67 research outputs found
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 structures for semigroup constructions
We survey results concerning automatic structures for semigroup
constructions, providing references and describing the corresponding automatic
structures. The constructions we consider are: free products, direct products,
Rees matrix semigroups, Bruck-Reilly extensions and wreath products.Comment: 22 page
Finite complete rewriting systems for regular semigroups
It is proved that, given a (von Neumann) regular semigroup with finitely many
left and right ideals, if every maximal subgroup is presentable by a finite
complete rewriting system, then so is the semigroup. To achieve this, the
following two results are proved: the property of being defined by a finite
complete rewriting system is preserved when taking an ideal extension by a
semigroup defined by a finite complete rewriting system; a completely 0-simple
semigroup with finitely many left and right ideals admits a presentation by a
finite complete rewriting system provided all of its maximal subgroups do.Comment: 11 page
Hopfian and co-hopfian subsemigroups and extensions
This paper investigates the preservation of hopficity and co-hopficity on
passing to finite-index subsemigroups and extensions. It was already known that
hopficity is not preserved on passing to finite Rees index subsemigroups, even
in the finitely generated case. We give a stronger example to show that it is
not preserved even in the finitely presented case. It was also known that
hopficity is not preserved in general on passing to finite Rees index
extensions, but that it is preserved in the finitely generated case. We show
that, in contrast, hopficity is not preserved on passing to finite Green index
extensions, even within the class of finitely presented semigroups. Turning to
co-hopficity, we prove that within the class of finitely generated semigroups,
co-hopficity is preserved on passing to finite Rees index extensions, but is
not preserved on passing to finite Rees index subsemigroups, even in the
finitely presented case. Finally, by linking co-hopficity for graphs to
co-hopficity for semigroups, we show that without the hypothesis of finite
generation, co-hopficity is not preserved on passing to finite Rees index
extensions.Comment: 15 pages; 3 figures. Revision to fix minor errors and add summary
table
Markov semigroups, monoids, and groups
A group is Markov if it admits a prefix-closed regular language of unique
representatives with respect to some generating set, and strongly Markov if it
admits such a language of unique minimal-length representatives over every
generating set. This paper considers the natural generalizations of these
concepts to semigroups and monoids. Two distinct potential generalizations to
monoids are shown to be equivalent. Various interesting examples are presented,
including an example of a non-Markov monoid that nevertheless admits a regular
language of unique representatives over any generating set. It is shown that
all finitely generated commutative semigroups are strongly Markov, but that
finitely generated subsemigroups of virtually abelian or polycyclic groups need
not be. Potential connections with word-hyperbolic semigroups are investigated.
A study is made of the interaction of the classes of Markov and strongly Markov
semigroups with direct products, free products, and finite-index subsemigroups
and extensions. Several questions are posed.Comment: 40 pages; 3 figure
Right noetherian semigroups
A semigroup S is right noetherian if every right congruence on S is finitely generated. In this paper we present some fundamental properties of right noetherian semigroups, discuss how semigroups relate to their substructures with regard to the property of being right noetherian, and investigate whether this property is preserved under various semigroup constructionsPostprintPeer reviewe
Automatic semigroups : constructions and subsemigroups
In this thesis we start by considering conditions under which some standard semigroup constructions preserve automaticity. We first consider Rees matrix semigroups over a semigroup, which we call the base, and work on the following questions: (i) If the base is automatic is the Rees matrix semigroup automatic? (ii) If the Rees matrix semigroup is automatic must the base be automatic as well? We also consider similar questions for Bruck-Reilly extensions of monoids and wreath products of semigroups. Then we consider subsemigroups of free products of semigroups and we study conditions that guarantee them to be automatic. Finally we obtain a description of the subsemigroups of the bicyclic monoid that allow us to study some of their properties, which include finite generation, automaticity and finite presentability
Graph automatic semigroups
In this thesis we examine properties and constructions of graph automatic semigroups, a generalisation of both automatic semigroups and finitely generated FA-presentable semigroups.
We consider the properties of graph automatic semigroups, showing that they are independent of the choice of generating set, have decidable word problem, and that if we have a graph automatic structure for a semigroup then we can find one with uniqueness.
Semigroup constructions and their effect on graph automaticity are considered. We show that finitely generated direct products, free products, finitely generated Rees matrix semigroup constructions, zero unions, and ordinal sums all preserve unary graph automaticity, and examine when the converse also holds. We also demonstrate situations where semidirect products, Bruck-Reilly extensions, and semilattice constructions preserve graph automaticity, and consider the conditions we may impose on such constructions in order to ensure that graph automaticity is preserved.
Unary graph automatic semigroups, that is semigroups which have graph automatic structures over a single letter alphabet, are also examined. We consider the form of an automaton recognising multiplication by generators in such a semigroup, and use this to demonstrate various properties of unary graph automatic semigroups. We show that infinite periodic semigroups are not unary graph automatic, and show that we may always find a uniform set of normal forms for a unary graph automatic semigroup. We also determine some necessary conditions for a semigroup to be unary graph automatic, and use this to provide examples of semigroups which are not unary graph automatic.
Finally we consider semigroup constructions for unary graph automatic semigroups. We show that the free product of two semigroups is unary graph automatic if and only if both semigroups are trivial; that direct products do not always preserve unary graph automaticity; and that Bruck-Reilly extensions are never unary graph automatic
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