On one-relator inverse monoids and one-relator groups

A prefix monoid is a finitely generated submonoid of a finitely presented group generated by the prefixes of its defining relators. Important results of Guba (1997), and of Ivanov, Margolis and Meakin (2001), show how the word problem for certain one-relator monoids, and inverse monoids, can be reduced to solving the membership problem in prefix monoids of certain one-relator groups. Motivated by this, in this paper, we study the class of prefix monoids of finitely presented groups. We obtain a complete description of this class of monoids. All monoids in this family are finitely generated, recursively presented and group-embeddable. Our results show that not every finitely generated recursively presented group-embeddable monoid is a prefix monoid, but for every such monoid, if we take a free product with a suitably chosen free monoid of finite rank, then we do obtain a prefix monoid. Conversely, we prove that every prefix monoid arises in this way. Also, we show that the groups that arise as groups of units of prefix monoids are precisely the finitely generated recursively presented groups, whereas the groups that arise as Schützenberger groups of prefix monoids are exactly the recursively enumerable subgroups of finitely presented groups. We obtain an analogous result classifying the Schützenberger groups of monoids of right units of special inverse monoids. We also give some examples of right cancellative monoids arising as monoids of right units of finitely presented special inverse monoids, and we show that not all right cancellative recursively presented monoids belong to this class.

The word problem for inverse monoids generated by a set Γ subject to relations of the form e = f, where e and f are both idempotents in the free inverse monoid generated by Γ, is investigated. It is shown that for every fixed monoid of this form the word problem can be solved in polynomial time which solves an open problem of Margolis and Meakin. For the uniform word problem, where the presentation is part of the input, EXPTIME-completeness is shown. For the Cayley-graphs of these monoids, it is shown that the first-order theory with regular path predicates is decidable. Regular path predicates allow to state that there is a path from a node x to a node y that is labeled with a word from some regular language. As a corollary, the decidability of the generalized word problem is deduced. Finally, it is shown that the Cayley-graph of the free inverse monoid has an undecidable monadic second-order theory.

The word problem for inverse monoids generated by a set $Gamma$ subject to relations of the form $e=f$, where $e$ and $f$ are both idempotents in the free inverse monoid generated by $Gamma$, is investigated. It is shown that for every fixed monoid of this form the word problem can be solved in polynomial time which solves an open problem of Margolis and Meakin. For the uniform word problem, where the presentation is part of the input, EXPTIME-completeness is shown. For the Cayley-graphs of these monoids, it is shown that the first-order theory with regular path predicates is decidable. Regular path predicates allow to state that there is a path from a node $x$ to a node $y$ that is labeled with a word from some regular language. As a corollary, the decidability of the generalized word problem is deduced. Finally, some results on free partially commutative inverse monoids are presented.

Free partially commutative inverse monoids are investigated. Analogously to free partially commutative monoids (trace monoids), free partially commutative inverse monoid are the quotients of free inverse monoids modulo a partially defined commutation relation on the generators. An O(n log(n)) algorithm on a RAM for the word problem is presented, and NP-completeness of the generalized word problem and the membership problem for rational sets is shown. Moreover, free partially commutative inverse monoids modulo a finite idempotent presentation are studied. For these monoids, the word problem is decidable if and only if the complement of the commutation relation is transitive.KeywordsWord ProblemInverse SemigroupCayley GraphDependence GraphDependence RelationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Inverse monoids: Decidability and complexity of algebraic questions

The word problem for inverse monoids presented by one idempotent relator

E-unitary inverse monoids and the Cayley graph of a group presentation

There is an important class of finitely presented groups known as automatic groups, in which many computations can be undertaken efficiently. In particular, there is a normal form for group elements which is the language of a finite state automation, and arbitrary words in the group generators can be reduced to normal form in quadratic time. The normal form can also be used to enumerate group elements, and to compute the (rational) growth function of the group. The theory of automatic groups is developed [2], and the algorithms are described in more detail in [4]. Implementations of the algorithms are available in a software package of the author ([3]).The word-hyperbolic groups form an important subclass of the automatic groups. They are defined by the property that geodesic triangles in the Cayley graph of the group are uniformly thin. In other words, there is a constant e, such a that any point on one of the sides of such a triangle is within distance e of the union of the other two sides of the triangle. Small cancellation groups and groups acting discretely and cocompactly on hyperbolic space are examples of word-hyperbolic groups. Their theory is developed in [1].The word problem in a word-hyperbolic group can be solved by a Dehn algorithm in linear time. Furthermore, Epstein has shown that the conjugacy problem can be solved in time O(n ln n). However, both of these algorithms require a prior knowledge of the constant e, and it does not seem to be easy to compute this constant in general.In practice, the quadratic time algorithm for the word problem coming from the automatic groups procedures performs very effectively, and it would be useful to have alternatives for the conjugacy problem.The automatic groups machinery does provide such an algorithm (which works in the the intermediate class of biautomatic groups), but that unfortunately is exponential.Alternative algorithms for conjugacy testing of elements and subgroups are currently being developed and implemented by a research student of mine, Joe Marshall. They have the minor disadvanatage that their basic versions work only in torsion-free groups, but adaptations to the more general case seem possible. They include also a test for malnormality of a subgroup H of G, which means that g-1Hg∩H = 1 for all g ∈ G\H. This test is important in topological applications.They make essential use of the boundary of the group. The points on the boundary can be defined as the equivalence classes of infinite geodesic rays starting from the base point of the Cayley graph, where two such rays are equivalent if they remain a bounded distance apart. The precise bound involved here turns out to be another universal constant d of the Cayley graph, but unlike e it can be computed in practice relatively easily. Roughly speaking, a certain linear function of d provides us with a bound on the length of the elements g that we need to consider as possible conjugating elements when testing for conjugacy or malnormality. To make the algorithm practical, we need to restrict this set of potential conjugators considerably more than this, and Marshall has come up with various tricks for doing this.

In this article the word problem for certain Burnside semigroups is shown to be decidable. For each word W ∈ A* a nondeterministic, finite-state automaton is constructed. This automaton accepts a word iff it is equivalent to W under the relations Ta = Ta+b, where a and b are fixed positive integers and T is an arbitrary word in A*. The method decides the word problem for those cases where a ≥ 6. The maximal subgroups are shown to be cyclic groups of order b.

Generic-case complexity, decision problems in group theory, and random walks

We study the language-theoretic properties of the word problem, in the sense of Duncan & Gilman, of weakly compressible monoids, as defined by Adian & Oganesian. We show that if C is a reversal-closed super-, as defined by Greibach, then M has word problem in C if and only if its compressed left monoid L(M) has word problem in C. As a special case, we may take C to be the class of context-free or indexed languages. As a corollary, we find many new classes of monoids with decidable rational subset membership problem. Finally, we show that it is decidable whether a one-relation monoid containing a non-trivial idempotent has context-free word problem. This answers a generalization of a question first asked by Zhang in 1992.

We study inverse semigroup amalgams [S 1,S 2;U], where S 1 and S 2 are finitely presented inverse semigroups with decidable word problem and U is an inverse semigroup with decidable membership problem in S 1 and S 2. We use a modified version of Bennett’s work on the structure of Schützenberger graphs of the ℛ-classes of S 1* U S 2 to state sufficient conditions for the amalgamated free products S 1* U S 2 having decidable word problem.

This paper presents several advances in the understanding of dynamic data structures in the bit-probe model: We improve the lower bound record for dynamic language membership problems to $\Omega((\frac{{\rm n}{\rm lg lg {\it n}})^{2})$. Surpassing $\Omega({\rm lg} {\it n})$ was listed as the first open problem in a survey by Miltersen. We prove a bound of $\Omega(\frac{{\rm n}{\rm lg lg lg {\it n}})$ for maintaining partial sums in ${\mathbb Z}/2{\mathbb Z}$. Previously, the known bounds were $\Omega(\frac{{\rm n}{\rm lg lg {\it n}})$ and $O({\rm n)$. We prove a surprising and tight upper bound of $O(\frac{{\rm lg} {\it n}}{\rm lg lg {\it n}})$ for predecessor problems. We use this to obtain the same upper bound for dynamic word and prefix problems in group-free monoids.

We establish a universal approach to solutions of the word problem in the varieties of di- and tri-algebras. This approach, for example, allows us to apply Gröbner–Shirshov bases method for Lie algebras to solve the ideal membership problem in free Leibniz algebras (Lie di-algebras). As another application, we prove an analogue of the Poincaré–Birkhoff–Witt Theorem for universal enveloping associative tri-algebra of a Lie tri-algebra (CTD!-algebra).

We give a number of new lower bounds in the cell probe model<br />with logarithmic cell size, which entails the same bounds on the random access computer with logarithmic word size and unit cost operations. We study the signed prefix sum problem: given a string of length n of zeroes and signed ones, compute the sum of its ith prefix during updates. We show a<br />lower bound of Omega(log n/log log n) time per operations, even if the prefix sums are bounded by log n/log log n during all updates. We also show that if the update time is bounded by the product of the worst-case update time and the<br />answer to the query, then the update time must be Omega(sqrt(log n/ log log n)).<br /> These results allow us to prove lower bounds for a variety of seemingly unrelated<br />dynamic problems. We give a lower bound for the dynamic planar point location in monotone subdivisions of <br />Omega(log n/ log log n) per operation. We give<br />a lower bound for the dynamic transitive closure problem on upward planar graphs with one source and one sink of <br />Omega(log n/(log logn)^2) per operation. We give a lower bound of Omega(sqrt(log n/log log n)) for the dynamic membership problem of any Dyck language with two or more letters. This implies the same<br />lower bound for the dynamic word problem for the free group with k generators. We also give lower bounds for the dynamic prefix majority and prefix equality problems.