Relations Among Algebras

Transcription

1 Itroductio to leee Algebra Lecture 6 CS786 Sprig 2004 February 9, 2004 Relatios Amog Algebras The otio of free algebra described i the previous lecture is a example of a more geeral pheomeo called adjuctio. A adjuctio is a way of describig a particular relatioship betwee categories of algebraic structures. A adjuctio typically arises whe some category C of algebras has more structure tha aother category D, but there is a caoical way to exted ay D-algebra to a C-algebra. The costructio ormally costitutes a fuctor F : D C, ad there is ormally a correspodig forgetful fuctor : C D goig i the opposite directio that forgets the extra structure. We will show that adjuctios characterize the relatioships amog the followig categories: A, the category of star-cotiuous leee algebras ad leee algebra morphisms; CS, the category of closed semirigs ad ω-cotiuous semirig morphisms (semirig morphisms that preserve suprema of coutable sets); ad SA, the category of S-algebras ad cotiuous semirig morphisms (semirig morphisms that preserve arbitrary suprema). Adjuctio Formally, let F : D C ad : C D be fuctors betwee two categories C ad D (thik of D as the category with less structure). We say that F is a left adjoit of ad that is a right adjoit of F if for ay D-algebra A ad C-algebra B, there are maps ι A : A F A ad η B : B B such that (i) for ay D-homomorphism h : A B, there is a uique C-homomorphism ĥ : FA B such that h = η B ĥ ι A; (ii) for ay C-homomorphism g : F A B, the map η B g ι A : A B is a D- homomorphism; ad (iii) for ay D-homomorphism h : A B, (F h) ι A = ι B h, ad similarly for η ad. 1

2 C F D ι A F A A ĥ h B η B B I all istaces we will cosider, is a forgetful fuctor, which meas that g is settheoretically the same fuctio as g, ad η is the idetity fuctio. I the free costructio of the last lecture, the two categories are the category A of star-cotiuous leee algebras ad leee algebra homomorphisms ad the category Set of sets ad set fuctios. The free costructio is the left adjoit of the forgetful fuctor that associates to every star-cotiuous leee algebra its uderlyig set. A F Set R Σ Reg Σ Σ ĥ h id I geeral, a category of algebras has free algebras iff the forgetful fuctor to Set, which forgets all algebraic structure, has a left adjoit. As observed, every closed semirig C gives a star-cotiuous leee algebra C by defiig x = x. Also, if h : C C is a ω-cotiuous semirig morphism betwee closed semirigs, the h must preserve, therefore is a leee algebra morphism h : S S. Similarly, every S-algebra S is a closed semirig S, ad every complete semirig morphism is ω-complete. Thus we have a forgetful fuctors : SA CS ad : CS A. I the other directio, ot every star-cotiuous leee algebra is a closed semirig. However, it is possible to costruct, i a caoical way, a closed semirig C extedig ay star-cotiuous leee algebra. Similarly, although ot every closed semirig is a S-algebra, every closed semirig ca be exteded to oe. Furthermore, ay leee algebra homomorphism h : exteds aturally to a ω-complete semirig morphism C g : C C, ad every ω-complete semirig morphism g : C C exteds to a complete semirig morphism S g : S C S C. The fuctors C : A CS ad S : CS SA are left adjoits to ad, respectively. A C CS S SA 2

3 Completio by Star-Ideals The basic costructio used here is kow as completio by star-ideals ad was used by Coway to exted a star-cotiuous leee algebra to a S-algebra [1, Theorem 1, p. 102]. Thus Coway s costructio is equivalet to the compositio S C. The costructio C, which shows that every star-cotiuous leee algebra is embedded i a closed semirig, ca be described as a completio by coutably geerated star-ideals. Defiitio 6.1 (Coway [1]) Let be a star-cotiuous leee algebra. A star-ideal is a subset I of such that I is oempty I is closed uder + I is closed dowward uder if ab c I for all 0, the ab c I. A oempty set A geerates a star-ideal I if I is the smallest star-ideal cotaiig A. We write <A> to deote the star-ideal geerated by A. A star-ideal is coutably geerated if it has a coutable geeratig set. If A is a sigleto {x}, the we abbreviate <{x}> by <x>. Such a ideal is called pricipal with geerator x. Let be a star-cotiuous leee algebra. We defie a closed semirig C as follows. The elemets of C will be the coutably geerated star-ideals of. For ay coutable set of coutably geerated star-ideals I, defie I = < I >. This ideal is coutably geerated, sice if A is coutable ad geerates I for 0, the A is coutable ad geerates I. The operator is associative, commutative, ad idempotet, sice is. For ay pair of elemets I, J, defie I J = <{ab a A, b B}>. This ideal is coutably geerated if I ad J are, sice <A> <B> = <{ab a <A>, b <B>}> = <{ab a A, b B}>, ad {ab a A, b B} is coutable if A ad B are (Exercise??). 3

4 The ideal <0> = {0} is icluded i every ideal ad is thus a additive idetity. It is also a multiplicative aihilator: <0> I = <{ab a <0>, b I}> = <{ab a {0}, b I}> = <0>. The ideal <1> is a multiplicative idetity: <1> I = <{ab a <1>, b I}> = <{ab a {1}, b I}> by Exercise?? = <I> = I. Fially, the distributive laws hold: I J = <{ab a I, b = <{ab a I, b < J }> J >}> = <{ab a I, b J }> by Exercise?? = < = < = < = {ab a I, b J }> <{ab a I, b J }>> {ab a I, b J }> I J, ad symmetrically. Closed Semirigs ad S-algebras The other half of the factorizatio of Coway s costructio embeds a arbitrary closed semirig ito a S-algebra. I compariso to the previous costructio, this costructio is much less iterestig. We give the mai costructio ad omit formal details. Recall that closed semirigs ad S-algebras are both idempotet semirigs with a ifiite summatio operator satisfyig ifiitary associativity, commutativity, idempotece, ad 4

5 distributivity laws. The oly differece is that closed semirigs allow oly coutable sums, whereas S-algebras allow arbitrary sums. Morphisms of closed semirigs are the ω-cotiuous semirig morphisms ad those of S-algebras are the cotiuous semirig morphisms. To embed a give closed semirig C i a S-algebra S C, we complete C by ideals. A ideal is a subset A C such that A is oempty A is closed uder coutable sum A is closed dowward uder. Take S C to be the set of ideals of C with the followig operatios: I α α = < I α > α I J = <{ab a I, b J}> 0 = <0> 1 = <1>. The argumets from here o are quite aalogous to those of the previous sectio. Refereces [1] Joh Horto Coway. Regular Algebra ad Fiite Machies. Chapma ad Hall, Lodo,