46 research outputs found
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
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
Unary FA-presentable semigroups
Automatic presentations, also called FA-presentations, were introduced to extend nite model theory to innite structures whilst retaining the solubility of interesting decision problems. A particular focus of research has been the classication of those structures of some species that admit automatic presentations. Whilst some successes have been obtained, this appears to be a dicult problem in general. A restricted problem, also of signicant interest, is to ask this question for unary automatic presentations: auto-matic presentations over a one-letter alphabet. This paper studies unary FA-presentable semigroups. We prove the following: Every unary FA-presentable structure admits an injective unary automatic presentation where the language of representatives consists of every word over a one-letter alphabet. Unary FA-presentable semigroups are locally nite, but non-nitely generated unary FA-presentable semigroups may be innite. Every unary FA-presentable semigroup satises some Burnside identity.We describe the Green's relations in unary FA-presentable semigroups. We investigate the relationship between the class of unary FA-presentable semigroups and various semigroup constructions. A classication is given of the unary FA-presentable completely simple semigroups.PostprintPeer reviewe
Ideals and finiteness conditions for subsemigroups
In this paper we consider a number of finiteness conditions for semigroups
related to their ideal structure, and ask whether such conditions are preserved
by sub- or supersemigroups with finite Rees or Green index. Specific properties
under consideration include stability, D=J and minimal conditions on ideals.Comment: 25 pages, revised according to referee's comments, to appear in
Glasgow Mathematical Journa