Free Inverse Semigroup Research Articles

Rational Growth of a Class of Inverse Semigroups

We give a sufficient condition for a finitely presented Rees quotient of a free inverse semigroup to have rational growth. Using related techniques we give a new proof that nonmonogenic free inverse semigroups have irrational growth. A new criterion for polynomial growth is proved and is used to show that polynomial growth implies rational growth. However we give an example of such a semigroup which has rational and exponential growth.

Degree of Growth of Some Inverse Semigroups

We show that if a finitely presented Rees quotient of a free inverse semigroup has polynomial growth, then its growth series is in a certain subsemiring of N[[z]]. We present a procedure for calculating the degree of growth of such a semigroup and use it to determine the possible values for the degree of growth.

Amalgams of free inverse semigroups

We study inverse semigroup amalgams of the formS *U T whereS andT are free inverse semigroups andU is an arbitrary finitely generated inverse subsemigroup ofS andT. 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 amalgamS *U T is in general undecidable, even ifS andT have decidable word problem,U is a free semigroup, and the membership problem forU inS andT is decidable. We also obtain a number of results concerning the structure of such amalgams. We obtain conditions for theD-classes of such an amalgam to be finite and we show that the amalgam is combinatorial in such a case. For example every one-relator amalgam of this type has finiteD-classes and is combinatorial. We also obtain information concerning when such an amalgam isE-unitary: for example every one relator amalgam of the formInv whereA andB are disjoint andu (resp.v) is a cyclically reduced word overA ∪A−1 (resp.B ∪B−1) isE-unitary.

HNN Extensions of Inverse Semigroups and Applications

Originally the concept of an HNN extension of a group was introduced by Higman, Neumann and Neumann in their study of embeddability of groups. Howie introduced the concept of HNN extensions of semigroups and showed embeddability in the case that the associated subsemigroups are unitary. On the other hand, T. E. Hall showed the embeddability of HNN extensions of inverse semigroups of a special type in his survey article on amalgamation of inverse semigroups. We introduce a more general definition of an HNN extension and show that free inverse semigroups and the bicyclic semigroup are HNN extensions of semilattices as examples of our new construction. We discuss weak HNN embeddability in several classes of semigroups and strong HNN embeddability in the class of inverse semigroups. One of our main purposes in the study of HNN extensions of inverse semigroups is to employ HNN extensions to examine some algorithmic problems. We prove the undecidability of Markov properties of finitely presented inverse semigroups using HNN extensions. This result was announced by Vazhenin in 1978, but no proof of it has been published to date. We also show undecidability of several non-Markov properties and discuss some undecidable problems on finitely generated inverse subsemigroups of finitely presented inverse semigroups.

On the Structure of Inverse Semigroup Amalgams

This paper is the second of two papers devoted to the study of amalgamated free products of inverse semigroups. We use the characterization of the Schützenberger automata given previously by the author to obtain structural results and preservational properties of lower bounded amalgams. Haataja, Margolis and Meakin have shown that if [S1,S2;U is an amalgam of regular semigroups in which S1∩ S2=U is a full regular subsemigroup of S1 and S2, then the maximal subgroups of the amalgamated free product S1*U S2 may be described by the fundamental groups of certain bipartite graphs of groups. In this paper we show that the maximal subgroups of a lower bounded amalgam [S1,S2;U] are either isomorphic copies of subgroups of S1 and S2 or can be described by the same Bass-Serre theory characterization. It follows, as for the regular case, that if S1 and S2 are combinatorial, then the maximal subgroups of S1*U S2 are free. By studying the endomorphism monoids of the Schützenberger graphs we obtain a number of results concerning when inverse semigroup properties are preserved under the amalgamated free product construction. For example, necessary and sufficient conditions are given for S1*U S2 to be completely semisimple. Under a mild assumption we establish necessary and sufficient conditions for S1*U S2 to have finite ℛ-classes. This enables us to reprove a result of Cherubini, Meakin and Piochi on amalgams of free inverse semigroups. Finally we give sufficient conditions for S1*U S2 to be E-unitary.

Compactness of Systems of Equations in Semigroups

We consider systems ui=vi(i∈I) of equations in semigroups over finite sets of variables. A semigroup (or a monoid) S is said to satisfy the compactness property (CP, for short), if each system of equations has an equivalent finite subsystem. We prove that all monoids in a variety [Formula: see text] satisfy CP if and only if the finitely generated monoids in [Formula: see text] satisfy the maximal condition on congruences. Consequently, all commutative monoids (and semigroups) satisfy CP. Also, if a finitely generated semigroup S satisfies CP, then S is necessarily hopfian and satisfies the chain condition on idempotents. It follows that the free inverse semigroups do not satisfy CP. Finally, we give two simple examples (the bicyclic monoid and the Baumslag-Solitar group) which do not satisfy CP, and show that the necessary conditions above are not sufficient.

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.

On the Bifree Locally Inverse Semigroup

A class of regular semigroups closed under taking direct products, regular subsemigroups, and homomorphic images is an existence variety, (or e-variety) of regular semigroups. For an e-variety V of locally inverse or E-solid regular semigroups, the bifree objectBFV(X) on a set X is the natural concept of a "free object" in V. Its existence has been proved by Y. T. Yeh. Using canonical forms of the elements of the bifree completely simple semigroup BFCS(X) we shall present two models of the bifree locally inverse semigroup BFLI(X). One is by means of a semidirect product of a semilattice by the bifree completely simple semigroup and is an analogue to Scheiblich′s model of the free inverse semigroup. The other description is in terms of canonical forms which are strongly related to Schein′s canonical forms for the elements of the free inverse semigroup. The proofs use the concept of bi-identities and are syntactical. As an application we get (new proofs of) the following results: (i) each locally inverse semigroup divides a semidirect product of a semilattice and a completely simple semigroup, (ii) the e-variety of all locally inverse semigroups is generated by either of the Mal′cev products S∘CS and I∘RB where S = semilattices, CS = completely simple semigroups, I = inverse semigroups, RB = rectangular bands.

Trace functions on inverse semigroup algebras

Let S be an inverse semigroup and let F be a subring of the complex field containing 1 and closed under complex conjugation. This paper concerns the existence of trace functions on F[S], the semigroup algebra of S over F. Necessary and sufficient conditions on S are found for the existence of a trace function on F[S] that takes positive integral values on the idempotents of S. Although F[S] does not always admit a trace function, a weaker form of linear functional is shown to exist for all choices of S. This is used to show that the natural involution on F[S] is special. It also leads to the construction of a trace function on F[S] for the case in which F is the real or complex field and S is completely semisimple of a type that includes countable free inverse semigroups.

Trace functions on the algebras of certain E-unitary inverse semigroups

A construction is given for a trace function on the semigroup algebra of a certain type of E-unitary inverse semigroup over any subfield of the complex field that is closed under complex conjugation. In particular, the method applies to the semigroup algebras of free inverse semigroups of arbitrary rank.

A note on primeness of semigroup rings

SynopsisThis note discusses primeness for semigroup rings of semigroups satisfying a certain condition, weaker than the u.p. property. These semigroups include free products of semigroups, semigroups presented by a single relator and at least three generators, and free inverse semigroups of infinite rank.

A class of varieties of inverse semigroups

Let Wr( U , V ) denote the variety of inverse semigroups generated by wreath products of semigroups in U with semigroups in V , where U and V are varieties of inverse semigroups. In this note we provide a description of the fully invariant congruence on the free inverse semigroup corresponding to Wr( U , V ) in terms of the fully invariant congruences corresponding to U and V . Some properties of varieties of this form are then presented.

The Szendrei expansion of a semigroup

MathematikaVolume 37, Issue 2 p. 251-260 Research Article The Szendrei expansion of a semigroup John Fountain, John Fountain Department of Mathematics, University of York, Heslington, York, YO1 5DDSearch for more papers by this authorGracinda M. S. Gomes, Gracinda M. S. Gomes Departamento de Matemàtica, Faculdade de Ciências, Universidade de Lisboa, 1700 Lisboa, PortugalSearch for more papers by this author John Fountain, John Fountain Department of Mathematics, University of York, Heslington, York, YO1 5DDSearch for more papers by this authorGracinda M. S. Gomes, Gracinda M. S. Gomes Departamento de Matemàtica, Faculdade de Ciências, Universidade de Lisboa, 1700 Lisboa, PortugalSearch for more papers by this author First published: 01 December 1990 https://doi.org/10.1112/S0025579300012973Citations: 9AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume37, Issue2December 1990Pages 251-260 RelatedInformation

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

Geometric methods have played a fundamental and crucial role in combinatorial group theory almost from the inception of that field. In this paper we initiate a study of the use of some of these methods in inverse semigroup theory. We modify a lemma of I. Simon and show how to construct E-unitary inverse monoids from the free idempotent and commutative category over the Cayley graph of the maximal group image. The construction provides an expansion from the category of X-generated groups to the category of X-generated E-unitary inverse monoids and specializes to a construction of certain relatively free E-unitary inverse monoids. We show more generally that this construction is the left adjoint of the maximal group image functor. Munn's solution to the word problem for the free inverse monoids and several of the results of McAlister and McFadden on the free inverse semigroup with two commuting generators may be obtained fairly easily from the construction. We construct the free product of E-unitary inverse monoids, thus providing an alternate construction to that of Jones.

Basis properties for semigroups

A universal algebra A is said to have the basis property (BP) if any two minimal generating sets (bases) for a subalgebra of A have the same cardinality. This property was studied by the author for inverse semigroups in [5, 6]. For instance free inverse semigroups have BP. When treated as universal algebras, a classical theorem of linear algebra states that vector spaces have BP. In this paper we study BP for semigroups.

Expansions, free inverse semigroups, and Schützenberger product

In this paper we shall present a new construction of the free inverse monoid on a set X. Contrary to the previous constructions of [9, 11], our construction is symmetric and originates from classical ideas of language theory. The ingredients of this construction are the free group on X and the relation that associates to a word w of the free monoid on X, the set of all pairs ( u, v) such that uv = w. It follows at once from our construction that the free inverse monoid on X can be naturally embedded into the Schützenberger product of two free groups of basis X. We shall also give some connections with the theory of expansions as developed by Rhodes and Birget [2, 3].

Properties of relatively free inverse semigroups

The objective of this paper is to study structural properties of relatively free inverse semigroups in varieties of inverse semigroups. It is shown, for example, that if S S is combinatorial (i.e., H \mathcal {H} is trivial), completely semisimple (i.e., every principal factor is a Brandt semigroup or, equivalently, S S does not contain a copy of the bicyclic semigroup) or E E -unitary (i.e., E ( S ) E(S) is the kernel of the minimum group congruence) then the relatively free inverse semigroup F V X F{\mathcal {V}_X} on the set X X in the variety V \mathcal {V} generated by S S is also combinatorial, completely semisimple or E E -unitary, respectively. If S S is a fundamental (i.e., the only congruence contained in H \mathcal {H} is the identity congruence) and | X | ⩾ ℵ 0 |X| \geqslant {\aleph _0} , then F V X F{\mathcal {V}_X} is also fundamental. F V X F{\mathcal {V}_X} may not be fundamental if | X | > ℵ 0 |X| > {\aleph _0} . It is also shown that for any variety of groups U \mathcal {U} and for | X | ⩾ ℵ 0 |X| \geqslant {\aleph _0} , there exists a variety of inverse semigroups V \mathcal {V} which is minimal with respect to the properties (i) F V X F{\mathcal {V}_X} is fundamental and (ii) V ∩ G = U \mathcal {V} \cap \mathcal {G} = \mathcal {U} , where G \mathcal {G} is the variety of groups. In the main result of the paper it is shown that there exists a variety V \mathcal {V} for which F V X F{\mathcal {V}_X} is not completely semisimple, thereby refuting a long standing conjecture.

Diophantine theories of free inverse semigroups

It is known that in a free inverse semigroup I the problem of equality of words is decidable [I]. It has been established in [2] that the elementary theory of I is decidable in the case when I is monogenic and inseparable in other cases. For finitely generated nonmonogenic free inverse semigroups the undecidability of positive theories was proved in [3]. The main goal of the present note is to prove that the problem of consistency of systems of equations on a nonmonogenic free inverse semigroup is undecidable.

RELATIVELY FREE INVERSE SEMIGROUPS

Journal Article RELATIVELY FREE INVERSE SEMIGROUPS Get access P. G. TROTTER P. G. TROTTER Department of Mathematics, University of TasmaniaHobart, Tasmania, Australia Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 37, Issue 3, September 1986, Pages 357–374, https://doi.org/10.1093/qmath/37.3.357 Published: 01 September 1986 Article history Received: 24 April 1985 Published: 01 September 1986

Regular expansions of completely simple semigroups

Corresponding to each completely simple semigroup S a semigroup P( S) is constructed. The semigroup P( S) is a perfect rectangular band of inverse monoids (that is, P( S) a disjoint union of inverse monoids S i, λ ( i ϵ I, λ ϵ Λ) with S iλ S jμ = S iμ for all i, j ϵ I and λ, μ ϵ Λ). The functor S → P( S), Φ → P( Φ) is an expansion in the sense of Birget and Rhodes (“Almost finite expansions of arbitrary semigroups”, J. Pure Appl. Algebra 32 (1984) 239–287). A representation for P( S) as a Rees matrix semigroup M (T; I, Λ; P) over an inverse monoid T with entries from P in the group of units of T is provided. The expansion provides a construction of the free inverse semigroup and the free rectangular band of inverse semigroups when applied to the free group and the free completely simple semigroup respectively.