For the property of having nite derivation type this result has been strengthened considerably by Otto and Sattler-Klein by showing that this property

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1 FP is undecidable for nitely presented monoids with word problems decidable in polynomial time Robert Cremanns and Friedrich Otto Fachbereich Mathematik/Informatik Universitat Kassel, 349 Kassel, Germany WWW: TH/ September 8, 998 Abstract In the rst part of this paper properties P of nitely presented monoids are considered that satisfy the following two conditions: () each monoid that can be presented through a nite convergent string-rewriting system has property P, and () each nitely presented monoid having property P is of homology type FP 3. Following an idea presented in (Otto and Sattler- Klein 997) it is shown that these properties are undecidable even if attention is restricted to those nitely presented monoids that have word problems decidable in polynomial time. In the second part it is shown that for nitely presented monoids the homology type FPn (n 3) is compatible with free products and with a restricted form of free products with amalgamation. Keywords: Introduction monoid-presentation, homology type FP n, word problem, decidability, free product, free product with amalgamation. By now quite a few conditions have been found that a monoid must necessarily satisfy if it is to have a presentation through some nite convergent (string-) rewriting system (see [OtKo97] for a recent survey). Historically one of the rst has been a homological niteness condition. In 987 Squier proved that a monoid that can be presented through a nite convergent rewriting system has homology type FP 3 [Squ87]. Since there were known examples of nitely presented monoids (and groups) that have decidable word problem, but that are not of type FP 3, this gave a negative answer to the long-standing open problem of whether each nitely presented monoid with a decidable word problem does admit such a presentation. Squier's result was then extended by Kobayashi who showed that in fact each monoid with a nite convergent presentation is of type FP [Kob9], a result that had been obtained before in a dierent setting independently of Squier's work by Anick [Ani87]. Later Squier developed still another condition, the homotopical niteness condition of having nite derivation type [SOK94], which is related to the homology type FP 3 in that each nitely presented monoid that has nite derivation type is of type FP 3 [CrOt94], while for nitely presented groups the property of having nite derivation type and FP 3 are even equivalent [CrOt96]. The property of having nite derivation type and the property of being of type FP n for n 3 are undecidable in general for nitely presented monoids. In [CrOt96] this is shown by using a well-known construction from combinatorial group theory, which reduces the word problem for a nitely presented group to the respective property under consideration.

2 For the property of having nite derivation type this result has been strengthened considerably by Otto and Sattler-Klein by showing that this property is even undecidable for the class of all nitely presented monoids that have polynomial-time decidable word problems [OtSa97]. Here we give a simpler proof for this result. In fact, we prove a more general result. Let P be a property of nitely presented monoids such that each monoid with a nite convergent presentation has property P, and each nitely presented monoid that has property P has homology type FP 3. Based on an idea outlined in the last section of the paper by Otto and Sattler-Klein [OtSa97] we show that property P is undecidable for the class of all nitely presented monoids with word problems solvable in polynomial time. Examples of such properties are the property of having nite derivation type and, for each n 3, the property of having homology type FP n. In a second part we consider the class of all nitely presented monoids that are of type FP n for some n 3. We will show that this class is closed under the operation of free product. In fact, we will prove that for two nitely presented monoids M and M and an integer n 3 or n =, the free product of M and M is of type FP n if and only if M and M both are of type FP n. Finally we prove an analogous result for a restricted form of free products of nitely presented monoids with amalgamated submonoids. This paper is organized as follows. In the next section the notation regarding monoids and monoid-presentations is introduced, and with each monoid-presentation an innite graph describing the underlying reduction relation is associated. In Section 3 the homology type and standard sequences are dened, and some results of Squier are restated in short, before in Section 4 the announced undecidability result is proved in detail. The nal two sections then deal with the operation of free product and the operation of free product with amalgamation. Monoid-presentations In this section we restate some basic denitions and results on string-rewriting systems and monoidpresentations that we will need throughout this paper in short. For details we refer to [BoOt93]. Let be an alphabet. Then denotes the set of words over including the empty word. As usual the concatenation of two words u and v is simply written as uv. A rewrite rule on is an ordered pair (l; r). Usually we will denote such a rule as (l! r), l is called its left-hand side, and r its right-hand side. A string-rewriting system (or rewriting system for short) R on is a set of rewrite rules on. It generates a single-step reduction relation! R on which is dened as follows: u! R v i there exist two words x; y and a rule (l! r) R such that u = xly and v = xry. A word u is called reducible if there is a word v such that u! R v; otherwise u is called irreducible. The reexive transitive closure! R of! R is the reduction relation generated by R. If u! R v and v is irreducible, then v is called a normal form of u. A rewriting system R on is called recursive if the process of reduction is an eective one, that is, given a word u, it is decidable whether or not u is reducible, and if it is, a word v can be determined eectively such that u! R v. Obviously, each nite rewriting system is recursive. The reexive, symmetric, and transitive closure $ R of! R is the Thue congruence generated by R. For a word w, the congruence class fu j u $ R wg of w is denoted by [w] R, and = $ R denotes the set of all congruence classes. By dening [u] R [v] R := [uv] R we obtain a binary operation on = $ R that is associative with identity element [] R. Thus, = $ R is a monoid, the factor monoid of modulo $ R. The pair (; R) is called a monoid-presentation with generators and dening relations R for the monoid =$ R and for each monoid isomorphic to it. A monoid is called nitely generated if it has a presentation (; R) with a nite set of generators, and a monoid is called nitely presented if it has a presentation (; R) with and R both being nite. Two rewriting systems on the same alphabet are called equivalent if they generate the same Thue congruence. A fundamental decision problem for a rewriting system R (a monoid-presentation) is its word problem: given two words u; v, decide whether u and v are congruent modulo R, that

3 is, decide whether u $ R v holds. Let ( ; R ) and ( ; R ) be two nitely generated monoidpresentations of the same monoid. Then the word problem for ( ; R ) is decidable if and only if the word problem for ( ; R ) is decidable. Therefore, we can speak of a nitely generated monoid as having a solvable or an unsolvable word problem. In fact, the same is true when we talk about the complexity of the word problem. If the word problem for ( ; R ) can be solved in time T (n) for some function T : N! N, then the word problem for ( ; R ) can be solved in time T (c n) for some constant c N [AvMa77]. Hence, if the function T is super-additive, that is, T (m + n) T (m) + T (n) for all m; n N, then we can say that the monoid presented by ( ; R ) has a word problem that is decidable in time O(T ). In particular, we can speak of nitely generated monoids with word problems decidable in polynomial time. A rewriting system R is noetherian if no innite reduction sequence of the form u! R u! R u 3! R : : : exists. Two words x and y are called joinable if there is a word z such that x! R z and y! R z both hold. The system R is called conuent if, for all u; v; w, u! R v and u! R w imply that v and w are joinable. If R is both noetherian and conuent, then we say that R is convergent. If R is a convergent rewriting system, then each word has a unique normal form, and two words u and v are congruent modulo R if and only if their respective normal forms modulo R coincide. Thus, if R is a recursive rewriting system that is convergent, then the word problem for R can simply be solved by reduction. Let (; R) be a monoid-presentation. Following Squier [SOK94] we associate a graph? =?(; R) := (V; E; ; ) with this monoid-presentation as follows: (a) V := is the set of vertices, (b) E := f(u; (l; r); v; ") j (l; r) R; u; v ; " f;?gg = R f;?g is the set of edges, (c) the mappings ; : E! V, which associate with each edge e E its initial vertex (e) and its terminal vertex (e), respectively, are dened through and (u; (l; r); v; ") := ulv if " = urv if " =? (u; (l; r); v; ") := urv if " = ulv if " =?: The graph? describes the single-step reduction relation! R on that is induced by the rewriting system R. Accordingly paths in? correspond to sequences of reduction steps and their inverses. By P(?) we denote the set of all paths in?, where a path of length from u to u is included for each u. The mappings and can easily be extended to paths. Further, the concatenation of paths is an associative binary operation on P(?) that we will denote by. By P + (?) we denote the set of all those paths in? that only contain edges of the form (u; (l; r); v; ") with " =. Finally, by P () (?) we denote the set of all parallel paths, that is, P () (?) := f(p; q) j p; q P(?) such that (p) = (q) and (p) = (q)g: Corresponding to the notion of a critical pair of the rewriting system R (see, e.g., [BoOt93]) we dene two types of critical pairs of edges in?:. Let (xyz! u) and (y! v) be two dierent rules in R, where x; y; z; u; v. Then xyz! R u and xyz! R xvz. Let e be the edge (; (xyz; u); ; ), and let e be the edge (x; (y; v); z; ). In this situation, the pair (e ; e ) is called a critical pair.. Let (xy! u) and (yz! v) be two rules in R, where x; y; z; u; v and x; y; z 6=. Then xyz! R uz and xyz! R xv. Let e be the edge (; (xy; u); z; ), and let e be the edge (x; (yz; v); ; ). In this situation, the pair (e ; e ) is called a critical pair. Let (e ; e ) be a critical pair. A resolution of (e ; e ) is a pair (p ; p ) of paths in P + (?) such that (p ) = (e ), (p ) = (e ), and (p ) = (p ) hold. It is well-known that a noetherian rewriting system R is conuent if and only if a resolution exists for each of its critical pairs. 3

4 3 The homological niteness conditions FP n Here we introduce the homological niteness conditions FP n, n N [ fg, that are the main topic of this paper. In the next section we will prove that these conditions are undecidable even for fairly restricted classes of nite monoid-presentations. Let R be a ring (with identity), and let S be a set. By R[S] we denote the set of all mappings f : S! R for which the set fs S j f(s) 6= g is nite. An element f R[S] will be written as f = P ss f(s)[s], that is, each f R[S] is expressed as a formal polynomial P ss r s[s], where fs S j r s 6= g is a nite set. On R[S] the operations of addition and scalar multiplication are dened as follows: if f = P ss r s[s], f = P ss r s[s], and r R, then f + f := (r s + rs)[s] and r f := r s )[s]: ss ss(r It is easily veried that R[S] is a (left) R-module. It is called the free R-module generated by S. If S is a nite set, then jsj is called the rank of R[S]. If the set S is a monoid M, we dene an operation of multiplication on R[M] through f f := mm B@ m ;m M m m =m r m rm CA [m]: It is easily veried that (R[M]; +; ) is a ring with identity. It is called the monoid ring of M over R. The monoid ring Z[M] of M over Z is called the integral monoid ring of M. It is simply denoted by ZM. The elements of ZM are written as P mm z mm, that is, we omit the square brackets. The ring Z itself can be interpreted as a ZM-module by simply dening a scalar multiplication as ( P mm z mm) z := P mm z mz for all P mm z mm ZM and z Z. Let R be a ring, let A; B; C be R-modules, and let ' : A! B and : B! C be R-modulehomomorphisms. The sequence A! ' B! C is called exact at B if Im ' = Ker. Here Im ' denotes the image '(A), and Ker denotes the kernel of. A sequence A n+?! 'n+ A n?! 'n : : :?! ' A?! ' A is exact if it is exact at A ; : : : ; A n, and an innite sequence : : :?! 'i+ A i?! 'i : : :?! '3 A?! ' A?! ' A is exact if it is exact at A i for all i. A resolution of an R-module A over R is an exact sequence of R-module-homomorphisms : : :?! i+ F i?! i : : :?! 3 F?! F?! F?! A?! : If each F i is free, then this is called a free resolution. If each F i is a free R-module of nite rank, then this resolution is called free-of-nite-rank. For n N, a partial resolution of length n of A over R is an exact sequence of R-modulehomomorphisms F n?! n : : :?! 3 F?! F?! F?! A?! : If each F i is free, then this is called a partial free resolution. If each F i is a free R-module of nite rank, then this partial resolution is called free-of-nite-rank. Now we are prepared to give the denition of the announced homological niteness conditions. A monoid M is said to be of type FP, if Z has a free-of-nite-rank resolution over ZM. M is said to be of type FP n for some integer n, if Z has a partial free-of-nite-rank resolution of length n over ZM. The following fundamental result will be used repeatedly in subsequent sections. 4

5 Proposition 3. [Bro8] For every monoid M and every integer n, the following conditions are equivalent: (i) M is of type FP n. (ii) M is nitely generated, and for every partial free-of-nite-rank resolution F k! : : :! F! Z! of Z over ZM with k < n, the kernel of the homomorphism F k! F k? is nitely generated. It follows that for every monoid M, M is of type FP if and only if M is of type FP n for all integers n. Let (; R) be a monoid-presentation of a monoid M. From (; R) we will dene a particular partial free resolution C?! C?! C?! Z?! of Z over ZM. Let C be the free ZM-module generated by the single formal symbol ;, let C be the free ZM-module generated by, and let C be the free ZM-module generated by R. We dene the ZM-module-homomorphism : C! Z through ([;]) :=. Thus, for each element ( P mm z mm)[;] C = ZM[f;g], (( z m m)[;]) = ( z m m) = z m = mm mm mm mm z m : The homomorphism : C! C is dened by setting ([a]) := (a? )[;] for each a : In the expression (a? ), a is used to denote the element of the monoid M that is presented by the letter a. To dene : C! C we need an auxiliary function :! C, which is dened as follows: () := (wa) := (w) + w[a] for all w and a : Here the word w is used to denote the element of the monoid M that is presented by w. Now the ZM-module-homomorphism : C! C is dened by ([`; r]) := (`)? (r) for each (`; r) R: It is well-known that the sequence C?! C?! C?! Z?! is exact [Squ87]. We call it the standard sequence for (; R). We will need standard sequences throughout this paper. When we write ZM[R]! ZM[], then we will always mean the homomorphism from the corresponding standard sequence. If and R are both nite, then all the modules of the standard sequence are of nite rank. Thus, for every nitely presented monoid M, there is a partial free-of- nite-rank resolution of Z over ZM which has length two. This reects the well-known fact that each nitely presented monoid is of type FP. Proposition 3. implies that M is of type FP 3 if and only if the kernel of the homomorphism is nitely generated. We will need an auxiliary function : P(?)! C, where? is the graph associated with the monoid-presentation (; R). This function is dened as follows: for each path p of length, (p) :=, and for each path p P(?) and each edge e = (u; (`; r); v; ") of? satisfying (p) = (e), (p (u; (`; r); v; ")) := (p) + "u[`; r]: Concerning nitely generated presentations involving a convergent rewriting system, Squier has established the following important result. 5

6 Lemma 3. [Squ87] Let (; R) be a nitely generated monoid-presentation such that the rewriting system R is convergent, and let B be the subset of P () (?) that consists of all pairs (e p ; e p ), where (e ; e ) is a critical pair of R, and (p ; p ) is a xed resolution of (e ; e ). Then the kernel of the homomorphism is generated by the set f (p)? (q) j (p; q) Bg. It follows that each monoid that can be presented by a nite convergent rewriting system is of type FP 3. In fact, each monoid with a nite convergent presentation is even of type FP [Ani87, Kob9]. For each path p P(?), ( (p)) = ((p))? ((p)) [Squ87, CrOt94]. This gives the following technical result. Lemma 3.3 [Squ87] For all pairs of parallel paths (p; q) P () (?), ( (p)) = ( (q)). We close this section with a technical result concerning the relationship between the standard sequences of equivalent presentations of a monoid. Let be a nite alphabet, let R and R be two rewriting systems on that are equivalent, and let R denote the union of R and R. Then the presentations (; R ), (; R ), and (; R) all present the same monoid M. Further, let? be the graph that is associated with the presentation (; R), let be the corresponding function : P(?)! ZM[R] as dened above, and let : ZM[R]! ZM[], : ZM[R ]! ZM[], and : ZM[R ]! ZM[] be the corresponding homomorphisms from the respective standard sequences. For each rule (l; r) R, let p (l; r) P(?) denote a path from l to r that contains only edges of the subgraph?(; R ), and for each rule (l; r) R, let p (l; r) P(?) denote a path from l to r that contains only edges of the subgraph?(; R ). Such paths do exist as the rewriting systems R and R are equivalent. Using these paths we dene the following homomorphisms: h : ZM[R ]! ZM[R ] : h ([l; r]) := (p (l; r)); and h : ZM[R ]! ZM[R ] : h ([l; r]) := (p (l; r)); and we take B to denote the following subset of ZM[R ]: With these notions the following result holds. B := f[l; r]? h (h ([l; r])) j (l; r) R g: Lemma 3.4 If B ZM[R ] is a generating set for Ker, then B := h (B )[B is a generating set for Ker. Proof. By hb i and hb i we denote the submodules of ZM[R ] that are generated by B and B, respectively, and by hb i we denote the submodule of ZM[R ] that is generated by B. We need the following auxiliary results. Note that the restriction of the homomorphism to ZM[R ] and to ZM[R ] coincides with the homomorphism and, respectively. Claim. For all x ZM[R ], (h (x)) = (x), and for all x ZM[R ], (h (x)) = (x). Proof. Let x = P (l;r)r t l;r [l; r] ZM[R ]. Then we have (h (x)) = (l;r)r t l;r ((p (l; r))) = (l;r)r t l;r (((; (l; r); ; ))) = (l;r)r t l;r ([l; r]) = (x); where the second equality follows from Lemma 3.3, and the third one follows from the denition of the mapping. Thus, for all x ZM[R ], (h (x)) = (x). By symmetry it follows that, for all x ZM[R ], (h (x)) = (x). Claim implies that B is contained in the kernel of. Claim. For all x ZM[R ], x? h (h (x)) hb i. 6

7 Proof. Let x = P (l;r)r t l;r [l; r] ZM[R ]. Then we have x? h (h (x)) = (l;r)r t l;r ([l; r]? h (h ([l; r]))) hb i: We have already seen that B Ker, and so hb i Ker. To prove the converse inclusion, let x Ker. By Claim, (h (x)) = (h (x)) = (x) =. Thus, h (x) Ker = hb i, which implies that h (h (x)) hh (B )i hb i. By Claim, x? h (h (x)) hb i hb i. It follows that x = h (h (x)) + (x? h (h (x))) hb i. Thus, hb i = Ker. 4 The undecidability result In [OtSa97] a nite alphabet with a distinguished letter o, and a family S n (n N) of nite rewriting systems on are dened that have the following properties. First of all, the word problem for S n is decidable in polynomial time for each n N. Secondly, for each n N, if o $ S n, then s $ S n holds for each letter s, and hence, S n is equivalent to the nite convergent rewriting system fs! j s g, but if o 6$ S n, then there is an innite, recursive, and convergent rewriting system Sn that is equivalent to S n such that S n is a subsystem of Sn, and the left-hand side of no rule of Sn begins with the letter o. Moreover it is shown that, given the system S n as input, it is undecidable whether or not o $ S n holds. For n N, we denote the monoid presented by ( ; S n ) as Mn. In addition to the above family of monoids, we consider a nitely presented monoid from [Squ87]. Let = fa; b; t; x ; x ; y ; y g, where we assume without loss of generality that and are disjoint. We consider the following rewrite rules on, where each rule is given a name for future reference: at n b! (P n ) x i a! atx i (A i ) x i t! tx i (T i ) x i b! bx i (B i ) x i y i! (Q i ) Here, n ranges over all non-negative integers, while i ranges over the set f; g only. We dene the following rewriting systems on : T = fp ; A i ; T i ; B i ; Q i j i = ; g, and T = fp n ; A i ; T i ; B i ; Q i j n ; i = ; g: In [Squ87] Squier shows that the systems T and T are equivalent, that their word problems are solvable in polynomial time, that the innite system T is recursive and convergent, and that the monoid N presented by ( ; T ) is not of type FP 3. For each n N we now dene a new monoid M n as an extension of the free product of the monoids M n and N as follows. Let H = [ be a new letter, let := [ [ fhg, and let R n := S n [ T [ fho! o; Hb! Hg, and let M n denote the monoid presented by (; R n ). Lemma 4. For each n N, if o $ S n, then the monoid M n is presented by a nite convergent rewriting system, and its word problem is solvable in polynomial time. Proof. If o $ S n, then s $ S n holds for all s. Hence, H $ R n, which in turn implies that b $ R n, a $ R n, and t $ R n. Thus, R n is equivalent to the nite rewriting system fs! j s g [ fh! ; a! ; b! ; t! g [ fx i y i! j i = ; g; which is length-reducing and convergent. The result follows. 7

8 Hence, if o $ S n, then the monoid M n is of type FP. On the other hand, we will see that the monoid M n is not of type FP 3, if o $ S n does not hold. Accordingly, let us assume for the following considerations that n N is chosen such that o $ S n does not hold. Obviously, the system R n is equivalent to the innite rewriting system Rn := Sn [ T [ fho! o; Hb! Hg. Since the letter H does neither occur in the rules of Sn nor in the rules of T, it is easily seen that Rn is noetherian. As the systems Sn and T are both convergent, and as they have no letter in common, their union Sn [ T is convergent, too. The left-hand side of no rule of Sn begins with o, and the left-hand side of no rule of T begins with b. Hence, there are no critical pairs between the rules in Sn [ T and the rules Ho! o and Hb! H. Thus, the system Rn = Sn [ T [ fho! o; Hb! Hg is also convergent. It follows that the monoids Mn and N are both embedded in M n by the corresponding identity mapping. Since the systems Sn and T are both recursive, it follows that Rn is recursive, too. Thus, the monoid M n has a solvable word problem. In fact, the word problem for M n is solvable in polynomial time, since occurrences of the letter H are not introduced by applications of rules of Rn, and Mn as well as N have word problems that are decidable in polynomial time. Thus, we have the following. Lemma 4. For each n N, the word problem for the monoid M n is solvable in polynomial time. We claim that the monoid M n is not of type FP n, since o $ S n does not hold. To prove this we want to apply Lemma 3. to the convergent rewriting system Rn. Let?,?, and? be the graphs that are associated with the monoid-presentations ( ; Sn ), ( ; T ), and (; Rn ), respectively. Note that? and? both are subgraphs of?. The only critical pairs of Rn are those of Sn and those of T. For each of these critical pairs we x an arbitrary resolution. Let C be the subset of P () (?) that consists of all pairs (e p ; e p ), where (e ; e ) is a critical pair of Sn and (p ; p ) is the chosen resolution of (e ; e ). Obviously, C P() (? ). Analogously, let C be the subset of P () (?) that consists of all pairs (e p ; e p ), where (e ; e ) is a critical pair of T and (p ; p ) is the chosen resolution of (e ; e ). Then C P () (? ). Let : P(?)! ZM n [Rn ] be the corresponding function dened as in Section 3, and let B := f(p)? (q) j (p; q) C g and B := f(p)? (q) j (p; q) Cg. Then B ZMn[S n ] and B ZN[T ]. By Lemma 3. B [ B generates the kernel of the homomorphism ZM n [Rn ]! ZM n []. Since R n and Rn are equivalent, we can apply Lemma 3.4, which implies that there exist sets B ZMn[S n ] and B ZN[T ] such that B [ B generates the kernel of the homomorphism ZM n [R n ]! ZM n []. For the sake of our argument, let us assume that the monoid M n is of type FP 3. It follows that the kernel of the homomorphism ZM n [R n ]! ZM n [] is nitely generated. Thus we can assume that the sets B and B are both nite. Consider the homomorphisms : ZM n [R n ]! ZM n [] and N : ZN[T ]! ZN[ ]. Note that the restriction of to ZN[T ] equals N. The set B [ B generates the kernel of. We want to show that this implies that the kernel of N is generated by B. So let p Ker N. Obviously this implies that p Ker. Since Ker is generated by B [ B, and since p ZN[T ], it follows that p is in the submodule of ZM n [R n ] that is generated by B. To complete our argument we need the following auxiliary result. Lemma 4.3 For all x M n r N and all y N, xy M n r N. Proof. Let x M n r N, let y N, let u be the word that represents the monoid element x and that is irreducible modulo Rn, and let v be the word that represents the monoid element y and that is irreducible modulo T. As x = N, we have u =, that is, there exist u, s [ fhg, and u such that u = u su. Let w be the normal form of u v modulo T. If s, then the normal form of uv modulo Rn is u sw. Since u sw =, we see that xy = N. Finally suppose that s = H. Then there exist k and w such that w = b k w, and w does not begin with the letter b. In this case the normal form of uv modulo Rn is u Hw. Since u Hw =, we see that xy = N. P mmn bb Since p is in the submodule of ZM n [R n ] that is generated by B, p can be written as p = z m;b mb, where z m;b Z. Obviously p can uniquely be decomposed as p = p + p, where 8

9 p = P P mn z m;b mb and p = P mmnrn z m;b mb. Now p and p can uniquely be written as bb bb p = mmn z m[l; r] and p (l;r)t m;(l;r) = mmn z m[l; r], where (l;r)t m;(l;r) z ; m;(l;r) z Z. Since N is m;(l;r) a submonoid of M n, it follows that, for all m M n and (l; r) T, if z 6=, then m N holds. m;(l;r) By Lemma 4.3 it follows that, for all m M n and (l; r) T, if z 6=, then m M m;(l;r) n r N holds. Since p = p + p and p ZN[T ], it follows that p =, that is, p = p. Thus, we see that p is in the submodule of ZN[T ] that is generated by B. Hence, Ker N is generated by the nite set B. It follows that N is of type FP 3, contradicting the results on the monoid N stated before. Hence, we have shown the following result. Lemma 4.4 For each n N, if o 6$ S n, then the monoid M n is not of type FP 3. Finally, let P be a property of nitely presented monoids such that each monoid with a nite convergent presentation has property P, and each nitely presented monoid with property P is of type FP 3. Then by Lemma 4. and Lemma 4.4, it follows for all n N that o $ S n holds if and only if the monoid M n has property P. However, given the system S n, it is undecidable whether or not o $ S n holds. Hence, given the system R n, it is undecidable whether or not the monoid M n presented by (; R n ) has property P. Thus we have the following undecidability result. Theorem 4.5 Let P be a property of nitely presented monoids such that each monoid with a nite convergent presentation has property P, and each nitely presented monoid with property P is of type FP 3. Then for the class of all nitely presented monoids with word problems decidable in polynomial time the property P is undecidable. Examples of such properties are the homological niteness conditions FP n for each n 3 and FP. Another property that a nitely presented monoid satises if it admits a presentation through some nite convergent rewriting system is the homotopical niteness condition of having nite derivation type [SOK94]. In [CrOt94] it is shown that each nitely presented monoid that has nite derivation type is of type FP 3. So our result also applies to this property, thus giving an alternative proof for the undecidability of this property for the class of all nitely presented monoids with word problems decidable in polynomial time [OtSa97]. 5 Free products In the remaining part of the paper we consider the class of nitely presented monoids that are of type FP n for some n 3, and we ask for the closure properties of this class with respect to various algebraic constructions. For the special case of groups corresponding results are well-known [Bie76], but since until recently the case of monoids did not receive that much attention in the literature, we feel it worth-while to investigate this case in detail. In this section we will show that, for nitely presented monoids M and M and an integer n 3, the free product of M and M is of type FP n if and only if M and M both are of type FP n. In the next section we will then consider the operation of free product with amalgamated submonoids. Let ( ; R ) and ( ; R ) be two nite monoid-presentations, where we assume that and are disjoint. Let M be the monoid presented by ( ; R ), let M be the monoid presented by ( ; R ), and let M be the monoid that is presented by (; R), where := [ R := R [ R. Then M is the free product of M and M. Each element x M r fg can uniquely be written as a product x = x x n of elements x ; : : : ; x n (M [ M ) r fg, where n, and for all i = ; : : : ; n?, x i M implies that x i+ M, and x i M implies that x i+ M. Let M be the subset of M that contains the identity and all x M r fg such that, if x = x x n as above, then x n M. Analogously, let M be the subset of M that contains the identity and all x M r fg such that, if x = x x n as above, then x n M. We will have to compare exact sequences of ZM -module-homomorphisms and ZM -modulehomomorphisms with exact sequences of ZM-module-homomorphisms. For doing so we introduce the following notion. and 9

10 For some nite sets of generators ; ; ; Y ; Y ; and Y, let : ZM [ ]! ZM [Y ]; : ZM [ ]! ZM [Y ]; and : ZM[]! ZM[Y ] denote a ZM -, a ZM -, and a ZM-module-homomorphism, respectively. We say that extends and if all of the following conditions are satised: () ; are disjoint sets, and = [. () Y ; Y are disjoint sets, and Y = Y [ Y. (3) For all x, ([x]) = ([x]). (4) For all x, ([x]) = ([x]). The following technical result will be helpful below. Lemma 5. If : ZM [ ]! ZM [Y ], : ZM [ ]! ZM [Y ], and : ZM[]! ZM[Y ] are module-homomorphisms such that extends and, then the following statements hold: () Im is generated by Im [ Im. () Ker is generated by Ker [ Ker. (3) Im \ ZM [Y ] Im and Im \ ZM [Y ] Im. Proof. Let x ZM[]. Then x can uniquely be written as x = up u + vq v ; um vm where p u ZM [ ] and q v ZM [ ]. Applying we obtain (x) = u(p u ) + v(q v ); um vm which shows that Im is generated by Im [ Im. To prove (), let x Ker, that is, (x) =. This element x can be written as a sum as above, and it follows that (p u ) = for all u M and (q v ) = for all v M. Hence, x is in the submodule of ZM[] that is generated by Ker [ Ker, implying that Ker is included in this submodule. The converse inclusion is obvious, as Ker [ Ker Ker. This completes the proof of (). To prove (3), suppose that (x) ZM [Y ]. Then (p u ) = for all u M r fg and (q v ) = for all v M. Thus, (x) = (p ) = (p ) Im. This shows that Im \ ZM [Y ] Im. By symmetry it follows that Im \ ZM [Y ] Im. Based on this technical result we will now establish the following result relating exact sequences of ZM -module-homomorphisms and ZM -module-homomorphisms with exact sequences of ZMmodule-homomorphisms. Lemma 5. Let () ZM [ ] () ZM [ ] (3) ZM[]?! ZM [Y ]?! ZM [Y ]?! ZM[Y ]?! ZM [Z ];?! ZM [Z ]; and?! ZM[Z] be sequences of module-homomorphisms such that extends and, and extends and. Then the sequences () and () are both exact if and only if sequence (3) is exact.

11 Proof. Suppose that the sequences () and () are exact. By Lemma 5. Im is generated by Im [ Im, and Ker is generated by Ker [ Ker. Since Im = Ker and Im = Ker, it follows that Im = Ker, that is, sequence (3) is exact, too. To prove the converse implication, assume that sequence (3) is exact, and let x Im. Then x Im = Ker. Since (x) = (x) =, we see that x Ker. Thus, Im Ker. Conversely, if x Ker, then x Ker = Im. By Lemma 5. (3), it follows that x Im. Hence, Im = Ker, that is, sequence () is exact. Analogously, it follows that Im = Ker, that is, sequence () is exact as well. A scheme of length n (n ) consists of three exact sequences ZM [ (n) ZM [ (n)?! : : : ] (n)?! : : : ] (n) ZM[ (n) ]?! (n) : : : (4)?! ZM [ (3) (4)?! ZM [ (3)?! ZM [R ] ()?! ZM [ ]; ] (3)?! ZM [R ] ()?! ZM [ ]; ] (3) (4)?! ZM[ (3) ] (3)?! ZM[R] ()?! ZM[] of ZM -, ZM -, and ZM-module-homomorphisms, respectively, where the homomorphisms () (), and () are those from the standard sequences for ( ; R ), ( ; R ), and (; R), respectively, such that, for all i = 3; : : : ; n, the sets (i), (i), and (i) are nite, and the homomorphism (i) is the extension of the homomorphisms (i) and (i) (). We will also write, (), and () to denote R, R, and R, respectively. Observe that there is exactly one scheme of length. It is easily seen that the sequences of a scheme of length n can be combined with the standard sequences for ( ; R ), ( ; R ), and (; R), respectively, to form partial free-of-nite-rank resolutions of length n for Z over ZM, ZM, and ZM, respectively. The following two lemmata show that a scheme of length n can be extended to a scheme of length n +, if either M and M both are of type FP n+, or if the monoid M is of type FP n+. Lemma 5.3 For each n, if M and M both are of type FP n+, then a scheme of length n can be extended to a scheme of length n +. Proof. Consider a scheme of length n. By combining the rst two sequences with the standard sequences we obtain partial free-of-nite-rank resolutions of length n for Z over ZM and ZM, respectively. Since M and M both are of type FP n+, it follows by Proposition 3. that Ker (n) and Ker (n) both are nitely generated. Hence, there exist nite sets (n+) Ker (n) and, i = ;. We extend the given scheme of length n by one step by dening the ZM -module-homomorphism (n+) : ZM [ (n+) ]! (n+) Ker (n) such that (n+) i generates Ker (n) i ZM [ (n) ] through (n+) ([x]) = x for all x (n+), the ZM -module-homomorphism (n+), : ZM [ (n+) ]! ZM [ (n) ] through (n+) ([x]) = x for all x (n+), and the ZM-modulehomomorphism (n+) : ZM[ (n+) ]! ZM[ (n) ] as the extension of (n+) and (n+). Since (n+) generates Ker (n), Im (n+) = Ker (n). Analogously, since (n+) generates Ker (n), Im (n+) = Ker (n). By Lemma 5. it follows that Im (n+) = Ker (n). Thus, the resulting three sequences are exact, that is, we have a scheme of length n +. Lemma 5.4 For each n, if M is of type FP n+, then a scheme of length n can be extended to a scheme of length n +. Proof. Consider a scheme of length n. By combining the third sequence with the standard sequence we obtain a partial free-of-nite-rank resolution of length n for Z over ZM. Since M is of type FP n+, it follows by Proposition 3. that Ker (n) is nitely generated. On the other hand, Ker (n) is generated by Ker (n) [ Ker (n) (Lemma 5.). It follows that there exist nite sets (n+) Ker (n) and (n+) Ker (n) such that (n+) := (n+) [ (n+) generates Ker (n). We extend the given scheme of length n by one step by dening the ZM -module-homomorphism

12 (n+) : ZM [ (n+) ]! ZM [ (n) ] through (n+) [(x)] = x for all x (n+), the ZM -modulehomomorphism (n+) : ZM [ (n+) ]! ZM [ (n) ] through (n+) [(x)] = x for all x (n+), and the ZM-module-homomorphism (n+) : ZM[ (n+) ]! ZM[ (n) ] as the extension of (n+) and (n+). Since (n+) generates Ker (n), Im (n+) = Ker (n). By Lemma 5. it follows that Im (n+) = Ker (n) and Im (n+) = Ker (n). Thus, the resulting three sequences are exact, and we have a scheme of length n +. Let n 3. If M and M both are of type FP n, then, using Lemma 5.3, the scheme of length can be extended to a scheme of length n. Analogously, if M is of type FP n, the scheme of length can be extended to a scheme of length n by Lemma 5.4. Conversely, if a scheme of length n exists, then by combining the sequences of this scheme with the standard sequences for ( ; R ), ( ; R ), and (; R), respectively, we obtain partial free-of-nite rank resolutions of length n for Z over ZM, ZM, and ZM, respectively. Thus, we see that M, M, and M all are of type FP n. This proves the following result. Theorem 5.5 Let M and M be two nitely presented monoids, and let M be the free product of M and M. Then, for each n 3, M and M both are of type FP n if and only if M is of type FP n. In particular, M and M both are of type FP if and only if M is of type FP. 6 Free products with amalgamated submonoids In this section we consider the operation of free product with amalgamated submonoids. We will show that, for nitely presented monoids M and M with a common submonoid M and an integer n 3, the free product of M and M with the submonoid M amalgamated is of type FP n if and only if M and M both are of type FP n. However, in order to prove this equivalence we have to place rather serious restrictions on the presentations of M and M considered, which in turn translate into rather serious restrictions on the type of amalgamation covered by our result. Let ;, and be three nite alphabets that are pairwise disjoint, and let := [ [. Further, let R be a nite string-rewriting system on, let R be a nite string-rewriting system on [ such that the left-hand side and the right-hand side of each rule of R begins with a letter from, let R be a nite string-rewriting system on [ such that the left-hand side and the right-hand side of each rule of R begins with a letter from, and let R := R [ R [ R. Finally, let M be the monoid presented by ( ; R ), let M be the monoid presented by ( [ ; R [R ), let M be the monoid presented by ( [ ; R [ R ), and let M be the monoid presented by (; R). It is easily seen that M is a submonoid of both M and M. Further, M is the free product of M and M with the submonoid M amalgamated. We want to show, for all n 3, that under the above restrictions M is of type FP n if and only if M and M both are of type FP n. Let ; ; and be pairwise disjoint sets, and let := [ [. Consider the module ZM [ [ ], and let m M and x [. We say that the module element m[x] has a proper prex in M, if either m itself has a prex in M r M, that is, m M, or m = and x. By ZM [ [ ] we denote the subset of ZM [ [ ] that consists of all elements x [ mm z m;x m[x]; where, for all x [, and m M, the integer z m;x Z is either or the module element m[x] has a proper prex in M. Analogously, consider the module ZM [ [ ], and let m M and x [. We say that the module element m[x] has a proper prex in M, if either m itself has a prex in M r M, that is, m M, or m = and x. By ZM [ [ ] we denote the subset of ZM [ [ ] that consists of all elements x [ mm z m;x m[x];

13 where, for all x [ and m M, the integer z m;x Z is either or the module element m[x] has a proper prex in M. In the following we will follow the development of the previous section. Let : ZM [ [ ]! ZM [Y [ Y ]; : ZM [ [ ]! ZM [Y [ Y ]; : ZM[]! ZM[Y ] be a ZM -, a ZM -, and a ZM-module-homomorphism, respectively. We say that extends and if the following conditions are satised simultaneously: () ; ; are pairwise disjoint sets, and = [ [. () Y ; Y ; Y are pairwise disjoint sets, and Y = Y [ Y [ Y. (3) For all x, ([x]) = ([x]) = ([x]) ZM [Y ]. (4) For all x, ([x]) = ([x]) ZM [Y [ Y ]. (5) For all x, ([x]) = ([x]) ZM [Y [ Y ]. If extends and, then we denote by the restriction of, respectively, to ZM [ ], by we denote the restriction of to ZM [ [ ], and by we denote the restriction of to ZM [ [ ]. Note the restrictions that are placed on and in (3) to (5). They imply that Im ZM [Y ], and from them it is straightforward to deduce that Im ZM [Y [Y ] and Im ZM [Y [Y ]. Lemma 6. If : ZM [ [ ]! ZM [Y [ Y ], : ZM [ [ ]! ZM [Y [ Y ], and : ZM[]! ZM[Y ] are module-homomorphisms such that extends and, then all of the following statements hold: () Im is generated by Im [ Im [ Im. () Ker is generated by Ker [ Ker [ Ker. (3) Im \ ZM [Y [ Y ] Im and Im \ ZM [Y [ Y ] Im. (4) Im is generated by Im [ Im. (5) Ker is generated by Ker [ Ker. (6) Im is generated by Im [ Im. (7) Ker is generated by Ker [ Ker. Proof. Let x ZM[]. Then x can uniquely be written as x = x + up u + vq v ; um [M M vm [M M where x ZM [ ], p u ZM [ [ ], and q v ZM [ [ ]. Applying we obtain (x) = (x ) + u(p u ) + v(q v ); um [M M vm [M M which shows that Im is generated by Im [ Im [ Im. To prove (), let x Ker, that is, (x) =. The element x can be decomposed as above, and for this decomposition we obtain that (x ) =, (p u ) = for all u M [ M M, and (q v ) = for all v M [ M M. Hence, x is in the submodule of ZM[] that is generated by Ker [ Ker [ Ker, which implies that Ker is included in this submodule. Since the converse inclusion is obvious, this proves (). 3

14 To prove (3), suppose that (x) ZM [Y [ Y ]. Then (p u ) = for all u M M, and (q v ) = for all v M [ M M. Thus, (x) = (x ) + P um u(p u ) = (x + P um up u ) = (x + P um up u ) Im. Hence, Im \ ZM [Y [ Y ] Im. By symmetry it follows that Im \ ZM [Y [ Y ] Im. Now let x ZM [ [ ]. Then x can uniquely be written as x = x + um up u ; where x ZM [ ] and p u ZM [ [ ]. Applying we obtain (x) = (x ) + u(p u ); um which shows that Im is generated by Im [ Im. This proves (4). To prove (5), let x Ker, that is, (x) =. It follows that (x ) = and (p u ) = for all u M. Hence, x is in the submodule of ZM [ [ ] that is generated by Ker [ Ker. Thus, Ker is included in this submodule. Since the converse inclusion is obvious, this proves (5). The proof of (6) is analogous to the proof of (4), and the proof of (7) is analogous to the proof of (5). Lemma 6. If () ZM [ [ ] () ZM [ [ ] (3) ZM[]?! ZM [Y [ Y ]?! ZM [Y [ Y ]?! ZM[Y ]?! ZM [Z [ Z ];?! ZM [Z [ Z ]; and?! ZM[Z] are sequences of module-homomorphisms such that extends and, and extends and, then the sequences () and () are both exact if and only if sequence (3) is exact. Proof. Suppose that the sequences () and () are exact. From Lemma 6. (), (4), and (6), it follows that Im is generated by Im [ Im, and from Lemma 6. (), (5), and (7), it follows that Ker is generated by Ker [ Ker. Since Im = Ker and Im = Ker, it follows that Im = Ker, that is, sequence (3) is exact. To prove the converse implication, suppose that sequence (3) is exact. If x Im, then x Im = Ker. Since (x) = (x) =, we see that x Ker. Thus, Im Ker. Conversely, if x Ker, then x Ker = Im. By Lemma 6. (3), it follows that x Im. Thus, Im = Ker, that is, sequence () is exact. Analogously, it follows that Im = Ker, that is, sequence () is exact as well. A scheme of length n (n ) consists of three exact sequences ZM [ (n) [ (n) ZM [ (n) [ (n)?! : : : (4)?! ZM [ (3) [ (3) ] (3)?! ZM [R [ R ] ()?! ZM [ [ ]; ] (n)?! : : : (4)?! ZM [ (3) [ (3) ] (3)?! ZM [R [ R ] ()?! ZM [ [ ]; ] (n) ZM[ (n) ]?! (n) : : :?! (4) ZM[ (3) ]?! (3) ZM[R]?! () ZM[] of ZM -, ZM -, and ZM-module-homomorphisms, respectively, where the homomorphisms () (), and () are those from the standard sequences for ( [ ; R [ R ), ( [ ; R [ R ), and (; R), respectively, such that, for all i = 3; : : : ; n, the sets (i), (i), (i), and (i) are nite, and the homomorphism (i) is the extension of the homomorphisms (i) and (i). We will also denote the sets R, R, R, and R, by (), (), (), and (), respectively. Note that there is exactly one scheme of length. It is easily seen that the sequences of a scheme of length n can be combined with the standard sequences for ( [ ; R [R ), ( [ ; R [R ), and (; R), respectively, to give partial free-of-nite-rank resolutions of length n for Z over ZM, ZM, and ZM, respectively., 4

15 Lemma 6.3 For each n, if M and M both are of type FP n+, then a scheme of length n can be extended to a scheme of length n +. Proof. Consider a scheme of length n. By combining the rst two sequences with the standard sequences we obtain partial free-of-nite-rank resolutions of length n for Z over ZM and ZM, respectively. Since M and M both are of type FP n+, it follows by Proposition 3. that Ker (n) and Ker (n) both are nitely generated. On the other hand, Ker (n) is generated by Ker (n) [Ker (n), and Ker (n) is generated by Ker (n) [ Ker (n) (Lemma 6.). It follows that there exist nite sets (n+), and Ker (n) (n+) such that [ (n+) generates. We extend the given scheme of length n by one step by dening the ZM -module-homomorphism (n+) : ZM [ (n+) [ (n+) ]! ZM [ (n) [ (n) ] (n+) Ker (n), (n+) Ker (n) Ker (n), and (n+) [ (n+) generates Ker (n) through (n+) ([x]) = x for all x (n+) [ (n+), the ZM -module-homomorphism (n+) :, ZM [ (n+) [ (n+) ]! ZM [ (n) [ (n) ] through (n+) ([x]) = x for all x (n+) [ (n+) and the ZM-module-homomorphism (n+) : ZM[ (n+) ]! ZM[ (n) ] as the extension of (n+) and (n+). Since (n+) [ (n+) generates Ker (n), Im (n+) = Ker (n). Analogously, since (n+) [ (n+) generates Ker (n), Im (n+) = Ker (n). By Lemma 6. it follows that Im (n+) = Ker (n). Thus, the resulting three sequences are exact. Hence, we have a scheme of length n +. Lemma 6.4 For each n, if M is of type FP n+, then a scheme of length n can be extended to a scheme of length n +. Proof. Consider a scheme of length n. By combining the third sequence with the standard sequence we obtain a partial free-of-nite-rank resolution of length n for Z over ZM. Since M is of type FP n+, it follows by Proposition 3. that Ker (n) is nitely generated. On the other hand, (n) [ Ker (Lemma 6.). It follows that there exist (n+), and Ker (n) such that (n+) := generates Ker (n). We extend the given scheme of length n by one step by dening the ZM -module-homomorphism (n+) : ZM [ (n+) [ (n+) ]! ZM [ (n) [ (n) Ker (n) is generated by Ker (n) [ Ker (n) nite sets (n+) Ker (n), (n+) Ker (n) (n+) [ (n+) [ (n+) ] :, through (n+) ([x]) = x for all x (n+) [ (n+), the ZM -module-homomorphism (n+) ZM [ (n+) [ (n+) ]! ZM [ (n) [ (n) ] through (n+) ([x]) = x for all x (n+) [ (n+) and the ZM-module-homomorphism (n+) : ZM[ (n+) ]! ZM[ (n) ] as the extension of (n+) and (n+). Since (n+) generates Ker (n), Im (n+) = Ker (n). By Lemma 6. it follows that Im (n+) = Ker (n), and Im (n+) = Ker (n). Thus, the resulting three sequences are exact. Hence, we have a scheme of length n +. Let n 3. If M and M both are of type FP n, then by Lemma 6.3 the scheme of length can be extended to a scheme of length n. Analogously, if M is of type FP n, then by Lemma 6.4 the scheme of length can be extended to a scheme of length n. Conversely, if a scheme of length n exists, then by combining the sequences with the standard sequences for ( [ ; R [ R ), ( [ ; R [ R ), and (; R), respectively, we obtain partial free-of-nite rank resolutions of length n for Z over ZM, ZM, and ZM, respectively. Thus, M, M, and M all are of type FP n. This yields the following result. Theorem 6.5 For each n 3, the monoids M and M are both of type FP n if and only if the free product M of M and M with the submonoid M amalgamated is of type FP n. In particular, M and M both are of type FP if and only if M is of type FP. A corresponding result for the homotopical niteness condition of having nite derivation type is presented in [Ott97]. It remains the question of whether these results can be extended to free products with amalgamation that are less severely restricted. 5

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