Algebraic and Locally Algebraic Functors
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Veuillez prendre connaissance des accompany this thesis. formules d'autorisation qui accompagnent cette thkse. - THIS DISSERTATION LA THESE A ETE HAS BEEN MICROFILMED - MICROFCLMEETELLE QUE EXACTLY AS RECEIVED NOUS L'AYONS RECUE- Ottawa, Canzda KIA ON4 r+ ALGEBRAIC Al?D LOCRUY WX;EBRAIC FUNCTORS by James J. Dukarm B.A., Michigan State University, 1969 M.S. , St. ?4ary1sUniversity (Texas) , 1973 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREL?ENTS FOR THE DEGREE OF DOerOR OF PHILOSOPHY D in the Department of Nathematics @ James J. Warm 1980 s i SIMON FRASER UNIVERSITY . April 1980 All ;ights reserved. This thesis may not be reproduced in whole or in part, by photocopy or other means, without permission of the author. I hereby grant to Simon Eraser Unfversity the right to lend my thesis or dissertation (the title of which is shown below) to users of.the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library _ of any other university, or other educational institution, on its 'own behalf or for one of its users. I further agree that permission for multiple copying of this thesis for scholarly purposes may be granted by me or the Dean of Graduate Studies. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission Title of Thesis/Dissertation: ALGERRAIC. A LOCALLY &.GEi%?~ir Fd~c7oKS Author : (signature ) (name ) (date J Degree: Doctor of Philosophy Title of Thesis:* Algebraic and Locally Algebraic Furxtors Examining Committee: Chairman: S. K.. Tnomason 5. H. Lachlan Senior Supervisor B, S. Thomson A. R. Freedman B. Banaschewski External Examiner Professor Department of Mathematics Mcllaster University (ii) \ Many zl~ebrsicconstncticns cm be vieved as algebra-valued functors. Using a category-theoretic formulation of universal algebra- .-., and first-order iqic ~riginatedby F. W. Lawvere, we obtain algebraic and lo-gical results concerning functors which correspond to important kinds,of algebraic constructions--in particular, to Boolean powers and E bounded Boolean. powers. The notion of an equational interpretation of an equational theory T' in an equational l$eory T is'introduced and shown to be the syntactical counterpart to coalgebras. By means of equational interpre- a tations T' -+ T, the represehtable functors MocUTf --+ ModfTff are shwm to be obtainable as TI-algebras defined -"within" the underlying-set . functor UT: Mod (T) +Set when UT is treated as a T-algebra in 1 .=om-,c,scnt tfT,.G is ;r,sna2ic, are characterized sMlarly. The latter result is associatefi with a sp.tactica1 characterization of L 211 the equational ~~eoriesT' such that Mod(Tg) is equivalent as d a category to I?od!Tj. Finltary and infinitary versions of the I 3cclearl power constnrtior! are described as algebraic functors which zorres~ondto Post alge3ras in a functor category. k functor-:heoretic characterization of locally equational categories is given which is analogous to F. E. J. Linton's characterization of equational categories. The characterization of locally equatio~alcategories leads naturally to the notion of a locally algebraic functor, -Bounded Boolean powers are described as locally algebraic functors, and a new proof of T. -K. Hugs theorem characterizing the category of Boolean algebras as a locally equational category is sketched. My superrisor, Prof. Alistair Lachlan, has earned my deep - 5 appreciation and thanks for his tireless help and friendship aver the 2' last several years. Very special thanks also go to Doc. dr. Antoni t Wiweger of the Mathematics Institute of the Polish Academy of Sciences, Warsaw, for his valuable-help idencouragement during my stays in Warsaw in 1975-76 and in 1977. Prof. Helena Rasiowa arranged my extended stay in Warsaw in 1975-76 and supervised my work at Warsaw Gniversity, where I began to be a mathematician instead of a student largely as a result of opportunities which she provided. Countless other young mathematicians have been helped similarly by'Rasiowals -- -- - - - - - -- - - ppppp- --p gesture of appreciation for her help Ad example to all of us. The content of this thesis has been affected significantly by all three of the individnafs named above. Prof. Lachlan convinced ~se#at locally equational classes were worthy of investigation and - also encouraged and helped my study of Bmlean pwers. Docent Wiweger - also played an inqmrtant role in directing my attention to categorially * ----- - - -- - - interesting aspects of 3uulean pwers, and he provided much useful * criticism and many very helpful suggestions conce~i?gmy work on those functors and other algebraic problems. Prof. Rasiowa suggested the relevance of Cat-Ho Nguyen's generalized Post algebras to my work on -lean powers. Thanks are also offered to the many persuns uko cmtributed references, advice, and opportunities for discussing my research, -, Prauinent ammg tfiese ~feare 8- maschewski, M. Barr, S- Burris, fviif APPWAL ........ ABSTRACT ........ -- DEDICATION ....... ACKUOWLEDGEMENTS .... ?a?= 8P 4xxwEF%r *-. CXAFTER 1. IMT3tODUCTIa c33AFxR 2. FUNDmALS LIST OF REFERENCES ...................... (IfAPTER 1. INTRODUCTION , The revalued Post algebras, for finite m22, were intrcduced by P, C. msenblcan f381 as "many-valued" analorjues to the Boolean algebras in connection with a kind of many-valued propositional logic which was designed to h functionally complete with respect to a ... semantics of %valued truth tables. It was natnral for logiciks and algebraists to try to detennine the extent and nature of the * similarities between the Post algebras and the Boolean algebras. In search of "Boolean" properties of Post algebras, p-rings, and other -. analogues to Boolean algebras, A. L. Poster invented the Boolean Ijower construction and the notion of a norrmal subdirect-- power.* One of his a results was the following the~-- (A. L. Foster 1141, -- 0 A 8 - - - P i-- 0 . ' algebra. The f$llowjng classes of dlgebrak are coextensive: . - , * t i) ' The equational class generated' iij The class of all iskhic copies of noriadl sdxlipcrt d " Here, a primal algebra (in Foster's terminol6gy, a "functionally strictly cuq~lete"finite algebra) is a finite universal algebra A; having at least two elements, such that, each finite n, every firnceon. ---+ A is a polyncaial of A. Faster explicitly recqnired that the ael* wvalued Pest algCbra was primal, and in his proof he showed that the Boolean algebra involved in canstrucfiing any given algebra in a primally-generated proof amtshws that the category of Boolean algebras is equivalent to the equational cateqary (with algebras as objects ** 3 P"' *. a arrows) generated by any primal algebra, with the Boolean power , construction providing the equivalence functor. In a s&went paper fl6], Foster generalized the ~00lean power constructim, defining "hunded" Boolean powers of an infinite algebra and shming that the bounded Boolean power construction enjoyed 2roperties.vhich were "lucafized" versiona of corresponding properties of the original ~dleanpower construction. Much of Foster's work concerning Boolean powers bounded Boolean powers was tidied up and 1 - extended by M. I. Go-dd andPG. dtzer El81 , with most of the material # of the latter paper appearing in ~rztzer'sbook [19]. Foster's notions of "local" properties were studied and refined, in a succession of papers by T. K. liu, most notably 1223, 't233, 1241, and f251 , in which Bu formufated the definition of a locally equational keginning of Cha-der 5 of this thesis. The next theorem, one of Hu's -st sipificmt results, seems t3 have remained Sanewhat obscure, possibly because it never appeared in print with both a correct proof . e and the final definition of a locally equational class. s A finitary universal algebra A having at least two elements - --- A - -- is locally primal (see 1163) if, for every finite n and every finite subset 'X of A, each partial function X" -A- is the restriction . - of a polynomial -A of A-. -1.2. Theorem (T. K. Hu 1221, 1241, [251). Let K be a locally equational class of finitary universal algebras, regarded as a category whose arrows are all homomorphisms between algebras of K. Then K is equivalent to the category of Boolean algebras if and only 9b if K is generated as a locally equational class by a locally primal 3 algebra.