Mar 12, 1996 - Thus an e ective construction of each automaton A(X; R; w) provides a ..... with the property that there is at least one edge labelled over A A?1.
AMALGAMS OF FREE INVERSE SEMIGROUPS Alessandra Cherubini, John Meakin, Brunetto Piochi March 12, 1996 Abstract
We study inverse semigroup amalgams of the form S U T where S and T are free inverse semigroups and U is an arbitrary nitely generated inverse subsemigroup of S and T . We make use of recent work of Bennett to show that the word problem is decidable for any such amalgam. This is in contrast to the general situation for semigroup amalgams, where recent work of Birget, Margolis and Meakin shows that the word problem for a semigroup amalgam S U T is in general undecidable, even if S and T have decidable word problem, U is a free semigroup, and the membership problem for U in S and T is decidable. We also obtain a number of results concerning the structure of such amalgams. We obtain conditions for the D-classes of such an amalgam to be nite and we show that the amalgam is combinatorial in such a case. For example every one-relator amalgam of this type has nite D-classes and is combinatorial. We also obtain information concerning when such an amalgam is E unitary: for example every one relator amalgam of the form Inv < A [ B : u = v > where A and B are disjoint and u (resp. v ) is a cyclically reduced word over A [ A?1 (resp. B [ B ?1 ) is E -unitary.
1 Introduction
If S and S are semigroups such that S \ S = U is a non-empty subsemigroup of both S and S , then [S ; S ; U ] is called an amalgam of semigroups and U is the core of the amalgam. The amalgam [S ; S ; U ] is said to be strongly embeddable in a semigroup if there exists a semigroup S and injective homomorphisms 1
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i : Si ! S such that
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Research of all authors is supported by a grant from the Italian CNR. The rst and third authors' research was partially supported by MURST. The second author's research was also partially supported by NSF and the Center for Communication and Information Science of the University of Nebraska at Lincoln.
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S \ S = U = U : A semigroup S is a regular semigroup if for each a 2 S there exists a0 2 S such that a = aa0a and a0 = a0aa0 : such an element a0 is called an inverse of a. If each element of S has a unique inverse, S is called an inverse semigroup : equivalently, an inverse semigroup is a regular semigroup whose idempotents commute. Such semigroups may be faithfully represented as semigroups of partial one - one maps on a set X . We refer the reader to Petrich [18] for this result and many other standard results and ideas about inverse semigroups. In particular, we denote by a? the unique inverse of the element a in an inverse semigroup S . It is well known that a semigroup amalgam [S ; S ; U ] is not necessarily (strongly) embeddable. However it was shown by Howie in 1962 [7] that the amalgam is strongly embeddable if U is unitary in S and S . There is a large literature devoted to the question of when a semigroup amalgam is (strongly) embeddable. We refer to the paper of Howie [8] for some references to the literature on this problem and for an introduction to the connection between this problem and the notion of tensor product of semigroup actions. In particular, a very important theorem of T.E. Hall [6] shows that the category of inverse semigroups has the strong amalgamation property. That is, every amalgam of inverse semigroups may be strongly embedded in an inverse semigroup. It follows that such an amalgam may be strongly embedded in the free product with amalgamation in the category of inverse semigroups. We shall be concerned exclusively with inverse semigroups in this paper and will denote by S U S the free product of S and S amalgamating the inverse subsemigroup U in the category of inverse semigroups. This object is de ned by the usual universal diagram in the category of inverse semigroups. While there is a large literature concerned with the embeddability question for semigroup amalgams, there appears to have been until very recently little work concerned with the structure of semigroup amalgams or even with the word problem for such semigroups. During the past year several important developments along these lines have occurred. In [5] Haataja, Margolis and Meakin showed how the ideas of the Bass-Serre theory of groups acting on graphs may be used to study the structure of the maximal subgroups of an inverse semigroup amalgam S U S in the case when U contains all of the idempotents of both S and S . In [4] Birget, Margolis and Meakin showed that the word problem for a semigroup amalgam S U S is in general undecidable even if S and S have decidable word problem, the membership problem for U in Si is decidable and U is a free semigroup which is unitary in each Si. Thus the situation is very di erent from the situation for group amalgams, where the word problem is trivially decidable if the ambient groups have decidable word problem and the subgroup has decidable membership problem in each group. Recently, Sapir [19] has proved that the embeddability problem is in general undecidable for semigroup amalgams, even if all semigroups are nite. In his Ph.D thesis, Bennett [1] has introduced a class of inverse semigroup amalgams and has developed an algorithm for constructing the corresponding Schutzenberger graphs. We shall make use of his ideas throughout this paper and will provide a brief outline of parts of his work in the next section. An even more recent paper by Stephen [21] provides additional information about the Schutzenberger graphs of inverse semigroup amalgams and enables him to 1
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obtain an alternative proof of Hall's embeddability theorem and considerable structural information. Our concern in the present paper is with inverse semigroups of the form S = FIS (A)U FIS (B ) where A and B are disjoint non-empty sets and U is a nitely generated inverse subsemigroup of FIS (A) and FIS (B ). Here FIS (A) denotes the free inverse semigroup on A. We refer the reader to the book of Petrich [18] for much information about free inverse semigroups - i.e. free objects in the category of inverse semigroups. We recall here Munn's solution of the word problem for the free inverse semigroup since we shall have frequent occasion to use this throughout the paper. Denote by FG(A) the free group on A and by ?(A) the Cayley graph of FG(A) relative to the usual presentation. (Thus ?(A) is a tree of course.) For each word w 2 (A [ A? ) denote by MT (w) the nite subtree of ?(A) traversed when the path labelled by the word w is read in ?(A) starting at the vertex 1 and ending at the vertex r(w) (the reduced form of w in the usual group-theoretic sense). Then Munn's theorem [17] says that two words u and v in (A [ A? ) are equal in FIS (A) if and only if MT (u) = MT (v) and r(u) = r(v). This result is the starting point for a large amount of work that has been done over the past decade on presentations of inverse semigroups by generators and relations. We refer the reader to the papers of Stephen [20], Margolis and Meakin [12], Birget, Margolis and Meakin [3] and Margolis, Meakin and Stephen [14], [15] for some papers along these lines. Much of the work on presentations of inverse semigroups that had been done up until the last two or three years is surveyed in the paper of Meakin [16]. Central to all of this work is the notion of the Schutzenberger automaton A(X; R; w) of a word w 2 (X [ X ? ) relative to a presentation < X : R > of an inverse semigroup. The underlying graph of A(X; R; w) is the Schutzenberger graph S ?(X; R; w). The vertices of S ?(X; R; w) are the elements of the inverse semigroup S = Inv < X : R > that are related via Green's R relation to the image of w in S . There is an edge in S ?(X; R; w) from the vertex u to the vertex v if u and v are R-related to w in S and v = ux for some x 2 X [ X ? . (Thus S ?(X; R; w) is the restriction of the Cayley graph of the inverse semigroup presentation S = Inv < X : R > to the R-class of S that contains the image of w.) The automaton A(X; R; w) is obtained from S ?(X; R; w) by designating ww? as the initial state and w as the terminal state of the automaton. The automaton A(X; R; w) may be viewed as a birooted graph with initial root = ww? and terminal root = w. It is occasionally convenient to use the notation ( ; ?; ) to refer to such an automaton. We refer to [20] or [16] for more details and more precision about Schutzenberger automata and their role in inverse semigroup theory. We also refer to these papers and to [13] for a description of Stephen's iterative construction of A(X; R; w) from the linear automaton of the word w. These automata are crucial in the study of inverse semigroup presentations since from [20] we note that for any two words u; v 2 (X [ X ? ), u = v in S if and only if A(X; R; u) = A(X; R; v) (or equivalently if and only if these automata accept the same language). Thus an e ective construction of each automaton A(X; R; w) provides a solution to the word problem for S . Conversely, if S has decidable word problem and if w 2 (X [X ? ), then for each word u 2 (X [X ? ) there is an algorithm to decide whether ww? = uu? in S , i.e. to decide whether wRu in S . Hence for each letter x 2 X [ X ? and each word u 2 (X [ X ? ), there is an algorithm to decide whether the edge of the Cayley graph of S that starts at u, ends at ux and is labelled by x, belongs to the 1
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Schutzenberger automaton A(X; R; u) or not. We summarize this discussion by saying that the word problem for S is decidable if and only if each Schutzenberger automaton A(X; R; w) is e ectively constructible. Again we refer to [20] for details. In his thesis [1] and subsequent paper [2], Bennett provides a construction of the Schutzenberger automata corresponding to a lower bounded amalgam. We provide a brief analysis of Bennett's algorithm in the next section and we shall use this in subsequent parts of the paper to study inverse semigroup amalgams of the form FIS (A) U FIS (B ) where U is nitely generated.
2 An analysis of Bennett's algorithm In this section we provide a brief description of the algorithm developed by Bennett [2] for constructing the Schutzenberger automata corresponding to a class of inverse semigroup amalgams that Bennett refers to as lower bounded amalgams. We rst need to introduce some notation in order to discuss Bennett's ideas. Let [S ; S ; U ] be an amalgam in the class of inverse semigroups with amalgamated free product S U S . Denote the free product of S and S in the category of inverse semigroups by S S . Free products of inverse semigroups have been studied extensively by Jones [9], [10] and by Jones, Margolis, Meakin and Stephen [11]. We shall use the notation and constructions of [11] explicitly and consistently in this section. In particular we shall assume familiarity with the construction given in [11] of the Schutzenberger automata corresponding to the free product S S . E (S ) will denote the semilattice of idempotents of an inverse semigroup S. Let S = Inv < X : R > and S = Inv < X : R > where X and X are disjoint. Then we may write S S = Inv < X : R > and S U S = Inv < X : R [ W > where X = X [ X ; R = R [ R and W = f(w (u); w (u)) : u 2 U g. (Here wi(u) 2 (Xi [ Xi? ) de nes u under the given presentation for Si). We let i denote the natural partial order on Si, for i = 1; 2. Then for e 2 E (Si), put 1
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Ui(e) = fu 2 U : eiug: If the set Ui (e) contains a least element relative to i we shall denote it by f (e). For e 2 E (Si) we let eU denote the left coset feu : u 2 U g. Note that if e and e are two idempotents in E (Si) for which Ui (e ) and Ui(e ) are both non-empty then e U e U i e = e :f for some f 2 E (U ). 1
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De nition 2.1(Bennett) Let [S ; S ; U ] be an amalgam in the class of inverse semi1
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Bennett [2] provides an iterative procedure for constructing the Schutzenberger automata of S U S , relative to the usual presentation, for a lower bounded amalgam 1
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[S ; S ; U ]. We brie y summarize the steps in his procedure below and refer the reader to his work [2] for details and more precision. Given a word w 2 (X [ X ? ) we aim to construct the Schutzenberger automaton A(X; R [ W; w). 1
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Step 1 (Construction 1.4.7 of [2]) Apply the construction of [11] to build the Schutzenberger automaton A(X; R; w) of w in the free product S S . This is an automaton with nitely many lobes. Each lobe is monochromatic and is a Schutzenberger automaton A(Xi; Ri; u) for some word u 2 (Xi [ Xi? ) and i 2 f1; 2g. Two lobes intersect 1
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in at most one vertex and the lobe graph is a ( nite) tree. The resulting automaton is referred to in [1] and [11] as a cactoid automaton.
Step 2 (Construction 2.1.3 of [2]) Let w 2 (X [ X ? ) and A = ( ; ?; ) be isomorphic to the Schutzenberger automaton A(X; R; w). Let v 2 V (?) be an intersection 1
vertex at the intersection of two lobes (v) and (v). There are unique idempotents e (v) and e (v) such that (v; i(v); v) is isomorphic to the Schutzenberger automaton A(Xi; Ri; ei(v)) for i = 1; 2. Suppose that U (e (v)) 6= U (e (v)). Without loss of generality we may assume that U (e (v)) 6= and U (e (v)) is not contained in U (e (v)). Put f = f (e (v)). Form the automaton 1
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B = (v; ?; v) A(X ; R ; f ) obtained by taking a disjoint copy of A(X ; R ; f ) and forming the union of this automaton 1
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and the automaton (v; ?; v), both rooted at v. The union of the images of (v) and A(X ; R ; f ) forms a lobe in B and B is a cactoid inverse automaton. Apply Step 1 (repeated applications of Bennett's construction 1.4.7) to this automaton, to obtain an automaton B0 = (v0; ?0; v0) which is closed relative to < X : R >; here v0 is the image of v and is an intersection of B0. Finally, put A0 = ( 0; ?0; 0) where 0 and 0 denote the images of and in ?0. Bennett proves in [2] that if [S ; S ; U ] is a lower bounded amalgam and A = ( ; ?; ) is some Schutzenberger automaton relative to < X : R > then repeated applications of Step 2 terminate nitely in an automaton that has the lower bound equality property, that is: U (e (v)) = U (e (v)) at each intersection vertex v of the resulting automaton. This automaton is also cactoid. 1
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Step 3 (Construction 2.2.4 of [2]) Let w 2 (X [ X ? ) and A = ( ; ?; ) be isomorphic to the Schutzenberger automaton A(X; R; w). We say that a pair of intersection points u and v of a lobe of A are separate if they are not common to the same pair of lobes of A. Suppose that A has the lower bound equality property and that and are separate 1
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intersections of a lobe M in ?, colored by i such that there exists a path from to labelled by wi(u) for some u 2 U . Let M and M be the lobes colored by j = 3 ? i and adjacent to M that contain the intersections and respectively. There is a path in M labelled by w (u) from to some vertex . Form a graph ? by disconnecting ? at the vertex . Let T be the component of ? that contains the vertex and let T be 0
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the other component of ?. Now let B = ( ; T ; ) ( ; T ; ) The union of the images of M and M forms a lobe in B and B is a cactoid inverse automaton over X . Apply Step 1 (repeated application of Bennett's construction 1.4.7) to get a Schutzenber automaton that is closed relative to < X : R >. Then apply repeated applications of Step 2 to obtain a rooted Schutzenberger automaton B0 = (0; ?0; 0) which is closed relative to < X : R > and has the lower bound equality property. Then put A0 = ( 0; ?0; 0) where 0 and 0 are the natural images of and respectively in ?0. Bennett [2] proves that for lower bounded amalgams, repeated applications of Step 3 terminate, within nitely many steps, in an automaton which is isomorphic to a Schutzenberger automaton relative to < X : R >, that has the lower bounded equality property and that has what he refers to as the related pair separation property: that is the automaton is cactoid and has no lobe colored i = 1 or 2 such that for some u 2 U there is a path labelled by wi(u) connecting separate intersections of . 0
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Step 4 (Section 2.3 of [2]) Suppose that A = ( ; ?; ) has the lower bound equality
property and the related pair separation property. Let v be an intersection vertex and suppose that it is the only intersection common to the two lobes (v) and (v). Suppose that there are paths in (v) and (v) from v to v [resp. v ] labelled by w (u) [resp. w (u)] for some u 2 U . Adjoin the linear automaton of w (u) to ?, identifying its initial root with v and its terminal root with v and then fold the resulting two paths starting at v labelled by w (u), thus identifying the vertices v and v . The resulting graph ?0 has isomorphic copies of the lobes (v) and (v) that now have two common intersection points (v and the vertex v0 that is obtained when v and v are identi ed). It is not dicult to show that U (e (v0)) = U (e (v0)), so ?0 also has the lower bound equality property. This operation is equivalent to taking the quotient of ? by the equivalence relation generated by (v ; v ). Extend the operation by taking the quotient of ? by the equivalence relation generated by all such pairs. We say that the adjacent lobes (v) and (v) have been assimilated. The graph obtained from ? by simultaneously assimilating every pair of adjacent lobes is the assimilated form of ?. Bennett shows in [2] that the assimilated form of ? has the adjacent lobe assimilation property: that is, if and are adjacent lobes and v is any intersection vertex common to these lobes, then there is a path in from v to v0 labelled by some word of the form w (u) for some u 2 U if and only if there is a path in from v to v0 labelled by w (u): furthermore, if v0 is any other intersection vertex common to and then there is some word u 2 U such that wi(u) labels a path in i from v to v0 for i = 1; 2. 1
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It also follows from [2] that if [S ; S ; U ] is a lower bounded amalgam and w is any word in (X [ X ? ) then repeated application of Steps 1-4 yields an automaton A over X with the following properties: (1) A has the lower bound equality property; (2) A has the adjacent lobe assimilation property; (3) the lobe graph of A (the graph whose vertices are the lobes of A and whose edges correspond to adjacency of lobes in A) is a tree and each lobe of A is isomorphic to a 1
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Schutzenberger graph relative to either < X : R > or < X : R >. (4) A has nitely many lobes. 1
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A graph satisfying properties (1)-(3) above is referred to as an opuntoid graph in [2]. In particular, it is clear that A is a nite graph if each Schutzenberger graph relative to either < X : R > or < X : R > is nite. There is, however, one more step needed to complete the construction of the Schutzenberger automata for words relative to lower bounded amalgams. 1
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Step 5 (Construction 2.4.2 of [2]) Let ? be an opuntoid graph and let v 2 V (?) be a vertex belonging to a lobe colored by i 2 f1; 2g. Then v is a bud of ? if it is not an intersection and Ui(ei(v)) = 6 . The opuntoid graph is complete if it has no buds. If A = ( ; ?; ) is an opuntoid automaton which is not complete it has a bud v which is not an intersection and which belongs to a lobe colored by i 2 f1; 2g for which Ui(ei(v)) = 6 . Let j = 3 ? i and f = f (ei(v)). Form the automaton B = (v; ?; v) = (v; ?; v) A(Xj ; Rj ; f ): The images of and S ?(Xj ; Rj ; f ) are then lobes of B with common intersection v. Apply Step 4 at the vertex v and let A0 = ( 0; ?0; 0) be the resulting automaton. In [1] it is shown that A0 is also an opuntoid automaton and there is a natural embedding of A into A0. Repeated application of this construction yields a sequence A A0 A00 . . . of opuntoid automata. This gives a directed system of inverse automata that has a direct limit which is just the union of the automata in this sequence and is a complete opuntoid automaton. In [2], Bennett shows that if one starts with a lower bounded amalgam and a word w 2 (X [ X ? ) then repeated application of Steps 1-5 described above yields the Schutzenberger automaton A(X; R [ W; w). Furthermore, such automata are characterized as complete opuntoid automata that possess a nite host (a sub-opuntoid automaton containing nitely many lobes such that the whole automaton may be obtained from the host by repeated applications of Step 5). We refer to [2] for details. 1
In certain cases this procedure yields an e ective construction of the Schutzenberger automata corresponding to a lower bounded amalgam and hence provides a solution to the word problem for the corresponding presentation. We close this section by examining below conditions under which Bennett's construction is e ective.
Remark We rst make the observation that if an amalgam [S ; S ; U ] of inverse semi1
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groups satis es condition (1) of De nition 2.1, then it is possible to consider performing all of the steps in the iterative construction of the Schutzenberger automata relative to this presentation as outlined above. Condition (2) of De nition 2.1 guarantees that Step 2 of the procedure terminates in a nite number of steps, but termination of this step may possibly occur in some situations where Condition (2) is not satis ed (we shall provide a 7
class of examples of this situation in the next section). Consequently we shall consider amalgams for which Condition (1) of De nition 2.1 is satis ed but Condition (2) is not necessarily satis ed. We also remark that the input to the problem implicitly assumes that there are prescribed embeddings i : U ! Si for i = 1; 2. An analysis of Bennett's construction, as outlined in Steps 1-5 above leads to the following observation.
Theorem 1 Let [S ; S ; U ] be an amalgam of inverse semigroups with prescribed embeddings i : U ! Si for i = 1; 2. Then the word problem for the amalgam S U S relative 1
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to the natural presentation described above is decidable if the following conditions are satis ed. (1) The word problem is decidable for each inverse semigroup presentation Si = Inv < Xi : Ri > for i = 1; 2. (2) Each embedding i is e ectively calculable (this will always happen for example if U is nitely generated). Henceforth we shall simplify notation by assuming that U is a subsemigroup of each Si as usual. We shall also assume that U is nitely generated. (3) The membership problem for U in each Si is decidable. (4) For each w 2 Si (and each i = 1; 2) there is an algorithm that will decide whether fu 2 U : wiug is empty or not. (5) For each idempotent e 2 Si for which Ui (e) 6= (and each i = 1; 2) the element f (e) exists and there is an algorithm to compute f (e). (6) Step 4 of the above procedure is e ective (this will occur for example if there is an e ective bound on the size of all lobes of the relevant automata). (7) Step 2 of the above procedure must terminate after nitely many applications at all intersection points and there is an e ective bound on the number of applications of the procedure that need be applied in order for Step 2 to terminate. More precisely, for any cactoid automaton with n lobes associated with the given amalgam, there is an e ectively computable bound (n) such that if we build a sequence of cactoid automata by applying Step 2 successively at some intersection vertex of the previously constructed automaton in the sequence at which the lower bounded equality property fails, then this sequence will terminate after at most (n) steps in a cactoid automaton which has the lower bounded equality property. The bound (n) and the nal automaton constructed depends on n and on the structure of the lobes in the original cactoid automaton. (This is satis ed if the amalgam satis es Condition (2) of De nition 2.1 but may also be satis ed in other cases as well).
Proof Conditions (1)-(3) are clearly required to enable us to solve the word problem.
Recall that decidability of the word problem for an inverse semigroup presentation is equivalent to being able to e ectively construct each Schutzenberger graph relative to the given presentation. Condition (4) is needed to e ectively carry out Step 3 of the above procedure and also to check that the amalgam satis es Condition (1) of de nition 2.1. Condition (5) is also needed to check this condition and in many other places in the construction. Conditions (6) and (7) are clearly needed to e ectively apply the procedure described above for constructing the Schutzenberger graphs, and hence to solve the word problem. It is not dicult to check that Bennett's procedure, as described in Steps 1-5 8
above is e ective if all of these conditions are satis ed.
3 Decidability of the word problem
Throughout this section we consider an amalgam of the form S = FIS (A) U FIS (B ) where U is a nitely generated inverse subsemigroup of FIS (A) and FIS (B ). We show that Bennett's construction of the Schutzenberger graphs corresponding to this presentation, as outlined in the previous section, is e ective and hence the word problem for S is decidable. It suces of course to check that the conditions of Theorem 1 are satis ed. Conditions (1) and (2) of that theorem are clearly satis ed so we need only verify that the remaining ve conditions are satis ed. The veri cation of Condition (3) is accomplished via the following lemma.
Lemma 1 If U is a nitely generated inverse subsemigroup of a free inverse semigroup FIS (X ) then the membership problem for U in FIS (X ) is decidable. Proof There are various ways to prove this fact. We provide a proof using Munn trees below and in the process we establish some ideas and notation that will be used later in this section. Suppose that U is generated by the elements u ; u ; . . . ; un of FIS (X ) and that w is an arbitrary element of FIS (X ) (or an arbitrary word in (X [ X ? )). We need to nd an algorithm that will decide whether w 2 U or not. Let T be the Munn tree of w. It is convenient to consider the birooted tree (automaton) ( ; T; ) where [resp. ] is the initial [resp. terminal] root of T . It is clear that w 2 U if and only if there exist elements a ; a ; . . . ; ak such that w = a a . . . ak where each element aj is of the form ui or (ui )? for some generator ui of U . In terms of Munn trees this translates as a factorization of the form ( ; T; ) = ( ; T ; )( ; T ; ) . . . ( k ; Tk; k ) where Tj is a copy of MT (aj ), with roots j and j for each j and where = ; = ; . . . ; k = : Here, by de nition, ( i; Ti; i)( i ; Ti ; i ) denotes the birooted Munn tree ( i; Ti [ Ti ; i ) when i = i . Notice that if i = j for some i < j then the Munn tree of aiai . . . aj? has initial vertex and terminal vertex equal, so aiai . . . aj? is an idempotent of U . If v is a vertex of T such that v = s1 = s2 = . . . = s then this shows that the element ej = as . . . as +1? is an idempotent of U for each j . Now the Munn tree of each such idempotent is a tree that is rooted at v and that is contained in T . Since there are only nitely many such trees (T is nite), there are only nitely many distinct idempotents of the form ej for j = 1; . . . ; t ? 1. The fact that idempotents commute in an inverse semigroup then means that if ej = el for some j 6= l then we may remove the segment ej = as . . . as +1? from the product a a . . . ak and leave this product unchanged. This enables us to reduce the length of the sequence a ; a ; . . . ; ak without changing the resulting product. Hence there is an e ectively computable bound on the length of any such sequence needed to write w as a product of the resulting sequence of 1
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generators of U . Trying all sequences of length less than or equal to this bound tells us whether it is possible to write a given element w 2 FIS (X ) as a product of the generators of U or not. Hence the membership problem for U in FIS (X ) is decidable.
Remark The analogous result to this lemma for the free group is usually referred to
in group theory as the generalized word problem for the free group. This problem is well known to be decidable. The terminology \generalized word problem" seems less appropriate in the setting of inverse semigroups since the problem is not really a generalization of the word problem in this setting.
Lemma 2 Conditions (4),(5) and (6) of Theorem 1 are satis ed for an amalgam of the form [FIS (A); FIS (B); U ] where U is a nitely generated inverse semigroup. Proof Let w 2 FIS (A) and consider the Munn tree of w as a triple ( (w); T (w); (w)) as in the proof of the previous lemma. Note that if u is any other element of FIS (A) (including an element of U ) with Munn tree of the form ( (u); T (u); (u)) then w u in FIS (A) if and only if T (u) T (w) , (u) = (w) and (u) = (w). Since there are only nitely many subtrees of T (w) that satisfy these properties, one can simply try them all and apply Lemma 1 to see if any one of them corresponds to an element of U . In the case where w is an idempotent e of FIS (A), we have (e) = (e) and the maximal subtree of T (e) rooted at (e) corresponding to an element of U will yield an e ective construction of f (e) if there is any element u in U such that w u. This shows that conditions (4) and (5) of Theorem 1 are satis ed. Condition (6) is satis ed since all lobes of the appropriate automata are nite. The next example shows that Condition (2) of De nition 2.1 is not satis ed in general in the setting that we are considering, so it becomes necessary to check Condition (7) of Theorem 1.
Example 1. Let A = fa; bg; B = fc; dg and let U be the inverse subsemigroup of FIS (x; y) generated by fy; xx? ; xx? xg with embeddings of U in FIS (A) and FIS (B ) given by x ! a; y ! b and x ! c; y ! d respectively. Clearly this de nes an amalgam [FIS (A); FIS (B ); U ] of inverse semigroups. Let en = bnb?n a? a for each n. Then we have en = en:bn b?n? so en U enU for each n. Furthermore, it is routine to check that f (en) = bnb?n . For each n choose g = bn b?naa? . Then g 2 U and g f (en). Also, en:g = bnb?n aa? a 2 U so f (en:g) = 6 g. This shows that Bennett's condition (2) of 1
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De nition 2.1 is not satis ed.
Lemma 3 Condition (7) of Theorem 1 is satis ed for an amalgam of the form
[FIS (A); FIS (B ); U ] where U is a nitely generated inverse semigroup. Proof The proof follows closely along the lines of the proof of Lemma 2.1.5 of [2], the main di erence being that we are unable to use Condition (2) of De nition 2.1 since that condition is not satis ed in the present setting as the example above shows. Let w 2 ((A [ B ) [ (A [ B )? ) and let ( ; ?; ) be isomorphic to the Schutzenberger automaton of w in the free product of FIS (A) and FIS (B ). (Of course ? is just the 1
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Munn tree of w over the alphabet A [ B in this case.) We proceed by induction on the number of intersection vertices of this tree. In this setting an intersection vertex is any vertex v of the tree with the property that there is at least one edge labelled over A [ A? and at least one edge labelled over B [ B ? emanating from v: a lobe of the tree is a maximal subtree all of whose edges are labelled over A [ A? or B [ B ? . The rst case that needs to be checked (the basis for the induction) is when this tree has two lobes and . Let v be the vertex that is the intersection of the two lobes. Let g = e (v) and h = e (v). Without loss of generality assume that g is an idempotent in FIS (A) and h is an idempotent in FIS (B ). It is convenient to introduce the subscripted notation MTA (u) [resp. MTB (u); MTA[B (u)] for the corresponding Munn trees of an element u 2 FIS (A) [resp. FIS (B ); FIS (A[B)] and to consider the automata (lobes) (v; MTA(g ); v) = (v; ; v) and (v; MTB (h ); v) = (v; ; v), both rooted at v. Now assume that U (h ) 6= and that U (h ) is not contained in U (g ). Apply the construction described above in Step 2 of Bennett's algorithm. This construction builds the automaton (v; MTA[B(w); v) (v; MTA(f (h )); v) which has two lobes intersecting at v, namely MTB (h ) and MTA (g ) where g = g :f (h ). If f (g ) = f (h ) then the resulting automaton has the lower bounded equality property, so no further application of Step 2 is possible. So assume that f (g ) 6= f (h ). We must then have f (g ) < f (h ): equivalently, MTA(f (g )) properly contains MTA(f (h )). Since MTA (g ) = MTA (g ) [ MTA (f (h )) this implies that MTA(f (g )) must contain at least one edge of MTA(g ) that is not in MTA(f (h )). We may now apply Step 2 again at the vertex v, interchanging the roles of the two lobes. The resulting automaton has two lobes joined at v again, namely MTA (g ) and MTB (h ) where h = h :f (g ). As before, if f (h ) = f (g ) then the automaton has the lower bounded equality property and the algorithm stops. Otherwise f (h ) < f (g ) < f (h ) and as before this implies that MTB (f (h )) contains at least one edge of MTB (h ) that is not in MTB (f (h )). Continue this process, inductively de ning idempotents gi = gi:f (hi) and hi = hi:f (gi ). If f (gi ) = f (hi) or f (gi ) = f (hi ) for any i then the corresponding automaton has the lower bounded equality property and the algorithm stops. Otherwise, for each i, the number of elements in MTA (g ) that are not in MTA (f (gi )) is strictly less than the number of elements of MTA(g ) that are not in MTA(f (gi )). This latter condition is impossible since MTA (g ) is nite (a similar argument applies to MTB (h )). Hence the algorithm must terminate after nitely many steps (in fact after at most jMTA(g )j + jMTB (h )j steps). This completes the basis for the induction. Assume inductively that if x is any word in ((A [ B ) [ (A [ B )? ) such that MTA[B (x) has fewer than n intersection vertices, then there is an e ectively computable bound n(x) such that successive applications of Step 2 at intersection vertices where the lower bounded equality property fails will eventually produce after at most n (x) steps, an automaton that has the lower bounded equality property. Let w be a word in ((A [ B ) [ (A [ B )? ) such that MTA[B (w) has n intersection vertices where n > 1. Construct a sequence of Munn trees (Schutzenberger automata) M = MTA[B (w); M ; M ; . . . determined by repeated applications of Step 2 at various intersection vertices of the intermediate Munn trees. Note that no identi cation of distinct intersection vertices of the original Munn tree M occurs at any stage in this procedure. It is convenient to denote the natural image of an intersection vertex v of M in the tree 1
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Mi by vi. Recall that each tree (automaton) Mi may be viewed as a cactoid automaton, so its lobe graph is a tree. Hence Mi has at least two lobes each of which is adjacent to
precisely one other lobe (these are extremal vertices of the lobe tree). Thus let be a lobe in M that has precisely one intersection vertex v. Without loss of generality assume that = (v; MTB(e (v)); v). The lobe adjacent to is (v; MTA(e (v)); v). Denote by i the extremal lobe (vi; MTB (e (vi)); vi) and by ?i the cactoid subtree of Mi consisting of all of the lobes of Mi except i. Clearly ?i is the Munn tree of some word over the alphabet A [ B and ?i contains n ? 1 intersection vertices. Since Mi is obtained from Mi by an application of Step 2 above we must have one of the following three situations: 1
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Type (a) The application of Step 2 results from some intersection v0 6= vi with U (e (v0)) 6= U (e (v0)). In this case ?i is obtained from ?i by an application of Step 2 and e (vi ) e (vi); e (vi ) = e (vi). 1
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Type (b) The application of Step 2 results from U (e (vi)) 6= and U (e (vi)) is not contained in U (e (vi)). In this case we have e (vi ) = e (vi):f (e (vi)) < e (vi) and e (vi ) = e (vi). 2
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Type (c) The application of Step 2 results from U (e (vi)) 6= and U (e (vi)) is not contained in U (e (vi)). In this case we have ?i = ?i and e (vi ) = e (vi):f (e (vi)) < e (vi): 2
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Note that by the induction hypothesis there is an e ective bound on the number of successive applications of Type (a) (without intermediate applications of Type (b) or Type (c)) in the sequence of applications of Step 2 that are used to construct the Munn trees Mi. The idea now is to examine the sequence of occurrences of applications of Step 2 of Type (b) and Type (c) in our sequence of Munn trees. One may start by applying an e ectively bounded number of Type (a) applications. If the sequence of Munn trees has not terminated at this stage (i.e. if we have not reached a Munn tree that has the lower bounded equality property) then there must be a least integer l such that Ml is obtained from Ml by an application of Type (b) or Type (c). Assume temporarily that Ml is obtained from Ml by an application of Type (b). This yields U (e (vl )) U (e (vl )). Again, assuming that our sequence of Munn trees does not terminate, there is a least integer m l +1 such that Mm is obtained from Mm by an application of Type (b) or Type (c). The intermediate applications (if any) must be of Type (a) so we deduce that U (e (vm)) U (e (vm)) and hence that Mm is obtained from Mm by an application of Type (c). Thus let h = e (vm ) = e (vm):f (e (vm)) = e (v ):f (e (vm)) < e (v ) and g = e (vm ) = e (vm) < e (v ): Repeat this argument. Assuming that our sequence of Munn trees does not yet terminate, there exists an e ectively computable least positive integer k > m + 1 such that Mk is obtained from Mk by an application of Type (c). Since the intervening applications (at least one) must be of Type (a) or (b) it follows as before that if we de ne +1
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h = e (vk ) and g = e (vk ), then h < h and g < g . Again repeat the argument. Assuming that our sequence of Munn trees does not stop, we obtain e ectively computable sequences of idempotents g ; g ; . . . 2 FIS (A) and h ; h ; . . . 2 FIS (B ) such that hi = hi? :f (gi) < hi? and gi < gi? . Note that between two successive applications of Type (b) we must have an application of Type (c) (and as indicated above, between two successive applications of Type (c) we must have an application of Type (a) or Type (b)). Note also that if MTB (hi ) contains no more edges of MTB (e (v)) than MTB (hi) does, then by the same argument as was used in the proof of the basis for the induction, we see that we cannot apply an application of Type (b) between the two successive applications of Type (c) that resulted in the creation of the idempotents hi and hi . This means that we may in this case apply only applications of Type (a) between these successive applications of Type (c). By the induction hypothesis there is an e ective bound on the number of such successive applications of Type (a) that may be applied. Again, if also MTB (hi ) contains no more edges of MTB (e (v)) than MTB (hi ) does, then all intermediate applications must be of Type (a). Since no application of Type (b) occurred between the applications of Type (c) that created hi and hi , the Type (a) applications all applied to the sequence of trees ? ; ? ; . . .. Hence by the inductive assumption there is an e ectively bounded number of such applications. Continuing with this argument, one sees that for each i there must exist some e ectively computable integer j > i such that MTB (hj ) contains more edges of MTB (e (v)) than MTB (hi) does. The niteness of MTB (e (v)) makes this impossible and shows that any sequence of applications of Step 2 must terminate (in an e ectively computable number of steps). 2
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As an immediate corollary of Lemmas 1, 2 and 3 and Theorem 1 we deduce the following.
Theorem 2 The word problem is decidable for any inverse semigroup amalgam of the form S = FIS (A) U FIS (B ) where U is a nitely generated inverse subsemigroup of FIS (A) and FIS (B ).
4 Some structural results Throughout this section S will denote an inverse semigroup amalgam of the form S = FIS (A) U FIS (B ) where U is a nitely generated inverse subsemigroup of FIS (A) and FIS (B ). We turn to a study of some structural properties of such a semigroup S . We begin with a result which provides necessary and sucient conditions for the Schutzenberger graphs (R-classes) of such an amalgam to be nite under a mild assumption on the way in which U embeds in FIS (A) and FIS (B ). We say that the amalgam [FIS (A); FIS (B); U ] respects the J -order if, for all g; h 2 E (U ); Jg Jh in FIS (A) if and only if Jg Jh in FIS (B ).
Theorem 3 Let [FIS (A); FIS(B ); U ] be an amalgam of inverse semigroups which respects the J -order. Then each R-class of S = FIS (A) U FIS (B ) is nite if and only if, 13
for any pair of idempotents g; h 2 E (U ), either g and h are D-related in U or else g and h are not D-related in either FIS (A) or FIS (B ).
Proof Let w 2 (A [ A? [ B [ B ? ). Since all Munn trees are nite it is clear that the automaton A obtained from the linear automaton of w by closing under repeated 1
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applications of Steps 1-4 of Bennett's algorithm must be nite. Hence the Schutzenberger automaton of w in S will be nite if and only if any sequence of automata obtained from A by iteratively applying Step 5 must terminate after nitely many steps. Recall that an application of Step 5 may be carried out at each bud of A and the resulting automaton is again an opuntoid automaton with one more lobe sewed on so as to make the bud that was used an intersection vertex of the new automaton. Assume rst that for any pair of idempotents g; h 2 E (U ), either g and h are D-related in U or else g and h are not D-related in either FIS (A) or FIS (B ). Let w be any word in (A [ A? [ B [ B ? ) and let A be the nite opuntoid automaton obtained from the linear automaton of w by closing with respect to Steps 1-4. If A has no buds, then A is the required Schutzenberger automaton of w in S , so this automaton is nite. Otherwise, A has buds: refer to any bud on A as a rst generation bud. Let v be a rst generation bud of A. Without loss of generality we may assume that v belongs to a lobe which is the Munn tree of some word over the alphabet A [ A? . Let e (v) be the idempotent whose Munn tree birooted at v is (v; ; v) . As in Step 5, let f = f (e (v)) 2 U , build the automaton (v; ; v) MTB (f ) and apply Step 4 of the construction to obtain the automaton A0. The vertex v becomes an intersection vertex of A0 and the automaton A0 has one more lobe than A (namely the lobe (v; MTB(f ); v)). If A0 has no buds the procedure stops and the required Schutzenberger automaton is nite. Suppose that A0 has a bud v0 on the new lobe (v; MTB (f ); v). Call any bud obtained in this fashion a second generation bud. Denote by e (v 0) the idempotent whose Munn tree birooted at v 0 is (v0; MTB (f ); v0). Since v0 is a bud there is no path in MTB (f ) from v0 to v labelled by any word in U . This implies that the idempotents f and f 0 = f (e (v0)) are not D-related in U . The reason for this is that by de nition we have e (v0) f 0 and e (v0)Df in FIS (B ), so if f Df 0 in U then we would have f Df 0 in FIS (B ) and so f 0 = e (v0) since the D-classes of FIS (B ) are nite. This would imply that e (v0)Df in U and so there would be some element u0 2 U such that u0(u0)? = f and (u0)? u0 = e (v0), whence u0 labels a path in MTB (f ) from v to v0, contradicting the fact that v0 is a bud. Hence by hypothesis it follows that f and f 0 are not D-related in FIS (B ) (or in FIS (A)). This implies that Jf 0 > Jf in the natural partial order on the J -classes of FIS (B ). By hypothesis this implies that Jf 0 Jf in FIS (A). Since f is not D-related to f 0 in FIS (A), this forces Jf 0 > Jf in FIS (A) as well. The same argument may be applied if the original lobe is the Munn tree of some word over the alphabet B [ B ? of course. This shows that if an application of Step 5 at a rst generation bud v in A produces a second generation bud v0 in the new lobe created by applying Step 5 at v, then the corresponding idempotent f 0 of U associated with v0 (as described in the above argument) is in a J -class of FIS (A) and FIS (B ) that is strictly greater than that of f . The fact that the free inverse monoid is nite J -above implies that this procedure, when applied iteratively to construct higher generation buds on new lobes of the intermediate automata, must terminate in a nite number of steps. Since the original automaton (and all intermediate automata) had only 1
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nitely many buds, this shows that any sequence of automata obtained by iteratively applying Step 5 must terminate, and hence the Schutzenberger automaton of our original word w is nite. Suppose conversely that there exist idempotents g; h 2 U such that g is not D-related to h in U but g is D-related to h in FIS (A). Since the amalgam respects the J -order, we also have g is D-related to h in FIS (B ). Construct the automaton A = MTA(g), birooted at some vertex say. Since gDh in FIS (A), these words have the same Munn tree, so there is some vertex v in MTA(g) such that f (e (v)) = h. The vertex v must be a bud or else there would be a path in MTA (g) = MTA(h) from to v labelled by an element of U , contradicting the fact that g and h are not D-related in U . Construct the new automaton A0 obtained from A by applying Step 5 at the bud v. The automaton A0 has a new lobe at v, namely the lobe (v; MTB (h); v). Again, by the same argument as above, this lobe has a new bud v0 such that f (v0) = g. One may apply Step 5 at this lobe to construct a new automaton A00 with three lobes and another new bud on the new lobe etc. Continuing in this way produces an unbounded sequence of automata and so the Schutzenberger automaton of g in S is in nite. This completes the proof of the theorem. 1
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Remark The above result is in general false if we do not assume that the amalgam respects the J -order. For example, if S = Inv < a; b : aa? = b? b ; bb? = a? a > then S is an amalgam of FIS (a) and FIS (b) over a three-element semilattice U which does not respect the J -order. It is easy to check that the Schutzenberger graphs are in nite 1
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The next result shows that the maximal subgroups of an amalgam considered in the above theorem must be nite.
Theorem 4 If each R-class of an amalgam of the form S = FIS (A) U FIS (B ) is nite then S is combinatorial (i.e. all maximal subgroups of S are trivial). Proof From the results of Stephen [20] it suces to show that for each word w 2 (A [ A? [ B [ B ? ), the automorphism group of the Schutzenberger graph of w in S is trivial. 1
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Recall that an automorphism of such a graph is by de nition a graph automorphism that preserves labelling on edges. Also the Schutzenberger graphs of S are all nite opuntoid graphs by the above discussions and assumption. It follows immediately that any automorphism of such a graph must permute the intersection vertices of the graph, since these are the only vertices which have incident edges labelled over both alphabets A and B . Also an automorphism must take an edge labelled over A [resp. B ] to another edge with the same labelling. Note also that every lobe of the graph is a nite tree and that the automorphism group of any nite (labelled) tree is trivial (this is easy to prove and is well known). In addition, it is clear that if an automorphism of the whole graph maps a vertex to another vertex in the same lobe, then it must induce an automorphism of the lobe and hence must be the identity automorphism when restricted to that lobe. Hence any non-trivial automorphism of the graph must induce a non-trivial permutation of the lobes of the graph that are labelled over A [resp. B ]. 15
So let be an automorphism of our Schutzenberger graph S . We claim that induces an automorphism of the lobe tree T of the opuntoid graph S . Choose an extremal lobe of S , labelled over A without loss of generality, and suppose that the automorphism takes a vertex v in to a vertex v0 in some lobe 0 (which must also be labelled over A). If 0 is not an extremal lobe then it has at least two distinct adjacent lobes and 0 labelled over B . Choose a path p in 0 from v0 to an intersection vertex v0 2 0 \ and a path p in 0 from v0 to an intersection vertex v0 2 0 \ 0. Then the automorphism ? must map v0 to an intersection vertex v 2 \ ? where ? is the unique lobe adjacent to in S , and ? must map v0 to an intersection vertex v 2 \ ?. Now there is a path q in ? labelled over B from v to v , so (q) must be a path labelled over B from v0 to v0 . This is impossible since v0 and v0 are not intersection points of the same pair of lobes of S . Hence 0 must be an extremal lobe of S . Thus maps an extremal lobe of S to another extremal lobe 0 and also maps the lobe ? that is adjacent to to the lobe ?0 that is adjacent to 0. The orbit of under the action of consists of extremal lobes of S labelled over A. These can be removed from S to leave an opuntoid graph S 0 with fewer lobes and with a smaller lobe tree T 0 so by induction induces an automorphism of T 0 which can be extended to an automorphism of T . It is now easy to see by induction on the number of lobes of S that must be the identity automorphism. The basis for the induction (the case where S contains only one lobe) is easily established since the automorphism group of a nite tree is trivial. Assume inductively that our graph S 0 constructed above has trivial automorphism group. If is an automorphism of S then the argument given above shows that restricts to an automorphism of S 0 which must be the identity automorphism of S 0 by the induction assumption. Also by the induction assumption, any intersection vertex v connecting an extremal lobe of S to its neighbor ? in S 0 must be xed by . Thus must induce an automorphism of (for each extremal lobe of S ) and so the restriction of to is trivial. This shows that is the trivial automorphism. 1
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As an application of this result, we note that one-relator amalgams of the form under consideration are combinatorial and completely semisimple (with nite D-classes). Note that an inverse semigroup of the form S = Inv < A [ B : u = v > where A and B are disjoint sets and u 2 (A [ A? ) ; v 2 (B [ B ? ) is an amalgam if and only if either u and v are both idempotents or u and v are both non-idempotents, since any non-idempotent of a free inverse semigroup generates a free inverse subsemigroup. 1 +
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Theorem 5 Every one relator semigroup amalgam of the form S = Inv < A [ B : u = v > where A and B are disjoint sets and u 2 (A [ A? ) ; v 2 (B [ B ? ) has decidable word problem, has nite D-classes and is combinatorial. Proof Decidability of the word problem is of course immediate from Theorem 2. We need only show that all D-classes of S are nite. If either u or v is an idempotent, then so is 1 +
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the other element (since S is an amalgam). In this case the condition of Theorem 3 is trivially satis ed, so the D-classes of S are nite. So without loss of generality we may assume that u = eu and v = fv for some idempotents e 2 FIS (A) and f 2 FIS (B ) and some reduced words u 2 (A [ A? ) 1
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and v 2 (B [ B ? ). It is easy to see that this amalgam respects the J -order since the idempotents of U are of the form un u? m n um (in FIS (A)) or vnv? m n vm (in FIS (B )) and unu? m n umJ upu? p q uq in FIS (A) if and only if m + n p + q. We may write u = pu p? and v = qv q? for some reduced words p 2 (A [ A? ) and q 2 (B [ B ? ) and some cyclically reduced words u 2 (A [ A? ) and v 2 (B [ B ? ) . Consider the copy of the subsemigroup U =< u > that is embedded in FIS (A). Recall that every idempotent of U may be expressed in the form u?aua bu?b for some integers a; b 0. Let g = u?aua bu?b and h = u?c uc d u?d be two idempotents in U . Then gDua b in U and hDuc d in U . Thus if gDh in FIS (A) then ua bDuc d in FIS (A) and so ua b = uc d , which implies that gDh in U . Hence gDh in U if and only if gDh in FIS (A). Similarly gDh in U if and only if gDh in FIS (B ). Hence the conditions of Theorem 3 are satis ed and so the D-classes of S are nite. This completes the proof of the theorem. 1
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Remark We remark that it is easy to nd examples of amalgams of the form considered in this paper whose D-classes are not nite and whose maximal subgroups are
not trivial. The example considered in the remark preceding the previous theorem would suce. For another example, let S = Inv < a; b : aa? = bb? ; a? a = b? b >. It is easy to see that the Schutzenberger graph of the element a in S is in nite, with automorphism group the group Z of integers. This semigroup S is an amalgam of the form S = FIS (a) U FIS (b) where U is the (3-element) free semilattice on two generators. It would be interesting to provide a general structural characterization of the maximal subgroups of amalgams of the type considered in this paper. It seems likely that all such subgroups are free and that techniques from Bass-Serre theory analogous to those used by Haataja, Margolis and Meakin in [5] may be used to calculate their rank. 1
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We close the paper with some results and remarks relating to the question of when the amalgams under consideration are E-unitary. Recall that an inverse semigroup S is Eunitary if the natural map from S onto its maximal group image is idempotent pure (that is, the inverse image of the identity under this map consists precisely of the idempotents of S ). We focus on the one-relator case. Theorem 6 If S is an inverse semigroup amalgam of the form S = Inv < A[B : u = v > where A and B are disjoint sets and u 2 (A [ A? ) ; v 2 (B [ B ? ) then S is E-unitary if either both u and v are cyclically reduced words or both are idempotents. Proof The result is well-known in the case when u and v are idempotents - see, for example the paper of Margolis and Meakin [12]. So we consider the case where both u and v are cyclically reduced words. Let G = gp < A [ B : u = v > and let : S ! G be the natural map from S onto G. We need to show that is idempotent-pure. It is convenient to use the notation x =S y [resp. x =G y] to denote equality of two words x; y 2 (A [ A? [ B [ B ? ) in the semigroup S [resp. the group G]. If w 2 S then we may write w =S y y . . . yn where the words yi and yi are alternately from FIS (A) or FIS (B ). A standard argument involving the commuting of idempotents in an inverse semigroup then enables us to rewrite w in the form w =S ex x . . . xn where e is an idempotent in FIS (A [ B ) and the words xi and 1 +
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xi are reduced words alternately from (A [ A? ) or (B [ B ? ). Such an expression for w is of course not unique. We show by induction on the minimum length n of all such expressions for w that if (w) = 1, then w is an idempotent of S . The result is clearly true if n = 0. Also if n = 1 then we can write w =S ex for some idempotent e 2 FIS (A [ B ) and some reduced word x 2 (A [ A? ) say. If w =G 1 then x =G 1 so x is an idempotent in FIS (A) since FIS (A) is E -unitary and hence w =S ex =S w . This gives us a basis for the induction. Assume that the result is true for all words that may be written in the above form involving an alternating product of length less than n and suppose that the element w 2 S may be written as above in the form w =S ex x . . . xn where e is an idempotent in FIS (A [ B ) and the words xi and xi are reduced words alternately from (A [ A? ) or (B [ B ? ). If w =G 1 then we also have x x . . . xn =G 1. Now G is a group amalgam of the form G = FG(A) U FG(B ) where U is a cyclic subgroup (generated by u in FG(A) or by v in FG(B )). So by the normal form theorem for group amalgams we must have xk 2 U for some k with 1 k n. Without loss of generality assume that xk = up in FG(A) for some integer p 6= 0. Since xk is reduced and u is cyclically reduced we have xk = up in FIS (A) and hence xk =S up. It follows that w =S ex . . . xk? upxk . . . xn and so w =S ex . . . xk? vpxk . . . xn: Now xk? ; vp and xk are all in FIS (B ) so we may multiply them in that semigroup and then rewrite the product in the form fz for some idempotent f 2 FIS (B ) and some reduced word z 2 (B [ B ? ). Again applying the standard trick of commuting idempotents we may then rewrite w in the form w =S e0x . . . xk? zxk . . . xn where e0 is an idempotent of FIS (A [ B ). This is a shorter alternating product of the desired form and so by the induction hypothesis w is an idempotent of S . This completes the proof of the theorem. 1
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Concluding Remarks and Examples (1) It is easy to provide examples of one-relator amalgams of free inverse semigroups which are not E -unitary. For example let S = Inv < a; b; c; d : aba? = cdc? >. Here the words u = aba? and v = cdc? are reduced but not cyclically reduced. Note that w = ab a? cd? c? is equal to 1 in the group image of S but w is not an idempotent in S since the Schutzenberger graph of w in S is linear. Hence S is not E -unitary. 1
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(2) If S is an amalgam of the form S = FIS (A) U FIS (B ) and S is E -unitary, then any inverse semigroup T obtained from S by adding relations of the form ei = fi where each ei[resp. fi] is an idempotent of FIS (A) [resp. FIS (B )] must also be E -unitary since T is an idempotent- pure image of S . However the converse of this is false, as the following example shows. 18
Let S = Inv < a; b : a? a = b b? b > and T = Inv < a; b : a? a = b b? b; a? a = >. We claim that T is E -unitary but S is not. To see that S is not E -unitary it suces to note that ab? = 1 in the group image G of S but ab? is not an idempotent of S (again since its Schutzenberger graph in S is linear). Note also that the inverse subsemigroup < a? a > of FIS (a) is isomorphic to the inverse subsemigroup < b b? b > of FIS (b) since both elements are non-idempotents. Thus S is an amalgam of the desired form. The proof that T is E -unitary is similar to the proof of Theorem 6. Any word that is the identity in the group image G of T can be written in T as an idempotent of FIS (A [ B ) followed by a balanced word in the powers of the letters a and b, for example in the form w =T eai1 bj1 ai2 . . . ai bj where e is an idempotent of FIS (A [ B ) and the sum of the exponents i + j + . . . + jn = 0. One proves that w =T w by induction on the number of alternating factors. To get the induction started note that ab? =T aa? ab? bb? =T abb? b? bb? =T abb? b b? bb? =T abb? (b? b b? )bb? =T (abb? a? )(a? a)(bb? ), which is an idempotent of T . Similarly, a? b =T a? aa? bb? b =T a? aa? a? ab =T a? ab? b b? b =T bb? b? b b? b =T bb? b, which is an idempotent of T . It follows from these two facts that b? a and ba? are also idempotents of T , so we have a base for the induction. We next observe that a?nbn ; anb?n; bna?n and b?n an are all idempotents of T for each n > 0. This follows by induction on n. For example, a?n bn =T a? n? (a? b)bn? , and since a? b is an idempotent of T this may be rewritten in the form a?n bn =T fa? n? bn? for some idempotent f in FIS (A [ B ), so by induction a?nbn is an idempotent of T . A similar argument applies in the other cases. Returning now to our general word w =T eai1 bj1 ai2 . . . ai bj , we consider a factor of the form ai bj where ik > 0 and jk < 0. (There must either be a factor of this form or one of the form ai bj where ik < 0 and jk > 0 or one of the form bj ai +1 where jk > 0 and ik < 0 or one of the form bj ai +1 where jk < 0 and ik > 0). Assume also without loss of generality that ik < jjk j since the other case is dual. We may write w =T eai1 bj1 . . . (ai b?i )b?t . . . ai bj ; which may be rewritten by the usual methods of commuting idempotents in the form w =T fai1 bj1 . . . ai ?1 bj ?1 b?tai +1 bj +1 . . . ai bj ; where f is an idempotent of FIS (A [ B ), since ai b?i is an idempotent of T by the above. This implies that w is an idempotent of T by the induction hypotesis since the alternating product obtained is shorter than the original one. A similar argument applies in all the other cases. Hence T is E -unitary. We note also that T is in fact an amalgam of the form considered in this paper. This follows since a? a is an identity for the inverse subsemigroup of FIS (a) generated by a? a and bb? is an identity for the inverse subsemigroup of FIS (b) generated by b b? b, so the subsemigroups < a? a; a? a > of FIS (a) and < bb? ; b b? b > of FIS (b) are isomorphic. 1
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(3) In the case of one-relator amalgams of monogenic inverse semigroups (i.e. inverse semigroups of the form S = Inv < a; b : u = v >) it is easy to give a complete character19
ization of when the amalgam is E -unitary. A routine argument similar to the arguments used in the remarks and examples above shows that such a semigroup S is E -unitary if and only if one of the following four cases occurs. (a) u = an and v = bm for some non-zero integers n and m; (b) u = e and v = f for some idempotents e and f ; (c) u = a and v = fbm for some idempotent f and some integer m; (d) u = ean and v = b for some idempotent f and some integer m. We omit the details of the proof of this fact.
Acknowledgement The authors would like to thank Dr. P. Bennett for making available a copy of his thesis [1] and details from his forthcoming paper [2] prior to their publication and for a number of helpful comments about the results in this present paper. Several of the results contained in this paper (or special cases of these results) may be proved directly (for example by induction) without use of Bennett's thesis but the thesis provides a much clearer and more satisfactory framework in which to formulate these arguments.
References [1] P. A. Bennett, Amalgamated free products of inverse semigroups, Ph.D. Thesis, University of York (1994). [2] P.A. Bennett, Amalgamated free products of inverse semigroups, Technical report, LITP, University of Paris, in progress. [3] J-C. Birget, S.W. Margolis and J.C. Meakin, The word problem for inverse semigroups presented by one idempotent relator, Theor. Comp. Sci., 123 (1994), 273-289. [4] J-C. Birget, S.W. Margolis and J.C. Meakin, On the word problem for tensor products and amalgams of semigroups, Preprint. [5] S. Haataja, S. Margolis and J. Meakin, On the structure of full regular semigroup amalgams, Preprint. [6] T.E. Hall, Free products with amalgamation of inverse semigroups, J. Algebra 34 (1975), 375-385. [7] J.M. Howie, Embedding theorems with amalgamation for semigroups, Proc. London Math. Soc. (3), 12 (1962), 511-534. [8] J.M. Howie, Amalgamations: a survey, in \Semigroups: Algebraic theory and applications to formal languages and codes" pp. 125-132, C. Bonzini, A. Cherubini and C. Tibiletti (Eds.), World Scienti c, 1993. [9] P.R. Jones, A graphical representation for the free product of E-unitary inverse semigroups, Semigroup Forum 24 (1982), 195-221. 20
[10] P.R. Jones, Free products of inverse semigroups, Trans. Amer. Math. Soc. 282 (1984), 293-317. [11] P.R. Jones, S.W. Margolis, J.C. Meakin and J.B. Stephen, Free products of inverse semigroups II, Glasgow Math. J. 33 (1991), 373-387. [12] S.W. Margolis and J.C. Meakin, Inverse monoids, trees and context-free languages, Trans. Amer. Math. Soc. 335, No. 1 (1993), 259-276. [13] S.W. Margolis, J.C. Meakin and M.V. Sapir, Algorithmic problems in groups, semigroups and inverse semigroups, Proc. NATO conf. on semigroups and languages, Univ. of York, to appear. [14] S.W. Margolis, J.C. Meakin and J.B. Stephen, Some decision problems for inverse monoid presentations, in \Semigroups and their applications", D. Reidel (1987), 99110. [15] S.W. Margolis, J.C. Meakin and J.B. Stephen, Free objects in certain varieties of inverse semigroups, Canad. J. Math. XLII, 6 (1990), 1084-1097. [16] J. Meakin, An invitation to inverse semigroups, Proc. Hong Kong Conf. on Ordered Structures and Algebra of Computer Languages, K.P. Shum and P.C. Yuen (Eds.), World Scienti c (1993), 91-115. [17] W.D. Munn, Free inverse semigroups, Proc. London Math. Soc. 30 (1974), 385-404. [18] M. Petrich, Inverse semigroups, Wiley, New York, 1984. [19] M. Sapir, Algorithmic problems for amalgams of nite semigroups, in preparation. [20] J.B. Stephen, Presentations of inverse monoids, J. Pure and Applied Algebra 63 (1990), 81-112. [21] J.B. Stephen, Amalgamated free products of inverse semigroups, Preprint. Alessandra Cherubini Dipartimento di Matematica Politecnico di Milano Piazza Leonardo Da Vinci No. 32 I 20133 Milano - Italy John Meakin Department of Mathematics and Statistics University of Nebraska Lincoln Nebraska 68588, USA Dipartimento di Matematica 21
Brunetto Piochi Universita di Siena Via del Capitano 15 I 53100 Siena, Italy
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