On generators and presentations of semidirect products in inverse semigroups

In this paper we prove two main results. The first is a necessary and sufficient condition for a semidirect product of a semilattice by a group to be finitely generated. The second result is a necessary and sufficient condition for such a semidirect product to be finitely presented.Publisher PDFPeer reviewe

Automatic S-acts and inverse semigroup presentations

To provide a general framework for the theory of automatic groups and semigroups, we introduce the notion of an automatic semigroup act. This notion gives rise to a variety of definitions for automaticity depending on the set chosen as a semigroup act. Namely, we obtain the notions of automaticity, Schutzenberger automaticity, R- and L-class automaticity, etc. We discuss the basic properties of automatic semigroup acts. We show that if S is a semigroup with local right identities, then automaticty of a semigroup act is independent of the choice of both the generators of S and the generators of the semigroup act. We also discuss the equality problem of automatic semigroup acts. To give a geometric approach, we associate a directed labelled graph to each S-act and introduce the notion of the fellow traveller property in the associated graph. We verify that if S is a regular semigroup with finitely many idempotents, then Schutzenberger automaticity is characterized by the fellow traveller property of the Schutzenberger graph. We also verify that a Schutzenberger automatic regular semigroup with finitely many idempotents is finitely presented. We end Chapter 3 by proving that an inverse free product of Schutzenberger automatic inverse semigroups is Schutzenberger automatic. In Chapter 4, we first introduce the notion of finite generation and finite presentability with respect to a semigroup action. With the help of these concepts we give a necessary and sufficient condition for a semidirect product of a semilattice by a group to be finitely generated and finitely presented as an inverse semigroup. We end Chapter 4 by giving a necessary and sufficient condition for the semidirect product of a semilattice by a group to be Schutzenberger automatic. Chapter 5 is devoted to the study of HNN extensions of inverse semigroups from finite generation and finite presentability point of view. Namely, we give necessary and sufficient conditions for finite presentability of Gilbert's and Yamamura's HNN extension of inverse semigroups. The majority of the results contained in Chapter 5 are the result of a joint work with N.D. Gilbert and N. Ruskuc

Almost factorisable straight locally inverse semigroups

We introduce the notion of a factorisable and an almost factorisable straight locally inverse semigroup, and prove that every straight locally inverse semigroup can be obtained as a subsemigroup of an idempotent separating homomorphic image of a Pastijn product of a semilattice by a completely simple semigroup. Moreover, we give an alternative proof of the fact that each straight locally inverse semigroup has a weakly E-unitary cover, implicitely due to Pastijn

On the minimum weight bounded length circuit cover problem on grid graphs

This paper looks at the minimum weight bounded length circuit cover problem on rectangular grid graphs. We give a constructive procedure to determine an edge cover of the augemented Eulerian graph with circuits of fixed length

The tiling semigroups of one-dimensional periodic tilings

A one-dimensional tiling is a bi-infinite string on a finite alphabet, and its tiling semigroup is an inverse semigroup whose elements are marked finite substrings of the tiling. We characterize the structure of these semigroups in the periodic case, in which the tiling is obtained by repetition of a fixed primitive word

Changes in engineering students’ surface and deep approaches to learning in first- and second-year university courses

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Finite presentability of HNN extensions of inverse semigroups

HNN extensions of inverse semigroups, where the associated inverse subsemigroups are order ideals of the base, are defined by means of a construction based upon the isomorphism between the categories of inverse semigroups and inductive groupoids. The resulting HNN extension may conveniently be described by an inverse semigroup presentation, and we determine when an HNN extension with finitely generated or finitely presented base is again finitely generated or finitely presented. Our main results depend upon properties of the -preorder in the associated subsemigroups. Let S be a finitely generated inverse semigroup and let U, V be inverse subsemigroups of S, isomorphic via φ: U → V, that are order ideals in S. We prove that the HNN extension S*U,φ is finitely generated if and only if U is finitely -dominated. If S is finitely presented, we give a necessary and suffcient condition for S*U,φ to be finitely presented. Here, in contrast to the theory of HNN extensions of groups, it is not necessary that U be finitely generated