive, with the least element 0  L and with the greatest element 1  L Boolean algebra:  (L, , , –, 0, 1), – : L  L is an operation of complement in L Example:  (2 A, ,  ), A = {1,2,3,4,5,6,7}