Universal Algebra Jesse E. Jenks University of Puget Sound May 2019 Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 1 / 20
Motivation ◮ “A group is defined to consist of a nonempty set G together with a binary operation ◦ satisfying the axioms.” ◮ “A field is defined to consist of a nonempty set F together with two binary operations + and · satisfying the axioms . . . ” ◮ “A vector space is defined to consist of a nonempty set V together with a binary operation + and, for each number r , an operation called scalar multiplication such that . . . ” Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 2 / 20
Motivation How can we generalize the different structures we encounter in an abstract algebra course? Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 3 / 20
Some definitions Definition Given an equivalence relation θ on A , the equivalence class of a ∈ A is the set a /θ = { b ∈ A | � a , b � ∈ θ } . Definition The quotient set of A by θ is the set A /θ = { a /θ | a ∈ A } Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 4 / 20
Some Definitions Definition Given an equivalence relation θ on A , the canonical map is the function φ : A → A /θ where φ ( a ) = a /θ. Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 5 / 20
Some Definitions Nothing out of the ordinary so far. Definition The kernel of a function ϕ : A → B is the set Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 6 / 20
Some Definitions Nothing out of the ordinary so far. Definition The kernel of a function ϕ : A → B is the set ker( ϕ ) = {� a , b � ∈ A 2 | ϕ ( a ) = ϕ ( b ) } Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 6 / 20
Some Definitions Nothing out of the ordinary so far. Definition The kernel of a function ϕ : A → B is the set ker( ϕ ) = {� a , b � ∈ A 2 | ϕ ( a ) = ϕ ( b ) } Pros of this new definition Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 6 / 20
Some Definitions Nothing out of the ordinary so far. Definition The kernel of a function ϕ : A → B is the set ker( ϕ ) = {� a , b � ∈ A 2 | ϕ ( a ) = ϕ ( b ) } Pros of this new definition ◮ Assumes nothing about any structure on A or B . Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 6 / 20
Some Definitions Nothing out of the ordinary so far. Definition The kernel of a function ϕ : A → B is the set ker( ϕ ) = {� a , b � ∈ A 2 | ϕ ( a ) = ϕ ( b ) } Pros of this new definition ◮ Assumes nothing about any structure on A or B . ◮ The kernel is an equivalence relation Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 6 / 20
The First Theorem Theorem If ψ : A → B is a function with K = ker( ψ ) , then K is an equivalence relation on A. Let φ : A → A / K be the canonical map. Then there exists a unique bijection η : A / K → ψ ( A ) such that ψ = ηφ . ψ A B η φ A / ker( ψ ) Figure: Commutative Diagram Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 7 / 20
Algebras Definition An n -ary operation f on A is a function from A n to A , where n ≥ 0. We define A 0 to be {∅} . Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 8 / 20
Algebras Definition An n -ary operation f on A is a function from A n to A , where n ≥ 0. We define A 0 to be {∅} . Example If f is a binary operation on A = { a , b , c } . f a b c a a a a b b b b c c c c Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 8 / 20
Algebras Definition A signature F is a set of function symbols . Each symbol f ∈ F is assigned an integer called its arity . ◮ In Universal algebra signatures are sometimes called types . ◮ Sometimes signatures are defined only in terms of their arities. Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 9 / 20
Algebras Definition An algebra A with universe A and signature F is a pair � A , F � , where F is a set of functions corresponding to symbols in F . Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 10 / 20
Algebras Definition An algebra A with universe A and signature F is a pair � A , F � , where F is a set of functions corresponding to symbols in F . We distinguish between function symbols and the “actualy” function with a superscript. Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 10 / 20
Algebras Definition An algebra A with universe A and signature F is a pair � A , F � , where F is a set of functions corresponding to symbols in F . We distinguish between function symbols and the “actualy” function with a superscript. For example f ∈ F , and f A ∈ F . Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 10 / 20
Example of an Algebra The additive group Z 3 is an algebra with the signature { + , − 1 , 1 } . Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 11 / 20
Example of an Algebra The additive group Z 3 is an algebra with the signature { + , − 1 , 1 } . These symbols have arities 2, 1, and 0 respectively Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 11 / 20
Example of an Algebra The additive group Z 3 is an algebra with the signature { + , − 1 , 1 } . These symbols have arities 2, 1, and 0 respectively + Z 3 0 1 2 () − 1 Z 3 0 0 1 2 0 1 2 1 Z 3 : ∅ �→ 0 1 1 2 0 0 2 1 2 2 0 1 Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 11 / 20
Congruences A congruence is a special kind of equivalence relation. They are the equivalence relations which “respect” the operations of an algebra. ⇒ f A ( a 1 , a 2 ) ∼ f A ( b 1 , b 2 ) a 1 ∼ b 1 , a 2 ∼ b 2 = Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 12 / 20
Congruences A congruence is a special kind of equivalence relation. They are the equivalence relations which “respect” the operations of an algebra. ⇒ f A ( a 1 , a 2 ) ∼ f A ( b 1 , b 2 ) a 1 ∼ b 1 , a 2 ∼ b 2 = Example Cosets are equivalence classes of a congruence. Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 12 / 20
Congruences A congruence is a special kind of equivalence relation. They are the equivalence relations which “respect” the operations of an algebra. ⇒ f A ( a 1 , a 2 ) ∼ f A ( b 1 , b 2 ) a 1 ∼ b 1 , a 2 ∼ b 2 = Example Cosets are equivalence classes of a congruence. Consider the subgroup { 0 , 4 , 8 } of Z 12 . Define the relation a ∼ b to mean a and b are in the same coset. Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 12 / 20
Congruences A congruence is a special kind of equivalence relation. They are the equivalence relations which “respect” the operations of an algebra. ⇒ f A ( a 1 , a 2 ) ∼ f A ( b 1 , b 2 ) a 1 ∼ b 1 , a 2 ∼ b 2 = Example Cosets are equivalence classes of a congruence. Consider the subgroup { 0 , 4 , 8 } of Z 12 . Define the relation a ∼ b to mean a and b are in the same coset. Then we have a 1 = 0 ∼ 8 = b 1 a 2 = 7 ∼ 3 = b 2 f A ( a 1 , a 2 ) = (0 + 7) ∼ (8 + 3) = f A ( b 1 , b 2 ) The congruence relation is preserved under the operation + Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 12 / 20
Homomorphisms Homomorphism are defined as we would expect. Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 13 / 20
Homomorphisms Homomorphism are defined as we would expect. They are functions which “respect” the operations of an algebra. Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 13 / 20
Homomorphisms Homomorphism are defined as we would expect. They are functions which “respect” the operations of an algebra. Definition If A and B are two algebras with the same signature F , then ϕ : A → B is a homomorphism if for every n -ary function symbol f ∈ F and every a 1 , . . . , a n ∈ A , ϕ ( f A ( a 1 , . . . , a n )) = f B ( ϕ ( a 1 ) , . . . , ϕ ( a n )) . We can “push ϕ through” operations. Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 13 / 20
Congruences and Homomorphisms Theorem The canonical map of a congruence is a homomorphism Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 14 / 20
Congruences and Homomorphisms Theorem The canonical map of a congruence is a homomorphism Proof. Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 14 / 20
Congruences and Homomorphisms Theorem The canonical map of a congruence is a homomorphism Proof. (they were secretly defined that way) Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 14 / 20
Congruences and Homomorphisms This gives us a well defined notion of quotient algebras. Jesse E. Jenks (University of Puget Sound) Universal Algebra May 2019 15 / 20
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