Introduction to universal algebra Defn: signature Σ = - a set Σ of operation symbols - an arity function ar : Σ N . → Defn: A Σ -algebra (or a model for Σ ) A is: - a set A (the carrier), | | - for each f Σ , a function f A : A ar( f ) A . ∈ | | → | | Defn: A Σ -algebra homomorphism h : A B → is a function h : A B such that | | → | | A B h(f (a1, . , an)) = f (h(a1), . , h(an)) for each f Σ and a , . , a A . ∈ 1 n ∈ | | Fact: Σ -algebras and their morphisms form a category Σ -alg . 1 Terms, equations, theories Defn: The set T Σ X of Σ -terms over X is defined by induction: - if x X then x is a term, ∈ - if f Σ and t 1 , . , t are terms then ∈ ar(f) f(t1, . , tar(f)) is a term. Fact: for a Σ -algebra A , every g : X A extends uniquely to g ! : T X A . → | | Σ → | | Defn: A Σ -equation is a pair of Σ -terms, written s = t . A theory is a set of equations. Defn: An equation s = t holds in an algebra A if for every g : X A there is g ! ( s )= g ! ( t ) . → | | Defn: An algebra A is a model (algebra) of a theory if every equation in holds in A . T Defn: -alg - the full subcategoryT of Σ -alg with modelsT of as objects T 2 Generated and free algebras Defn: An algebra A is generated by g : X A if → | | a A . t T X. g!(t)=a ∀ ∈ | | ∃ ∈ Σ Defn: A -algebra A is free over X (with g : X A ) if: T → | | - A is generated by g , “no junk” - no equations hold in A except those provable from . T “no confusion ” Fact: For every and X , a free -algebra over X exists. T T Fact: If A is free over X (with g : X A ) then for any → | | -algebra B , every h : X B extends uniquely T → | | to a homomorphism from A to B . 3 Free monoids Defn. The free monoid over a set X is the set of finite sequences: n X∗ = X n !∈N with concatenation as multiplication and ! = as unit. !" Defn. The free commutative monoid over X is the set of functions f : X N → with pointwise addition as multiplication and the constantly zero function as unit. Similarly: free groups, free rings, free lattices etc. 4 Freedom vs. initiality Roughly: Monoid M is initial if for every monoid N there is a unique monoid morphism from M to N . Monoid M is free over a set X if every interpretation of X to a monoid N extends to a unique monoid morphism from M to N . For example, the monoid free over is initial. ∅ 5 Free objects Consider a functor G : D C . → Defn. Given an object X in C , a free object over X w.r.t. G is an object A in D with an arrow η X : X GA in C → (the unit arrow) such that for every B in D with an arrow f : X GB → there exists a unique arrow f ! : A B s.t. Gf ! η = f . → ◦ X G CDo ηX X / GA A C CC CC Gf ! ! f ! f CC ∃ C! GB B 6.