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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), A ar(f ) → |A| . - for each f ∈ Σ , a function f : |A| Defn: A Σ -algebra homomorphism h : A → B is a function h : |A| → |B| such that h(f A (a1 , . . . , an )) = f B (h(a1 ), . . . , h(an )) for each f ∈ Σ and a1 , . . . , an ∈ |A|. Fact: Σ -algebras and their morphisms form a category Σ-alg . 1 Terms, equations, theories Defn: The set TΣ X of Σ -terms over X is defined by induction: - if x ∈ X then x is a term, - if f ∈ Σ and t1 , . . . , tar(f ) are terms then f (t1 , . . . , tar(f ) ) is a term. Fact: for a Σ -algebra A , ! every g : X → |A| extends uniquely to g : TΣ X → |A| . 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 T if every equation in T holds in A . Defn: T -alg - the full subcategory of Σ-alg with models of T as objects 2 Generated and free algebras Defn: An algebra A is generated by g : X → |A| if ∀a ∈ |A|. ∃t ∈ TΣ X. g (t) = a ! Defn: A T -algebra A is free over X (with g : X → |A| ) if: - A is generated by g , “no junk” - no equations hold in A except those provable from T . “no confusion ” Fact: For every T and X , a free T -algebra over X exists. Fact: If A is free over X (with g : X → |A| ) then for any T -algebra B , every h : X → |B| extends uniquely 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 ηX : X → GA in C (the unit arrow) such that for every B in D with an arrow f : X → GB ! ! f : A → B there exists a unique arrow s.t. Gf ◦ ηX = f . C! ηX ! GA X! !! !! !! Gf ! f !" # GB G D A ∃! f ! ! B 6