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Algebraic Aspects of Orthomodular Lattices

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Algebraic Aspects of Orthomodular Lattices Algebraic Asp ects of Orthomo dular Lattices Gunter Bruns McMaster University Hamilton Ontario LS K Canada John Harding New Mexico State University Las Cruces NM USA In this pap er we try to give an uptodate account of certain asp ects of the the ory of ortholattices abbreviated OLs orthomo dular lattices abbreviated OMLs and mo dular ortholattices abbreviated MOLs not hiding our own research in terests Since most of the questions we deal with have their origin in Universal Alge bra we start with a chapter discussing the basic concepts and results of Universal Algebra without pro ofs In the next three chapters we discuss mostly with pro ofs the basic results and standard techniques of the theory of OMLs In the remaining ve chapters we work our way to the b order of present day research mostly no or only sketchy pro ofs Chapter deals with pro ducts and sub direct pro ducts chap ter with free structures and chapter with classes of OLs dened by equations In chapter we discuss emb eddings of OLs into complete ones The last chap ter deals with questions originating in Category Theory mainly amalgamation epimorphisms and monomorphisms The later chapters of this pap er contain an abundance of op en problems We hop e that this will initiate further research Basic Universal Algebra We assume a denition of natural numb ers which makes every natural numb er n the set of all natural numb ers k n An nary op eration on a set A is a map f of n n A into A An element a A gives rise to a sequence a a an of A more commonly written with indices a a a The numb er n is called 0 1 n1 0 the arity of the op eration It is imp ortant to allow the case n Since A fg a ary or nullary op eration is a map of fg into A Since such a map is completely determined by the element f we usually sp ecify ary op erations also called constants by giving the element f A similar remark applies to unary ary op erations We identify them with maps from A to itself A closure system over a set A is a set C of subsets of A satisfying T If B C then B C Here we make the convention that the intersection of the empty subset of C is A so that A C If C is a closure system over A and if X A we dene the closure T X of X with resp ect to the closure system C by X fC jX C C g X is obviously the smallest element of C containing X as a subset If for B C we W S V T dene B B and B B then C b ecomes a complete lattice is the smallest and A is the largest element of this lattice An algebraic closure system is a closure system C satisfying S If K C is a chain ie totally ordered by then K C This denition one of many equivalent ones is particularly useful in pro ofs in volving Zorns lemma An algebra is a pair A A f where A is a set called the underlying i iI set or the universe of the algebra and each f is for some n an n ary op eration i i i on A The family n is called the typ e of the algebra Whereas every n i iI i is nite it is necessary in order to cover imp ortant examples to admit innitely many op erations f i A subuniverse of an algebra A A f is a set B A which is closed i iI under all op erations f ie satises i n i then f a B If a B i or in more common and more cumb ersome notation If a a a B then f a a a B 0 1 n 1 i 0 1 n 1 i i Let S ubA b e the set of all subuniverses of A The most imp ortant and in fact characteristic prop erty is Prop osition If A is an algebra then S ubA is an algebraic closure system As a consequence of this we obtain that for any X A there exists a smallest subuniverse X containing X X is said to b e the subuniverse generated by X A subalgebra of an algebra A A f is an algebra B B g i iI i iI n i ie such that B A and g is the restriction of f to B i i n i then g a f a If a B i i Clearly this requires that B is a subuniverse of A Conversely if B is a subuniverse n i then B g b ecomes a subalgebra of of A and g is the restriction of f to B i iI i i A Because of this oneone corresp ondence b etween subuniverses and subalgebras we will not b e to o fussy ab out distinguishing them Thus we refer to S ubA as the closure system or lattice of subalgebras of A Many authors require the denition of algebra and subalgebra that the underlying set b e not empty We see no reason for this Let A A f and B B g b e algebras of the same typ e A i iI i iI homomorphism of A into B is a map A B satisfying for every i I and n i a A f a g a i i or in cumb ersome notation f a a a g a a a i 0 1 n 1 i 0 1 n 1 i 1 B is said to b e a homomorphic image of A i there exists a homomorphism of A onto B An emb edding of A into B is a oneone homomorphism of A into B An isomorphism is an emb edding which is onto Closely related to homomorphisms are congruence relations A congruence relation short congruence on an algebra A A f is an equivalence i iI relation on A ie a reexive symmetric and transitive relation R satisfying n i and ak R bk k n then f aR f b If a b A i i i We express this by saying that R is compatible with the op eration f If R is an i equivalence relation on A and a A dene the equivalence congruence if R is a congruence class of a mo dulo R by aR fbjaR bg and the quotient set of A mo dulo R by AR faR ja Ag n i If R is a congruence in A we may dene op erations g in AR If a A i n i by aR k ak R Then dene g by dene aR AR i n i then g aR f aR if a A i i in cumb ersome notation g a R a R a R f a a a R i 0 1 n 1 i 0 1 n 1 i i With these op erations AR g b ecomes an algebra of the same typ e as A i iI the so called quotient algebra AR of A mo dulo R The map A AR R dened by a aR b ecomes a homomorphism of A onto AR called the R canonical homomorphism of A onto AR We are now in a p osition to establish the basic relationship b etween homo morphisms and congruences known under various names Homomorphism Theorem Let A B be a homomorphism of A into B Dene a relation R ker the kernel of by aR b i a b Then R is a congruence in A and there exists a unique map AR B such that This map is a oneone homomorphism of AR into B R A B R AR If is onto then clearly is onto hence an isomorphism Thus every homo morphic image of A is isomorphic with a quotient algebra of A It is of some imp ortance that the set C onA of all congruences in A is an algebraic closure system over A A Q A of the family consists Let A b e a family of sets The pro duct k k k K k K of all choice functions on the family ie all maps with domain K such that Q A there come maps pr of k A for all k K With every pro duct k k k k K the pro duct onto A dened by pr k The map pr is called the k th k k k k pro jection If the A A f are algebras we dene op erations f in the k k iI i i k pro duct comp onentwise ie by f k f pr The resulting algebra i k i Q A is called the pro duct of the family A In case of a pro duct of k k k K k K algebras the pr are homomorphisms k The cartesian pro duct A B of two sets is dened in basic set theory as fa bja A b B g If A A f B B g we may dene op era i iI i iI tions h in A B by h a f pr a g pr a This gives a new algebra i i i A i B A B This can b e subsumed under the pro duct dened b efore We leave out the details An imp ortant prop erty of pro ducts which in more general settings can b e used to dene pro ducts is the Extension Prop erty of Pro ducts If A A k K are algebras and for every k k K is a homomorphism of A into A then there exists a unique homomor k k Q A satisfying pr for every k phism of A into k k k k K A sub direct pro duct of a family A of algebras is a subalgebra A of the k k K Q A such that every pr maps A onto A A sub direct representation pro duct k k k k K Q A such that the of an algebra A is an emb edding of A into a pro duct k k K maps pr map A onto A An algebra A is said to b e sub directly irreducible i k k Q for every sub direct representation A A one of the maps pr is one k k k K one hence an isomorphism The fundamental result ab out sub directly irreducibles is Birkho s Sub direct Representation Theorem Every algebra is isomorphic with a subdirect product of subdirectly irreducible algebras Sub direct irreducibility of an algebra has a neat and very useful character ization in terms of congruences The smallest congruence of an algebra A A f is obviously the diagonal fa aja Ag An algebra A i iI A A f is sub directly irreducible i there is a smallest congruence dierent i iI from A Let K b e a class of algebras of the same typ e We dene H K to b e the class of all homomorphic images of algebras in K S K the class of all subalgebras of algebras in K and P K the class of all pro ducts of algebras in K An equational class or variety of algebras is a class K with H K K S K K and P K K If K is any class of algebras of the same typ e then HSP K is the smallest variety containing K In our general considerations so far we have distinguished corresp onding op erations in algebras of the same typ e by giving them the same index This is prac tically never done in concrete cases In these cases one identies corresp onding op erations by denoting them by the same symb ol For illustration let us consider the case of groups In order to get subgroups as a sp ecial case of our general notion of subalgebras we have to consider a group as an algebra with the group multiplication as a binary op eration the forming of inverses as a unary op eration and the unit as a constant In order to make this t our general development so far
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