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A Remark About Algebraicity in Complete Partial Orders

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A remark ab out algebraicity in complete partial orders Draft Leonid Libkin Department of Computer and Information Science University of Pennsylvania Philadelphia PA USA Email libkinsaulcisupennedu Abstract I prove a characterization theorem for algebraic b ounded complete cp os similar to that for algebraic lattices It is wellknown that a lattice is algebraic i it isomorphic to a lattice of subalgebras of an algebra Algebraicity plays the central role in denotational semantics for programming languages but the structures used there are not exactly algebraic lattices they are complete algebraic partial orders In this note I shall characterize such p osets as p osets of certain subalgebras of partial algebras Let me recall the denitions A p oset is called complete and is usually abbreviated as a cpo if it contains least upp er b ounds or suprema of directed subsets I shall use t for supremum An element x of D is called compact if x tX where X D is directed implies x x for some x X A cp o is called algebraic if for any x D the set of compact elements b elow x is directed and its supremum equals x A cp o D is bounded complete if supremum of X D denoted by tX as well exists whenever X is b ounded ab ove in D ie there is a D such that a x for all x X I shall use a more convenient notation a a instead of tfa a g An element x of a b ounded complete cp o D is compact 1 n 1 n if whenever tX exists and x tX x tX where X X is nite In a b ounded complete cp o the set of compact elements b elow any element is always directed therefore a b ounded complete cp o is algebraic if any element is the supremum of all compact elements b elow it Algebraic b ounded complete cp os are also called Scottdomain Equivalently a Scottdomain is an algebraic cp o which is 1 a complete meetsemilattice Algebraic lattices are a particular example of Scottdomains namely they are Scottdomains with top element The wellknown characterization of algebraic lattices as lattices of subalgebras gives rise to a natural question can a similar characterization b e obtained for Scottdomains The answer is yes and a couple of denitions is needed b efore we formulate the result of this pap er Partially supp orted by NSF Grants IRI and CCR and ONR Grant NOOOK 1 I must notice that program semantics p eople usually require that the set of compact elements b e countable and this requirement is forced if we need certain computability conditions No cardinality restriction on Scottdomains is imp osed in this pap er Let hA i b e a partial algebra with carrier A and signature and let denote the set of nary n op erations in A partial subalgebra of hA i is B A such that for any n and n x x B x x B if it is dened It is also known that partial subalgebras of a partial 1 n 1 n algebra form an algebraic lattice I shall call B A a total subalgebra if for any n and x x B x x exists and n 1 n 1 n x x B The set of all total subalgebras of hA i under inclusion ordering is denoted by 1 n TSubA Theorem Let D be a poset Then D is a Scottdomain i there is a partial algebra hA i such that D is isomorphic to TSubA Pro of Prove that TSubA is a Scottdomain rst Obviously TSubA is closed under arbitrary in tersections and therefore it is a complete meetsemilattice If A is a directed family of total i iI S A Then it is easy to check that A is a total subalgebra again subalgebras of A let A i iI Hence TSubA is a cp o Let suppA fx A j a total subalgebra B A such that x B g Let A b e the minimal total subalgebra containing x suppA which exists since TSubA is a complete x S S A and therefore there A then x meetsemilattice If for a directed family A A i i i iI x iI iI is j I such that x A Then by the denition of A A A Therefore each A is compact j x x j x in TSubA Since for any B TSubA B suppA B is the supremum of all A where x B in x TSubA Hence compact elements form a basis of TSubA and TSubA is algebraic Conversely let D b e a Scottdomain Denote the set of its compact elements by KD We are going to dene an algebra whose carrier is A KD and partial op erations are dened as follows An op eration is given by for any a a KD if fa a g is not b ounded ab ove then a a n 1 n 1 n 1 n is undened if this set is b ounded ab ove ie a a exists then a a b where 1 n 1 n b a a and b KD We dene just enough op erations so that for any n b a a 1 n 1 n b a a KD there exists such that b a a 1 n n 1 n To nish the pro of we must show that B TSubA i there exists x D such that B x KD where x is the principal ideal of x The if part follows immediately from the denition of To prove the only if part let B TSubA Then for any a a B a a exists otherwise there 1 n 1 n would b e an nary op eration undened on a a Therefore M fa a j a a B g 1 n 1 n 1 n is a directed set and we can dene x as tM tB By the way x was dened B x KD If b x KD then b tB and since b is compact b a a for some a a B Then 1 n 1 n there is an op eration such that b a a ie b B This proves the reverse inclusion n 1 n and nishes the pro of of the theorem 2 A similar characterization theorem can b e proven if partiality is removed from the signature to the subsets which are allowed to play the role of subalgebras To b e more precise let me dene F as a family of subsets of a set A which is downward closed ie B F and C B imply C F and complete as a partial order under inclusion Such a family is sometimes called a qualitative domain Given a signature dene Sub A as the p oset of subalgebras of hA i which happ en to b e in F F Corollary A poset D is a Scottdomain i there exist an algebra hA i and a qualitative domain F of subsets of A such that D is isomorphic to Sub A F Pro of is essentially the same as the one given ab ove A is taken to b e KD and F is the family of subsets of KD b ounded ab ove 2 Arbitrary cp os can also b e viewed as p osets of certain subalgebras of nondeterministic algebras ie algebras with op erations whose results are not uniquely dened even if they are dened More precisely dene a partial nondeterministic algebra as hA i where each is partial function n n A from A to We say that B A is a total subalgebra of A if for any a a B and 1 n n a a is dened and a a B The p oset of total subalgebras under inclusion 1 n 1 n ordering will b e denoted by TSubA again An example can b e given showing that TSubA is not necessarily algebraic if A has nondeterministic op erations However Prop osition Let D be an algebraic cpo Then there is a partial nondeterministic algebra hA i such that D is isomorphic to TSubA Pro of A bunch of deterministic op erations are dened as in the pro of of the theorem with only one exception instead of saying b a a we say that b is less than any upp er b ound of fa a g 1 n 1 n The only nary nondeterministic op eration is undened for ntuples of compact elements which are unb ounded if fa a g is b ounded ab ove then a a fb KD j b a a g Notice 1 n 1 n 1 n that a a is nonempty It can b e easily shown that for such a signature TSubhKD i is 1 n isomorphic to D 2 This construction sheds some light on wellknown condition M The condition M says that for any nite b ounded set X of compact elements the set of its minimal upp er b ounds if nite and each A upp er b ound of X is greater than some minimal upp er b ound Let stand for the family of all f in nite subsets of A Prop osition Let D be an algebraic cpo satisfying M Then there is a partial nondeterministic n A algebra hA i in which any nary operation is of form A such that D is isomorphic to TSubA f in Pro of is the same as ab ove but we take a a to b e the set of minimal upp er b ounds if it 1 n exists and undened otherwise 2 References Gratzer Universal Algebra Qualitative domains Condition M .
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