Automata with Counters that Recognize Word Problems of Free Products

An M-automaton is a finite automaton with a blind counter that mimics a monoid M. The finitely generated groups whose word problems (when viewed as formal languages) are accepted by M-automata play a central role in understanding the family 𝔏(M) of all languages accepted by M-automata. If G1 and G2 are finitely generated groups whose word problems are languages in 𝔏(M), in general, the word problem of the free product G1 * G2 is not necessarily in 𝔏(M). However, we show that if M is enlarged to the free product M*P2, where P2 is the polycyclic monoid of rank two, then this closure property holds. In fact, we show more generally that the special word problem of M1 * M2 lies in 𝔏(M * P2) whenever M1 and M2 are finitely generated monoids with special word problems in 𝔏(M * P2). We also observe that there is a monoid without zero, denoted by CF2, that can be used in place of P2 for this purpose. The monoid CF2 is the rank two case of what we call a monoid with right invertible basis and its Rees quotient by its maximal ideal is P2. The fundamental theory of monoids with right invertible bases is completely analogous to that of free groups, and thus they are very convenient to use. We also investigate the questions of whether there is a group that can be used instead of the monoid P2 in the above result and under what circumstances P1 (or the bicyclic monoid) is enough to do the job of P2.

GROWTH OF REES QUOTIENTS OF FREE INVERSE SEMIGROUPS DEFINED BY SMALL NUMBERS OF RELATORS

We study the asymptotic behavior of a finitely presented Rees quotient S = Inv 〈A|ci = 0(i = 1, …, k)〉 of a free inverse semigroup over a finite alphabet A. It is shown that if the semigroup S has polynomial growth then S is monogenic (with zero) or k ≥ 3. The three relator case is fully characterized, yielding a sequence of two-generated three relator semigroups whose Gelfand–Kirillov dimensions form an infinite set, namely {4, 5, 6, …}. The results are applied to give a best possible lower bound, in terms of the size of the generating set, on the number of relators required to guarantee polynomial growth of a finitely presented Rees quotient, assuming no generator is nilpotent. A natural operator is introduced, from the class of all finitely presented inverse semigroups to the class of finitely presented Rees quotients of free inverse semigroups, and applied to deduce information about inverse semigroup presentations with one or many relations. It follows quickly from Magnus' Freiheitssatz for one relator groups that every inverse semigroup Π = Inv 〈a1, …, an|C = D 〉 has exponential growth if n > 2. It is shown that the growth of Π is also exponential if n = 2 and the Munn trees of both defining words C and D contain more than one edge.

GROWTH OF FINITELY PRESENTED REES QUOTIENTS OF FREE INVERSE SEMIGROUPS

We prove that a finitely presented Rees quotient of a free inverse semigroup has polynomial growth if and only if it has bounded height. This occurs if and only if the set of nonzero reduced words has bounded Shirshov height and all nonzero reduced but not cyclically reduced words are nilpotent. This occurs also if and only if the set of nonzero geodesic words have bounded Shirshov height. We also give a simple sufficient graphical condition for polynomial growth, which is necessary when all zero relators are reduced. As a final application of our results, we give an inverse semigroup analogue of a classical result that characterizes polynomial growth of finitely presented Rees quotients of free semigroups in terms of connections between non-nilpotent elements and primitive words that label loops of the Ufnarovsky graph of the presentation.

FINITELY BASED, FINITE SETS OF WORDS

For W a finite set of words, we consider the Rees quotient of a free monoid with respect to the ideal consisting of all words that are not subwords of W. This resulting monoid is denoted by S(W). It is shown that for every finite set of words W, there are sets of words U⊃W and V⊃W such that the identities satisfied by S(V) are finitely based and those of S(U) are not finitely based [regardless of the situation for S(W)]. The first examples of finitely based (not finitely based) aperiodic finite semigroups whose direct product is not finitely based (finitely based) are presented and it is shown that every monoid of the form S(W) with fewer than 9 elements is finitely based and that there is precisely one not finitely based 9 element example.

FINITELY BASED WORDS

Let W be a finite language and let Wc be the closure of W under taking subwords. Let S(W) denote the Rees quotient of a free monoid over the ideal consisting of all words that are not in Wc. We call W finitely based if the monoid S(W) is finitely based. Although these semigroups have easy structure they behave "generically" with respect to the finite basis property [6]. In this paper, we describe all finitely based words in a two-letter alphabet. We also find some necessary and some sufficient conditions for a set of words to be finitely based.

GROWTH AND EXISTENCE OF IDENTITIES IN A CLASS OF FINITELY PRESENTED INVERSE SEMIGROUPS WITH ZERO

We prove that a finitely presented Rees quotient of a free inverse semigroup has polynomial or exponential growth, and that the type of growth is algorithmically recognizable. We prove that such a semigroup has polynomial growth if and only if it satisfies a certain semigroup identity. However we give an example of such a semigroup which has exponential growth and satisfies some nontrivial identity in signature with involution.

Nilpotents and congruences on semigroups of transformations with fixed rank

In 1988, Howie and Marques-Smith studied Pm, a Rees quotient semigroup of transformations associated with a regular cardinal m, and described the elements which can be written as a product of nilpotents in Pm. In 1981, Marques proved that if Δm denotes the Malcev congruence on Pm, then Pm/Δm is congruence-free for any infinite m. In this paper, we describe the products of nilpotents in Pm when m is nonregular, and determine all the congruences on Pm when m is an arbitrary infinite cardinal. We also investigate when a nilpotent is a product of idempotents.

Inverse semigroups generated by nilpotent transformations

SynopsisLet X be a set with infinite cardinality m and let B be the Baer-Levi semigroup, consisting of all one-one mappings a:X→X for which ∣X/Xα∣ = m. Let Km=<B 1B>, the inverse subsemigroup of the symmetric inverse semigroup ℐ(X) generated by all products β−γ, with β,γ∈B. Then Km = <N2>, where N2 is the subset of ℐ(X) consisting of all nilpotent elements of index 2. Moreover, Km has 2-nilpotent-depth 3, in the sense that Let Pm be the ideal {α∈Km: ∣dom α∣<m} in Km and let Lm be the Rees quotient Km/Pm. Then Lm is a 0-bisimple, 2-nilpotent-generated inverse semigroup with 2-nilpotent-depth 3. The minimum non-trivial homomorphic image of Lm also has these properties and is congruence-free.

Certain homomorphisms of a compact semigroup onto a thread

Let S be a compact semigroup and f a continuous homomorphism of S onto the (compact) semigroup T. What can be said concerning the relations among S, f, and T? It is to one special aspect of this problem which we shall address ourselves. In particular, our primary considerations will be directed toward the case in which T is a standard thread. A standard thread is a compact semigroup which is topologically an arc, one endpoint being an identity element, the other being a zero element. The structure of standard threads is rather completely determined e.g. see [20]. Among the standard threads there are three which have a rather special rôle. These are as follows: A unit thread is a standard thread with only two idempotents and no nilpotent element. A unit thread is isomorphic to the usual unit interval [14]. A nil thread again has only two idempotents but has a non-zero nilpotent element. A nil thread is isomorphic with the interval [½, 1], the multiplication being the maximum of ½ and the usual product — or, what is the same thing, the Rees quotient of the usual [0, 1] by the ideal [0,½ ]. Finally there is the idempotent thread, the multiplication being x o y = mm (x, y). These three standard threads can often be considered separately and, in this paper, we reserve the symbols I1I2 and I3 to denote the unit, nil and idempotent threads respectively. Also, throughout this paper, by a homomorphism we mean a continuous homomorphism.