Power-Set Functors and Saturated Trees

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1 Power-Set Functors and Saturated Trees Jiří Adámek 1, Stean Milius 1, Lawrence S. Moss 2, and Lurdes Sousa 3 1 Institut ür Theoretische Inormatik, Technische Universität Braunschweig Germany 2 Department o Mathematics, Indiana University Bloomington, IN, USA 3 Departamento de Matemática, Instituto Politécnico de Viseu, Portugal Abstract We combine ideas coming rom several ields, including modal logic, coalgebra, and set theory. Modally saturated trees were introduced by K. Fine in We give a new purely combinatorial ormulation o modally saturated trees, and we prove that they orm the limit o the inal ω op - chain o the inite power-set unctor P. From that, we derive an alternative proo o J. Worrell s description o the inal coalgebra as the coalgebra o all strongly extensional, initely branching trees. In the other direction, we represent the inal coalgebra or P in terms o certain maximal consistent sets in the modal logic K. We also generalize Worrell s result to M-labeled trees or a commutative monoid M, yielding a inal coalgebra or the corresponding unctor M studied by P. Gumm and T. Schröder. We introduce the concept o an i-saturated tree or all ordinals i, and then prove that the i-th step in the inal chain o the power set unctor consists o all i- saturated trees. This leads to a new description o the inal coalgebra or the restricted power-set unctors P λ (o subsets o cardinality smaller than λ) ACM Subject Classiication F.4.1 Mathematical Logic, F.3.2 Semantics o Programming Languages, G.2.2 Graph Theory Keywords and phrases saturated tree, extensional tree, inal coalgebra, power-set unctor, modal logic 1 Introduction Final coalgebras play a undamental rôle in the theory o systems represented as coalgebras: J. Rutten [18] demonstrated that the inal coalgebra describes all possible behaviors o states o systems. For Kripke structures considered as the coalgebras or the inite power-set unctor P two beautiul descriptions o the inal coalgebra exist: as the set o all hereditarily inite sets in the non-wellounded set theory due to P. Aczel, see [2], and as the set o all strongly extensional, initely branching trees 1 due to J. Worrell [21]. He used metric spaces: he described the limit P ω1 o the inal chain o P as the set o all strongly extensional, compactly branching trees. From that he derived the above description o the inal coalgebra. We give below two new descriptions that do not need topology, one combinatorial and one using modal logic. We prove that the limit P ω 1 consists (a) o all saturated trees or (b) o all maximal consistent theories o the modal logic K. And an alternative description o the inal coalgebra is: the set o all hereditarily inite (maximal consistent) theories. Related 1 Throughout the paper trees are directed graphs with a distinguished node called the root rom which every other node can be reached by a unique directed path, and they are always considered up to isomorphism. Strong extensionality or trees is recalled in Section 2 below. J. Adámek, S. Milius, L. S. Moss, L. Sousa; licensed under Creative Commons License NC-ND Leibniz International Proceedings in Inormatics Schloss Dagstuhl Leibniz-Zentrum ür Inormatik, Dagstuhl Publishing, Germany

2 2 Power-Set Functors and Saturated Trees descriptions were provided by S. Abramsky [1], A. Kurz and D. Pattinson [14] and by J. Rutten [17, Theorem 7.4]. We also present a generalization in two directions: one uses inite multisets with multiplicities drawn rom a given commutative monoid M, as introduced by H. P. Gumm and T. Schröder [12]. Form the unctor M o all such inite multisets; its coalgebras are labeled transition systems with actions rom M \ {0}. We prove a direct generalization or all monoids or which M preserves weak pullbacks: the inal coalgebra or M consists o all initely branching, strongly extensional M-labeled trees. For general monoids this result is not true, but we prove that the inal coalgebra or M is the coalgebra o extensional, initely branching M-labeled trees modulo an equivalence generalizing M. Barr s equivalence or P, see [7]. The other direction o generalization o the inal coalgebra or P is rom inite subsets to subsets o cardinality less than λ, where λ is an ininite cardinal. The corresponding power-set unctor P λ has the inal coalgebra o all strongly extensional λ-branching trees, as proved by D. Schwencke [19]. We present a dierent proo based on the description o the inal chain P i 1 o the (ull) power-set unctor P. We introduce the concept o an i-saturated tree or every ordinal i (where ω-saturated is the above concept), and we describe P i 1 as the set o all strongly extensional i-saturated trees. 2 Extensional and saturated trees For an endounctor H o Set recall that a coalgebra is a set A together with a morphism a: A HA. A coalgebra homomorphism into b: B HB is a morphism : A B with b = H a. The inal coalgebra, i it exists, is denoted by νh; by Lambek s Lemma [15] its coalgebra structure is an isomorphism νh H(νH). For example Kripke structures (W, R, l) where R W W and l : W 2 AP are precisely the coalgebras or HX = PX 2 AP where AP is a ixed set o atomic propositions and P is the powerset unctor. In the present paper we restrict ourselves to the case AP =. Then Kripke structures are simply graphs, or coalgebras or P. And the initely branching graphs are coalgebras or the inite power-set unctor P. In this and the next section we describe the inal coalgebra or P. Lambek s Lemma implies that P does not have a inal coalgebra, but we describe the inal chain o P in Section 5. Recall rom [7], dualizing the initial chain o [4], the inal chain o H which is the chain W : Ord op Set determined (uniquely up-to natural isomorphism) by its objects W i, i Ord, and connecting morphisms w i,j : W i W j (i j) as ollows W 0 = 1, W i+1 = HW i, and W i = lim j<i W j or limit ordinals i and w i+1,j+1 = Hw i,j, whereas (w i,j ) j<i is a limit cone or limit ordinals i. I this chain converges at some ordinal i, i.e., the connecting map HW i W i is an isomorphism, then its inverse yields the inal coalgebra or H. The inite steps o the inal chain o H are called the inal ω op -chain o H. Remark 2.1. Recall that coalgebras or P are simply graphs. When speaking about morphisms between graphs (in particular trees) we always mean coalgebra homomorphisms : A B. That is, preserves edges, and or every edge rom (a) to b in B there exists an edge rom a to a in A with b = (a ). Quotients o graphs are, as usual, represented by epimorphisms, that is, surjective morphisms. Deinition 2.2. A tree is extensional i distinct children o any node deine non-isomorphic subtrees.

3 J. Adámek, S. Milius, L. S. Moss, L. Sousa 3 The extensional modiication o a tree t is the smallest quotient o t which is extensional. It is obtained rom t by recursively identiying isomorphic subtrees whose roots have a joint parent. Example 2.3. The extensional modiication o the complete binary tree is the single path. Notation 2.4. For every tree t denote by n t the extensional tree obtained by cutting t at level n (i.e. deleting all nodes o depth > n) and orming the extensional modiication. For all trees t and u, we write t n u to mean that n t = n u (remember that we identiy isomorphic trees). Remark 2.5. The inal ω op -chain o P can be described as ollows: P n 1 = all extensional trees o depth n with the connecting maps n : P n+1 1 P n 1. Indeed, the unique element o 1 can be taken to be the root-only tree. Given a set M P n 1, we identiy it with the tree tupling o its elements and obtain a tree in Pn+1 1. The irst connecting map rom P 1 to 1 is obviously 0, and given that the n-th connecting map is n : P n+1 1 P n1, it ollows that the next connecting map, P n, is (with the above tree tupling identiication) precisely n+1. Deinition 2.6. Two trees t and u are called Barr equivalent, notation t ω u, provided that t n u holds or all n < ω. Remark 2.7. The set B o all initely branching extensional trees is a coalgebra or P : the coalgebra map is the inverse o tree tupling. This coalgebra is weakly inal, and a inal coalgebra can be described as its quotient: Theorem 2.8 (see [7]). The inal coalgebra or P can be described as the quotient B/ ω o the coalgebra o all initely branching, extensional trees modulo Barr-equivalence. Deinition 2.9 (see [21]). For trees t and s a tree bisimulation is a relation R t s such that the roots are related, two related child nodes always have related parents, and R is a bisimulation w.r.t. P; i.e., given related nodes a R b then or every child a o a in t there exists a child b in s with a R b, and vice versa. Example The irst two trees in the picture below are Barr-equivalent trees. The third and ourth trees are two extensional trees (the third tree has n children o the n-th node) that are bisimilar. In act, every tree without leaves is bisimilar to the ininite path ω Deinition 2.11 (see [21]). A tree t is called strongly extensional i distinct children o any node are not bisimilar. Equivalently, every tree bisimulation R t t satisies R t, where t is the diagonal relation on t.

4 4 Power-Set Functors and Saturated Trees Example (1) Every inite extensional tree is strongly extensional. (2) The ininite path is a strongly extensional tree. This is the only strongly extensional tree without leaves: or every tree t without leaves the relation x R y i x and y have the same depth is a tree bisimulation. Thus, the third tree in Example 2.10 shows an extensional tree which is not strongly extensional. Deinition Given a tree t, the subtree o t rooted at the node x is denoted by t x. A tree t is called saturated provided that or all nodes x o t and all trees s, i or all n, there are children x n o x with s n t xn (n < ω), then there is some ixed child y o x with s ω t y. Example (1) Every inite tree is saturated. (2) More generally: all initely branching trees are saturated. Indeed, given x n as above, there exists k < ω with x n = x k or ininitely many n, and then s n t xk or ininitely many n, proving s ω t xk. (3) The let-hand tree o Example 2.10 is not saturated. We obtain a saturated tree by adding a new child whose subtree is the ininite path. (4) For every set A ω the ollowing tree r A is saturated and strongly extensional: take an ininite path and add a lea at depth n i n + 1 lies in A. We know rom item (2) that r A is saturated, and strong extensionality is obvious. Lemma Given saturated, strongly extensional trees t and u with t ω u, we have t = u. Thereore there exist precisely 2 ℵ0 saturated, strongly extensional trees and they have branching at most 2 ℵ0. Proo. (a) For every child z o the root o u there exists a child y o the root o t with t y ω u z. Indeed, since t n u or every n there exist children x n with t xn n u z (n < ω) and then y exists since t is saturated. Conversely, or every y there exists z with t y ω u z. By continuing to lower nodes we conclude that the relation R t u deined recursively by y R z i { either y and z are the roots or y and z have R-related parents and t y ω u z is a tree bisimulation. Clearly, the opposite relation R op is a tree bisimulation, too, and, as P preserves weak pullbacks, so are the composite relations R op R t t and R R op u u. Since t and u are strongly extensional, we conclude R op R t and R R op u. Finally, since R and R op are total relations, the last two inequalities are equalities, and this implies that R is the graph o an isomorphism rom t to u, i.e., t = u. (b) The number o saturated, strongly extensional trees is at least 2 ℵ0 by Example 2.14(4). It is at most 2 ℵ0 because every saturated strongly extensional tree t is determined by the set M = {t x ; x a child o the root o t}, and M is determined, due to (a), by the sequence o sets M n = { n s; s M} or n < ω. Since M n is inite, the number o these sequences is at most 2 ℵ0. The last statement ollows since every subtree o a saturated tree is saturated. Recall Worrell s description o the limit P ω1 = lim n<ω Pn 1 o the inal chain o P as the set o all compactly branching trees [21]. Here is a new combinatorial description: Theorem The limit P ω 1 o the inal ωop -chain o P can be described as the set o all saturated, strongly extensional trees. The limit cone is ( n ) n<ω.

5 J. Adámek, S. Milius, L. S. Moss, L. Sousa 5 Proo. Let S be the set o all saturated, strongly extensional trees. We prove that n : S P n 1 (see Remark 2.5) is a limit cone. For deiniteness, we denote the connecting morphism o the inal ω op -chain by n : P n+1 1 P n1. It is obvious that n = n n+1, thus ( n ) is a cone on the inal ω op -chain. (a) The cone n is collectively monic by Lemma (b) For every compatible amily r n P n1 we prove that there exists t S with nt = r n or every n. Compatibility means r n = n(r n+1 ) or n < ω. Let ˆr n+1 be the tree obtained by cutting r n+1 at depth n. Since r n is extensional, the above equation tells us that r n is a quotient o ˆr n+1. Let e n : ˆr n+1 r n be the corresponding epimorphism. Deine a tree t to have as nodes o depths k = 0, 1, 2... all sequences x = ( x k, x k+1, x k+2... ) o nodes x n r n o depth k with e n ( x n+1 ) = x n or all n k. Thus the sequence o roots o r 0, r 1, r 2... is the root o t. And edges are deined componentwise: there is an edge rom ( x k, x k+1, x k+2... ) to (ȳ k+1, ȳ k+2, ȳ k+3,... ) i ( x n, ȳ n ) is an edge o r n or all n k + 1. It is easy to veriy that t is a well-deined tree. (b1) We prove n t = r n. To this end it suices to establish that there is an epimorphism o graphs rom the cutting o t at level n to r n (the desired equality then ollows since r n is extensional). Consider the n-th projection. This is surjective: For every node z r n there exists a node x t with z = x n. Indeed, put x n = z, and since e n is an epimorphism, choose x n+1 e 1 n ( x n ), etc. Then x = ( x n, x n+1, x n+2... ) has the required property. It is clear that the projection is a graph morphism: it preserves edges by the deinition o t. And analogously to the argument o surjectivity above, or every x in t and every edge rom z = x n to z in r n there exists an edge rom x to x in t with z = ( x ) n. (b2) t is strongly extensional. Indeed, given a tree bisimulation R t t, we prove that x R ȳ implies x = ȳ. Let k be the depth o x and ȳ. From (b1) it ollows that r n x = n n(t x ) and rȳ n = n(tȳ) or all n k. But x R ȳ implies that R restricts to a tree bisimulation n between t x and tȳ, thus, t x n tȳ. Consequently, r n x = n rn ȳ n. This implies xn = ȳ n. (This is clear rom extensionality o r n in case k = 1. This inishes the proo o x = ȳ i k = 1. For k = 2, we conclude that x and ȳ have the same parent, z, and apply the above to t z in lieu o t, etc.) (b3) The tree t is saturated. Indeed, let s be a tree or which the condition o Deinition 2.13 holds, taking x to be the root o t. (The proo or all other nodes x o t is completely analogous.) That is, we have children x n o the root o t with s n t xn or n < ω. We prove that the node ȳ o t with components ȳ n = ( x ) n n or all n 1 ulils s ω t y. Firstly, we need to veriy that ȳ is a node: e n (ȳ n+1 ) = ȳ n. Both o these nodes are children o the root o r n, thus, by extensionality we only need to prove that they deine the same subtree o r n. Let ˆt xn be the cutting o t xn at level n 1, then rom (b1) we know that rȳ n is the image o n ˆt xn under the n-th projection t r n. Since rȳ n n is extensional, this proves n t xn = rȳ n n and analogously or n + 1. Moreover, rom t xn n s n+1 t xn+1 we deduce t xn n t xn+1. Consequently, r n ȳ n = nt xn+1 and r n+1 ȳ n+1 = n+1 t xn+1. We also have nr n+1 ȳ = r n n+1 e n(ȳ n+1 ) because the right-hand tree is extensional, and it is the image o ˆr n+1 ȳ under e n+1 n. Consequently, re n n(ȳ n+1 ) = n n+1 t xn+1 = n t xn+1. This proves re n n(ȳ n+1 ) = rn ȳ n, thus, e n (ȳ n+1 ) = ȳ n by extensionality o r n. Next, we need to veriy s n tȳ or every n < ω. Indeed, we have s n t xn, and to prove t xn n tȳ observe that the n-th projection t r n maps the cutting o tȳ onto rȳ n n, thus, rȳ n = t ȳ. We already observed that r n n ȳ = n nt xn, thus, n t xn = n tȳ.

6 6 Power-Set Functors and Saturated Trees Corollary 2.17 (J. Worrell). The inal chain o P converges in ω + ω steps with the step ω + n given by the set P ω+n 1 o all saturated, strongly extensional trees initely branching up to level n 1. Moreover, the inal coalgebra or P is given by P ω+ω 1 = all initely branching, strongly extensional trees. Indeed, or n = 1 we have P ω+1 = P (P ω 1) and we identiy, again, every inite set M P ω 1 o saturated trees with its tree-tupling. This is, by Example 2.14(2), a saturated, strongly extensional tree which is initely branching at the root and conversely, every such tree is a tree tupling o a inite subset o P ω 1. Analogously or n = 2: we have P ω+2 = P (P ω+1 1) and the resulting trees are precisely those trees in P ω 1 that are initely branching at levels 0 and 1, etc. The connecting maps are the inclusion maps. The limit P ω+ω 1 = lim n<ω P ω+n 1 is the intersection o these subsets o P ω 1 which consists o all initely branching, strongly extensional trees: they are saturated, see Example 2.14(2). 3 Modally saturated trees K. Fine [10] introduced the concept o modal saturatedness or Kripke structures in modal logic. In this section, we review all o the needed deinitions, and we prove that modally saturated trees are the same as saturated trees. (a) We work with modal logic ormulated without atomic propositions. The sentences ϕ o modal logic are then given by ϕ ::= ϕ ϕ ϕ ϕ We use the usual abbreviations: = ϕ ψ = ( ϕ ψ) ϕ ψ = ϕ ψ ϕ = ϕ. A sentence has depth n i n is the maximum o nested in it. (b) We interpret modal logic on Kripke structures. Since we have no atomic sentences, our Kripke structures are just graphs G = (G, ), where is a binary relation on the set G. The main semantic relation is the satisaction relation = between the node set o a given graph and the sentences o the logic. This is deined as ollows: a = always a = ϕ i it is not the case that a = ϕ a = ϕ ψ i a = ϕ and a = ψ a = ϕ i or all neighbors b o a, b = ϕ Given a tree t we write t ϕ i the root satisies ϕ. (c) A theory is a set S o sentences. We write a S i a ϕ or all ϕ S and call a a model o S. (d) Turning to the proo system, the modal logic K extends the propositional logic (Hilbert s style) by one axiom (ϕ ψ) ( ϕ ψ), called K, and one deduction rule: i ϕ K then ϕ K. We write ϕ i ϕ can be derived in this logic. This logic is sound and complete. That is, ϕ holds i or every node a o any graph, a ϕ. (e) A theory S is inconsistent i or some inite {ϕ 1,..., ϕ n } S, ϕ i. S is consistent i S is not inconsistent. Or, equivalently, S has a model. I, moreover, S {ϕ} is inconsistent or every sentence ϕ / S, then S is maximal consistent. () S denotes the theory { ϕ : ϕ S}, and k S = ( k 1 S) or k 2.

7 J. Adámek, S. Milius, L. S. Moss, L. Sousa 7 Deinition 3.1. We deine canonical sentences χ o depth n by recursion on n, as ollows: (a) is the only canonical sentence o depth 0, and (b) canonical sentences o depth n + 1 are precisely the sentences S = ( S) S where S is a set o canonical sentences o depth n. We use the conventions that =, =, and we oten identiy sentences ϕ and ψ when ϕ ψ in K. Example 3.2. We have two canonical sentences o depth 1. = = and { } = = distinguishing whether the given node has a neighbor or not. Theorem 3.3 (K. Fine [10] and L. Moss [16]). For every node a o a graph and every n N there exists a unique canonical sentence χ o depth n satisied by a. Moreover, or every canonical sentence χ o depth n and every sentence ψ o depth at most n, either χ ψ or χ ψ. Corollary 3.4. The sentences o depth at most n orm a inite set (up to logical equivalence in K. Proo. Observe irst that there are only initely many canonical sentences o depth n. Let ψ be any sentence o depth n. Let A be the set o all canonical sentences χ o depth n with χ ψ and let B be the canonical sentences χ o depth n with χ ψ. So we have A ψ and ψ. By Theorem 3.3, we have A B. So by propositional logic we have A ψ. Thus, every sentence o depth n is equivalent to a disjunction o canonical sentences o depth n rom which the desired result ollows. Notation 3.5. (a) For every tree t we denote by χ n (t) the unique canonical sentence o depth n satisied in the root. It is easy to prove that χ n+1 (t) = { χ n (t x ) : x child o the root o t }. (b) For any graph G, and any a G, we denote by S a the set o all sentences ϕ with a ϕ in G. For a tree t, we similarly denote by S t the set o sentences satisied by the root o t. (c) Recall rom [8] that the canonical model o K is the graph C whose nodes are the maximal consistent theories, and with S S i S S (equivalently, S S ). The Truth Lemma (see [8], Lemma 4.21) is the statement that or all S C, {ϕ : S = ϕ in C} = S. This lemma is easy to check by induction on ϕ. Corollary 3.6. For two trees t and s we have t n s i t χ n (s). Consequently, t ω s i S t = S s. Proposition 3.7. The limit P ω 1 can be described as the set C o all maximal consistent theories in K.

8 8 Power-Set Functors and Saturated Trees Proo. We have described P ω 1 as the set o all saturated, strongly extensional trees. We prove that t S t is a bijection between this set and C. This inishes the proo. (a) For every t P ω1 the theory S t is maximal consistent. Indeed, it is is obviously consistent. Given ϕ / S o depth n, we have t ϕ and t χ n (t), thus, χ n (t) ϕ. By Theorem 3.3 χ n (t) ϕ. Thereore, S t {ϕ} is inconsistent. (b) By the Truth Lemma, every maximal consistent theory S is o the orm S t or some t: let t be the expansion o the canonical graph C at S. Moreover, t can be taken as saturated and strongly extensional, since the saturation operation on trees preserves modal theories (see Corollary 3.6). Deinition 3.8. A theory S is called hereditarily inite i it is maximal consistent and or every k N there exist only initely many maximal consistent theories S with k S S. Theorem 3.9. The set o all hereditarily inite theories is a inal coalgebra or P via the coalgebra map S {S : S S}. Proo. We prove that the bijection t S t o Proposition 3.7 has the property that or t P ω1 we have that t is initely branching i S t hereditarily inite. From that our theorem ollows, since the coalgebra map above corresponds to the coalgebra map o νp. Indeed: (a) I S t is hereditarily inite, then t is initely branching. It is suicient to veriy that t is initely branching at the root. Given a node x o depth k, we then apply this to t x : the theory o this subtree is also hereditarily inite, since k S tx S t (indeed: i t x ϕ then t k ϕ). Every child a o the root o t ulils S ta S t. Thus, there are only initely many such theories S ta. Now let a and b be children o the root o t with S ta = S tb, whence t a ω t b by Corollary 3.6. So since t a and t b are saturated and strongly extensional, we have t a = t b by Lemma Thereore, the root has only initely many children. (b) I t is initely branching, then S t is hereditarily inite. Indeed, or every maximal consistent theory S with k S S t let s be a tree with S = S s (see Proposition 3.7). Then or every n N we have t k χ n (s), i.e., some node o t o depth k satisies χ n (s). Since we have only initely many such nodes, one o them, say a, satisies χ n (s) or all n. That is, t a n s or n N, hence, S ta = S, see Corollary 3.6. Since we have only initely many nodes a o depth k, we see that S t is hereditarily inite. Deinition 3.10 (see [10]). A graph is called modally saturated i or every node a, given a theory S such that a S 0 or every inite S 0 S (3.1) there exists a neighbor b o a satisying S. Theorem A tree is saturated i it is modally saturated. Proo. (a) Let t be modally saturated. Let a be a node in t, and let s be a tree with the property that there exist children x n o a with s n t xn (n < ω). We prove s ω t b or some child b. The theory S s ulils (3.1): given S 0 S s inite, let n be the maximum o the depths o all ψ S 0 ; then χ n (s) ψ or all ψ S 0 (see Theorem 3.3). By Corollary 3.6, s n t xn i x n = χ n (s), and this implies x n ψ or all ψ S 0. Thus, a S 0. Let b be a neighbor o a satisying S s. Then t b χ n (s) or all n; i.e., s ω t b by Corollary 3.6. (b) Let t be saturated. Let a be a node o t and S be a theory satisying (3.1). For every natural number n deine S n to be a set o representatives o all ψ S o depth at most n modulo logical equivalence in K. By Corollary 3.4 the sentences o depth n orm a inite set

9 J. Adámek, S. Milius, L. S. Moss, L. Sousa 9 (up to logical equivalence).so we have that S n is inite. By (3.1) we see that or every n, there exists a child b n o a such that b n ψ or all ψ S n. It is our task to prove that a has a child b satisying S. Let v be the graph whose nodes are all canonical sentences χ o depth any n = 0, 1, 2,... such that a χ and χ ψ or all ψ S n. We make v a graph using the converse o logical implication in K. So the neighbors o the node χ are all the nodes χ o depth n + 1 with χ χ. The root is, and every node χ o v has indeed a unique parent (so v is a tree): since a χ, we have a child c o a with c χ which by Theorem 3.3 implies χ = χ n+1 (t c ). Put χ = χ n (t c ), then χ χ. (This is because χ χ cannot happen due to c χ and c χ. Now use Theorem 3.3). Consequently, χ is a parent o χ. And the uniqueness o the parent is obvious: suppose χ χ where χ v has depth n, then t c χ, thereore χ = χ n (t c ). The tree v is obviously initely branching. And since each χ n (t bn ) lies in v and each o these ormulas has a dierent depth, they orm an ininite set o nodes o v. By König s Lemma, v has an ininite branch = χ 0 χ 1 χ 2... Each S {χ n } is consistent. Indeed, by compactness it is suicient to veriy that S k {χ n } is consistent or every k n: due to a χ k we have a child c o a satisying χ k, then t c is a model o S k (due to χ k ψ or all ψ S k ) and o χ n (due to χ k χ n ). Consequently, S {χ 0, χ 1, χ 2,... } is consistent: use compactness again. Let s be a tree which is model o the last theory. Then s χ n which by Theorem 3.3 implies χ n = χ n (s) or every n. On the other hand, since a χ n, we have a child c n o a with c n χ n, thus, χ n = χ n (t cn ). By Corollary 3.6 this proves s n t cn. Since t is saturated, there exists a child b o a with s ω t b. Then S S s = S tb which concludes the proo: b satisies S. 4 Finite multisets with multiplicities in a commutative monoid Here we ollow the approach o P. Gumm and T. Schröder [12] who investigated initely branching Kripke structures with transitions having weights rom a given commutative monoid (M, +, 0). These are just coalgebras or the unctor M : Set Set (denoted by M ω in [12]) assigning to every set X the set M X o all inite multisets in X, i.e. all unctions A : X M with A 1 [M \ {0}] inite. Given a unction h: X Y, the unctor M assigns to every inite multiset A: X M the inite multiset M h(a) sending y Y to x X,h(x)=y A(x). Example 4.1. The Boolean monoid P = {0, 1} yields the inite power-set unctor P. The cyclic group C = {0, 1} yields a unctor C which coincides with P on objects but is very dierent on morphisms. Deinition 4.2. By an M-labeled graph G is meant a graph whose edges are labeled in M \ {0}. We denote by w G : G G M the corresponding weight unction with w G (x, y) 0 i y is a neighbor o x. Remark 4.3. (a) The coalgebras or M are precisely the initely branching M-labeled graphs. Indeed, given such a graph G, deine the coalgebra structure G M G by assigning to every vertex x the inite multiset w G (x, ): G M. Conversely, every initely branching M-labeled graph is obtained rom precisely one coalgebra o M.

10 10 Power-Set Functors and Saturated Trees (b) Coalgebra homomorphisms between two initely branching M-labeled graphs G and H are precisely the unctions : G H between the vertex sets such that w H ( (x), y ) = x X,(x )=y w G (x, x ) or all x G, y H. (4.1) (c) We identiy, once again, two M-labeled trees whenever they are isomorphic (as coalgebras or M ). Deinition 4.4. An M-labeled tree is extensional i distinct children o any node deine non-isomorphic M-labeled subtrees. We use n and ω in an obvious analogy to Notation 2.4 and Deinition 2.6. Remark 4.5. The concepts o tree bisimulation and strong extensionality (see Deinitions 2.9 and 2.11) also immediately generalize to M-labeled trees. We now generalize Theorem 2.8, using B again or the coalgebra o all initely branching extensional M-labeled trees. Theorem 4.6. Let M be a commutative monoid. The coalgebra B/ ω o all initely branching, extensional, M-labeled trees modulo Barr equivalence is inal or M. Sketch o proo. 2 (1) B is weakly inal. Indeed, or every initely branching M-labeled graph (A, α) we deine a coalgebra homomorphism h: A B by assigning to every vertex a A the extensional modiication o the tree expansion o a. Recall that the nodes o the tree expansion o a are the paths a 0, a 1,..., a k o A starting in a, including the empty path, a, which is the root. A child o a 1,..., a k is any extension a 1,..., a k, a k+1 and its weight in the tree expansion o a is w G (a k, a k+1 ), see Deinition 4.2. (2) The inal coalgebra is obtained rom B by the quotient modulo the largest congruence. This ollows rom the act that the category o coalgebras or M is complete. Thus, by Freyd s Adjoint Functor Theorem a weakly inal object always has the inal object as the quotient modulo the greatest congruence. (3) The Barr equivalence is a congruence on B. That is, the quotient B/ ω carries a coalgebra structure or M such that the quotient map q : B B/ ω is a coalgebra homomorphism. (4) Every congruence on B is contained in ω. That is, our task is to prove the implication t t implies n t = n t. This ollows by induction on n since being a congruence means that or the quotient map q : B B/ we have a coalgebra structure B/ making q a homomorphism. Deinition 4.7 (See [12]). A commutative monoid M is called (a) positive i a + b = 0 implies a = 0 = b and (b) reinable i a 1 + a 2 = b 1 + b 2 implies that there exists a 2 2 matrix with row sums a 1 and a 2, respectively, and column sum b 1 and b 2, respectively. 2 Full proos are in the appendix.

11 J. Adámek, S. Milius, L. S. Moss, L. Sousa 11 Theorem 4.8. The ollowing conditions on a commutative monoid M are equivalent (a) The unctor M weakly preserves pullbacks (b) M is positive and reinable, and (c) whenever a a n = b b k, there exists an n k-matrix whose vector o row sums is a 1,..., a n and the vector o column sums is b 1,..., b k. In [12] this theorem is proved except that in lieu o (a) weak preservation o non-empty pullbacks is requested. However, the unctor M has a unique distinguished point in the sense o V. Trnková [20], namely, the empty set M X. Since M = { }, it ollows rom the result in [20] that M preserves weak pullbacks i it preserves the nonempty ones. Now or (a) (b), see [12], Theorem 5.13, and concerning (b) (c), Proposition 5.10 o loc. cit. states that reinability is equivalent to condition (c) with n, k > 1 and positivity o M is equivalent to condition (c) with n > 1 and k = 0. For n = 1, condition (c) is trivial. Example 4.9 (See [12]). The Boolean monoid P = {0, 1} and the monoids (N, +, 0) and (N,, 1) are positive and reinable. The cyclic group C = {0, 1} is reinable but not positive. For every lattice L the monoid L = (L,, 0) is positive, and it is reinable i L is a distributive lattice. Theorem Let M be a positive and reinable monoid. The coalgebra B s o all strongly extensional, initely branching M-labeled trees is inal or M. Remark For the coalgebra B o extensional M-labeled trees all strongly extensional trees clearly orm a subcoalgebra m: B s B. We prove that the composite o m with the quotient homomorphism q : B B/ ω is an isomorphism q m: B s B/ ω. This proves that B s is inal. Sketch o proo. Since q m is a homomorphism o coalgebras, it is suicient to prove that it is a bijection, then it is an isomorphism. In other words: we are to prove that B s is a choice class o ω on the set B. (1) Every tree t in B is Barr equivalent to a strongly extensional tree t/ R. Indeed, recall rom Theorem 4.8 that M weakly preserves pullbacks. Thus the greatest bisimulation R t t is an equivalence relation which is also the greatest congruence. And or the greatest tree bisimulation R contained in R, the strongly extensional tree t/ R is bisimilar to t. Since B/ ω is the inal coalgebra, this proves that t ω t/ R. (2) I two strongly extensional trees are Barr equivalent, then they are equal. Instead, we prove in items (3) and (4) below that given trees t, s B then i t ω s then t is bisimilar to s. Thus, we obtain a tree bisimulation R t s and, by symmetry, a tree bisimulation S s t. Since M weakly preserves pullbacks, S R t t is a tree bisimulation. This proves in the case t and s are strongly extensional, that S R =. By symmetry, R S =. Then R is a graph o an isomorphism rom t to s, i.e., t = s. (3) We consider the given trees t ω s as elements o the coalgebra B o Theorem 4.6. We know that ω is the greatest congruence, hence, the greatest bisimulation on B. As proved in [12], Lemma 5.5, this means that there exists a matrix m: B B M such that (a) w B (t, t ) = m(t, s ) or all t B s B (b) w B (s, s ) = m(t, s ) t B (c) m(t, s ) 0 implies t ω s. or all s B, and Since M is positive, whenever m(t, s ) 0, we have w B (t, t ) 0, that is, there exists a child x o the root x 0 o t with t = t x and w B (t, t ) = w t (x 0, x). Analogously, m(t, s ) 0

12 12 Power-Set Functors and Saturated Trees implies s = s y or some child y o the root y 0 o s with w B (s, s ) = w s (y 0, y). Since t and s are extensional, the trees t B with w B (t, t ) 0 are in bijective correspondence with the children x o x 0 in t via x t x. Analogously or s. Thus we can translate (a) (c) as ollows: (a ) w t (x 0, x) = y s m(t x, t y ) or all x t (b ) w s (y 0, y) = x t m(t x, t y ) or all y s, and (c ) m(t, s ) 0 implies that there exists a unique child x o x 0 in t and a unique child y o y 0 in s with t x ω t y, t = t x and s = s y. (4) We prove that given trees t, s B with t ω s, it ollows that the relation R t s deined recursively by x R y i t x ω s y and x and y are roots or have R-related parents is a tree bisimulation. Example The above theorem does not generalize to all positive monoids. Indeed, consider the monoid L = (L,, 0) or the lattice {0, a, b, c, 1} where a, b, c are pairwise incomparable. Then strongly extensional initely branching L -labeled trees do not orm a inal coalgebra, since they are not a choice class o the Barr equivalence. The ollowing trees are easily seen to be Barr equivalent: t : s : a a b b c a b b c Here s has as nodes the binary words, and the weights are, or all x {0, 1}, deined by w s (x0, x00) = a, w s (x1, x11) = c and w s (x0, x01) = b = w s (x1, x10). It is obvious that t is strongly extensional. To prove that so is s, let R s s be a tree bisimulation. Using the conditions (a) (c) in the preceding proo it is easy to veriy that R s. 5 The inal chain o P Although the power-set unctor P has no inal coalgebra, one can describe its inal chain concretely: we now prove that the i-th step P i 1 consists o all i-saturated, strongly extensional trees. This generalization rom saturated to i-saturated turns out the be quite nontrivial. In addition, this allows us to describe the inal chain o the restricted power set unctors P λ (see Corollary 5.13). Notation 5.1. Recall that the subtree o t rooted at the node x is denoted by t x. We deine equivalences i on the class o all trees or every ordinal i (c. Notation 2.4 and Deinition 2.6) by transinite induction: s 0 t holds or all pairs s, t; s i+1 t holds i or every child x o the root o s there is a child y o the root o t with s x i t y, and vice versa and or limit ordinals i, s i t means s j t or all j < i. c

13 J. Adámek, S. Milius, L. S. Moss, L. Sousa 13 Example 5.2. ω is Barr equivalence, see Deinition 2.6. The irst and second trees in Example 2.10 are trees s and t with s ω+1 t which are Barr equivalent. There exist, or every ordinal i, trees s and t with s i t but s i+1 t, see [5]. Deinition 5.3. We deine the concept o i-saturated tree or every ordinal i by transinite induction: A tree t is i-saturated i (a) i = 0: t consists o the root only (b) i = j + 1: t x is j-saturated or every child x o the root (c) i a limit ordinal: given a tree s and a node x o t having children x j with s j t xj (j < i), then x has a child y with s i t y. Examples 5.4. (a) For i inite, a tree is i-saturated i it has height at most i. And s i t holds i i s = i t. Thereore, a tree is ω-saturated i it is saturated in the sense o Deinition (b) An example o an (ω + 1)-saturated tree which is not ω-saturated is the let-hand tree in Example (c) For every ininite cardinal λ, all λ-branching trees t are λ-saturated. Remark 5.5. For every tree s there exists a Barr-equivalent tree which is ω-saturated and strongly extensional. We denote it by ω s and call it the ω-saturation o s. In act, or the sequence r n = n s o trees in P n 1, which is clearly compatible, apply the construction in the proo o Theorem 2.16: the resulting tree t in P ω 1 is Barr equivalent to s because n t = r n = n s or every n < ω. Put ω s = t. We generalize this to all ordinals: Deinition 5.6. By the i-saturation o a given tree s is meant an i-saturated, strongly extensional tree i s with s i i s. Example 5.7. (1) For i inite, Notation 2.4 yields the desired tree. (2) An example o ω-saturation can be seen in Example 2.10: or the let-hand tree s the second tree is ω s. Remark 5.8. I t and u are bisimilar trees, then they are equivalent under all o the above equivalences i. This is easy to see by transinite induction. Also, i t is i-saturated, then every tree bisimilar to t is i-saturated (in act, every tree equivalent under i is). In particular, the strongly extensional quotient o a tree t, which is the quotient modulo the largest tree bisimulation R t t, is i-saturated whenever t is. So even though a tree t is i-saturated it might not be its own i-saturation. Indeed, the third tree in Example 2.10 is ω-saturated but its ω-saturation is the ourth tree. Proposition 5.9. Every tree has or every ordinal i a unique i-saturation. Remark For ininite ordinals we will see that there is a canonical tree morphism d i rom a tree s to its i-saturation i s. Moreover, i i = α + n where α is a limit ordinal and n < ω, then d i is surjective when restricted to nodes o depths at most n. Sketch o proo. (1) Uniqueness. Given i-saturated, strongly extensional trees t and u, we need to prove that t i u implies t = u. The step rom i to i + 1 is easy. For i = ω this was established in Theorem For all other limit ordinals i, this is proved analogously. (2) Existence. For i < ω we have i s as in Section 2. The isolated steps are trivial: given i we deine i+1 by taking a tree s and letting s be the tree-tupling o all i s x, where x ranges over the children o the root o s. Then the strong extensional quotient o s is i+1 s, see Remark 5.8.

14 14 Power-Set Functors and Saturated Trees The canonical morphism is the composite d i+1 = e d i where d i : s s is the tree morphism acting on every maximal subtree s x as the corresponding canonical morphism d (sx) i : s x i s x, and e is the strong extensional quotient. This composite is, in the case i = α + n or a limit ordinal α, surjective on nodes o depths at most n + 1 since each d (sx) i is surjective on nodes o depths at most n. For limit ordinals i we construct, or every tree t, the i-saturation in two steps: irst t will be a possibly large tree (with a class o nodes) which is i-saturated and ulils t i t. Then i t is the strong extensional quotient. This is an i-saturation o t because Remark 5.8 holds clearly or large trees, too. And the resulting tree i t is small, in act, it is λ i -branching or a speciic cardinal λ i analogously to 2 ℵ0 or i = ω (see Lemma 2.15). Theorem The inal chain o P can be described or all ordinals i by P i 1 = all i-saturated, strongly extensional trees with the connecting maps into P j 1 given by j or all j < i. Sketch o proo. For i inite this is obvious since P i 1 = P i 1, or i = ω use Theorem We proceed by transinite induction or all ininite ordinals. I the statement holds or i then it holds or i + 1 provided that every set M P(P i 1) = P i+1 1 o trees is identiied with the tree-tupling t M o all members o M. Obviously, t M is (i + 1)-saturated and strongly extensional. Conversely, every (i + 1)-saturated strongly extensional tree is obtained by tree tupling via a unique set M. Thus, P i+1 1 = P(P i 1) is the set o all (i + 1)-saturated, strongly extensional trees. I j : P i 1 P j 1 is the given connecting map, then the connecting map P j corresponds to j+1 when the above identiication o M and t M is perormed. Thus, it remains to prove, or every limit ordinal α > ω, that the cone i : P α 1 P i 1 (i < α) is a limit cone. This is technically more involved than the proo o Theorem 2.16 but the ideas are similar. Remark J. Worrell [21] proved that since P λ preserves intersections o chains o subobjects and is λ-accessible, the inal chain converges in λ + ω steps, where all steps ater λ are given by monomorphisms. We can describe the individual steps Pλ i 1 or all i < λ in terms o trees. I i is an ininite ordinal then it has the orm i = α + n, n < ω and α a limit ordinal. Corollary The inal chain o P λ has the steps Pλ i 1 or all ordinals i < λ given by P i λ1 = all i-saturated, strongly extensional trees whose i-saturation is λ-branching at the irst n levels or all i = α + n. The connecting maps are j : Pλ i 1 Pj λ1 or all j < i. Corollary Let λ be an ininite cardinal. The inal coalgebra or P λ can be described as the coalgebra o all strongly extensional, λ-branching trees. Indeed, each such tree is λ-saturated, see Example 5.4. Thus, it lies in P λ+ω λ 1. Conversely, every tree in P λ+ω λ 1 is λ-branching because the connecting map P λ+n+1 λ 1 P λ+n λ 1 is a monomorphism or every n < ω: this ollows rom P λ being a λ-accessible unctor, see [21]. Consequently, the limit P λ+ω λ 1 = lim n<ω P λ+n λ 1 is the intersection o the subsets P λ+n λ 1 o Pλ λ 1. Since all tree in Pλ+n λ 1 are λ-branching at the irst n levels, it ollows that every tree in P λ+ω λ 1 is λ-branching. A completely dierent proo has been presented by D. Schwencke [19].

15 J. Adámek, S. Milius, L. S. Moss, L. Sousa 15 6 Conclusions and uture work We proved several results which generalize Worrell s description [21] o the inal coalgebra o P as the coalgebra o all strongly extensional, initely branching trees. We described the inal coalgebra or the unctor M o inite multisets with weights rom a given commutative monoid M. This inal coalgebra consists o all initely branching strongly extensional M- labeled trees. This holds or all positive and reinable monoids. Our proo is substantially dierent rom Worrell s, since it is based on congruences on the coalgebra o all extensional trees. We would like to generalize our work on saturated trees to the case o unctors M. And we plan to apply our methods to probabilistic transition systems. For P we also described the limit P ω1 o the inal ωop -chain as the set o all saturated strongly extensional trees, or the set o all maximal consistent theories in the modal logic K. We proved that saturated trees are precisely those trees which are modally saturated in the sense o K. Fine [10]. This also is related to (but quite dierent rom) Worrell s description o P ω 1 by means o compactly branching trees. We then generalized saturatedness to i- saturatedness and proved that the inal chain P i 1 o the ull power-set unctor consists o all strongly extensional i-saturated trees. From this we derived e.g. that the countable power-set unctor P c has the inal coalgebra consisting o all strongly extensional countably branching trees. We leave as open problem the decision whether ω 1 + ω is the smallest ordinal or the convergence o the inal chain o P c. We have characterizations o the initial algebras o the unctors M. There are open questions concerning the inal chains or these unctors: or the Boolean monoid, yielding P, the chain needs ω + ω steps, as proved by Worrell. However, or the monoid (N, +, 0), the corresponding unctor N is analytic, hence, the convergence needs only ω steps. We do not know what happens in the case o a general monoid. Reerences 1 Abramsky, S.: A cook s tour o the initary non-wellounded sets. In S. Artemov et al (eds), We Will Show Them: Essays in honour o Dov Gabbay, College Publications, Vol. 1, 1 18, Aczel, P.: Non-well-ounded sets. Lecture Notes 14, CSLI, Stanord Aczel, P., Mendler, P. F.: A inal coalgebra theorem. Lecture Notes in Comput. Sci. 389, Springer 1989, Adámek, J.: Free algebras and automata realizations in the language o categories. Comment. Math. Univ. Carolinæ 15 (1974) Adámek, J., S. Milius, S., Velebil, J.: On coalgebra based on classes. Theoret. Comput. Sci. 316 (2004) Adámek, J., Trnková, V.: Automata and algebras in categories. Kluwer Academic Publ., Dordrecht Barr, M.: Terminal coalgebras in well-ounded set theory Theoret. Comput. Sci. 114 (1993) Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press Borceux, F.: Handbook o categorical algebra 2. Cambridge University Press Fine, K.: Normal orms in modal logic. Notre Dame J. Formal Logic 16 (1975) Fine, K.: Some Connections Between Elementary and Modal Logic, in S. Kanger (ed), Proc. o the Third Scandinavian Logic Symposium. Amsterdam: North Holland, 1975, Gumm, H.-P., Schröder T.: Monoid-labelled transition systems. Electron. Notes Theor. Comput. Sci. 44 (2001)

16 16 Power-Set Functors and Saturated Trees 13 Joyal, A.: Foncteurs analytiques et espèce de structure. Lecture Notes in Math. 1234, Springer 1986, Kurz, A., Pattinson, D.: Coalgebraic modal logic o inite rank. Math. Structures Comput. Sci. 15:3 (2005), Lambek, J.: A ixpoint theorem or Complete Categories, Math. Z. 103 (1968), Moss, L. S.: Finite models constructed rom canonical ormulas, J. Philos. Logic 36 (2007), Rutten, J.: A calculus o transition systems, CSLI lecture notes 53, (1995) 18 Rutten, J.: Universal coalgebra: a theory o systems, Theoret. Comput. Sci. 249, 3 80 (2000) 19 Schwencke, D.: Coequational logic or accessible unctors. Inorm. and Comput. 208 (2010), Trnková, V.: On descriptive classiication o set unctors I, Comment. Math. Univ. Carolinæ 12 (1971), Worrell, J.: On the inal sequence o a initary set unctor. Theoret. Comput. Sci. 338 (2005)

17 J. Adámek, S. Milius, L. S. Moss, L. Sousa 17 A Proo o Theorem 4.6 (1) B is weakly inal. Indeed, or every initely branching M-labeled graph (A, α) we deine a coalgebra homomorphism h: A B by assigning to every vertex a A the extensional modiication t a o the tree expansion o a. Recall that the nodes o the tree expansion o a are the paths a 0, a 1,..., a k o A starting in a, including the empty path, a, which is the root. A child o a 1,..., a k is any extension a 1,..., a k, a k+1 and its weight in the tree expansion o a is w G (a k, a k+1 ), see Deinition 4.2. We need to prove that the square A α M A h B β M B M h commutes. Let a A be a vertex with α(a) = { (a 1, m 1 ),..., (a k, m k ) }. Then β h(a) is the extensional modiication o the ollowing tree r: m 1 m k t a1 t ak Since t ai are all extensional, the extensional modiication r o r is the tree whose neighbors in B are precisely the trees s B or which the sum k m i = i=1 s=t ai k i=1 h(a i)=s m i is nonzero; the sum is then the weight o the edge rom r to s. But this precisely describes the inite multiset M h(α(a)). (2) The inal coalgebra is obtained rom B by the quotient modulo the largest congruence. This ollows rom the act that the category o coalgebras or M is complete. Thus, by Freyd s Adjoint Functor Theorem a weakly inal object has always the inal object as the quotient modulo the greatest congruence. (3) The Barr equivalence is a congruence on B. That is, the quotient B/ ω carries a coalgebra structure or M such that the quotient map q : B B/ ω is a coalgebra homomorphism. To prove this, all we need to veriy is that given a tree t : m 1 m k t 1 t k then the class [t] modulo ω is independent o the choice o representatives o the classes [t 1 ],..., [t k ]. That is, given trees s 1 ω t 1,..., s k ω t k and orming their tree-tupling s with the given weights m 1,..., m k, then s ω t.

18 18 Power-Set Functors and Saturated Trees Indeed, or every n the extensional modiication n+1 t o the cutting o t at depth n + 1 is obtained by orming the extensional modiications n t 1,..., n t k o the cuttings o the children and then taking the extensional modiication o the ollowing tree m 1 m k n t 1 n t k Analogously or s. Thereore n t i = n s i or all i = 1,..., k implies n+1 t = n+1 s, as required. (4) Every congruence on B is contained in ω. That is, our task is to prove the implication t t implies n t = n t or all n N. To be a congruence means that or the quotient map q : B B/ there is a commutative square q B B/ β M B M q M (B/ ) In other words, M q β actorizes through q. More detailed: or every pair t t o trees o the orm t = m 1 m k t = m 1 m l t 1 t k t 1 t l the multiset given by [t i ] and m i is the same one as that given by [t j ] and m j. This means that or every tree s B the two sums below are equal: k m i = i=1 s t i l m j. j=1 s t j (A.1) From this we derive n t = n t as ollows. Case n = 0 is trivial: n t is the root-only tree. Case n = 1. We are to prove m 1 + +m k = m 1+ +m l. For every s = t i 0, i 0 = 1,..., k, we have the equality in (A.1). In case the let-hand sum is nonzero, we thus have some j with t i0 t j. And we can express m m k as the sum o all non-zero sums s t i m i where i 0 ranges over a set o representatives (or ) o all indexes 1,..., k making the above let-hand sum nonzero. By symmetry, this yields m m k = m m l, as desired. Analogously or n = 2: here we take any t i0 with t i0 t i m i 0 and ind a corresponding t j t i 0 (and vice versa). Then we apply the case n = 1 to the pairs t i0, t j. Etc.