64 research outputs found
On residual finiteness of monoids, their SchĆ¼tzenberger groups and associated actions
RG was supported by an EPSRC Postdoctoral Fellowship EP/E043194/1 held at the University of St Andrews, Scotland.In this paper we discuss connections between the following properties: (RFM) residual finiteness of a monoid M ; (RFSG) residual finiteness of SchĆ¼tzenberger groups of M ; and (RFRL) residual finiteness of the natural actions of M on its Green's R- and L-classes. The general question is whether (RFM) implies (RFSG) and/or (RFRL), and vice versa. We consider these questions in all the possible combinations of the following situations: M is an arbitrary monoid; M is an arbitrary regular monoid; every J-class of M has finitely many R- and L-classes; M has finitely many left and right ideals. In each case we obtain complete answers, which are summarised in a table.PostprintPeer reviewe
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
Subalgebras of FA-presentable algebras
Automatic presentations, also called FA-presentations, were introduced to
extend finite model theory to infinite structures whilst retaining the
solubility of fundamental decision problems. This paper studies FA-presentable
algebras. First, an example is given to show that the class of finitely
generated FA-presentable algebras is not closed under forming finitely
generated subalgebras, even within the class of algebras with only unary
operations. However, it is proven that a finitely generated subalgebra of an
FA-presentable algebra with a single unary operation is itself FA-presentable.
Furthermore, it is proven that the class of unary FA-presentable algebras is
closed under forming finitely generated subalgebras, and that the membership
problem for such subalgebras is decidable.Comment: 19 pages, 6 figure
Growth rates for subclasses of Av(321)
Pattern classes which avoid 321 and other patterns are shown to have the same growth rates as similar (but strictly larger) classes obtained by adding articulation points to any or all of the other patterns. The method of proof is to show that the elements of the latter classes can be represented as bounded merges of elements of the original class, and that the bounded merge construction does not change growth rates
On well quasi-order of graph classes under homomorphic image orderings
In this paper we consider the question of well quasi-order for classes defined by a single obstruction within the classes of all graphs, digraphs and tournaments, under the homomorphic image ordering (in both its standard and strong forms). The homomorphic image ordering was introduced by the authors in a previous paper and corresponds to the existence of a surjective homomorphism between two structures. We obtain complete characterisations in all cases except for graphs under the strong ordering, where some open questions remain.PostprintPeer reviewe
Automatic semigroups
AbstractThe area of automatic groups has been one in which significant advances have been made in recent years. While it is clear that the definition of an automatic group can easily be extended to that of an automatic semigroup, there does not seem to have been a systematic investigation of such structures. It is the purpose of this paper to make such a study.We show that certain results from the group-theoretic situation hold in this wider context, such as the solvability of the word problem in quadratic time, although others do not, such as finite presentability. There are also situations which arise in the general theory of semigroups which do not occur when considering groups; for example, we show that a semigroup S is automatic if and only if S with a zero adjoined is automatic, and also that S is automatic if and only if S with an identity adjoined is automatic. We use this last result to show that any finitely generated subsemigroup of a free semigroup is automatic
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
On regularity and the word problem for free idempotent generated semigroups
The category of all idempotent generated semigroups with a prescribed structure E of their idempotents E (called the biordered set) has an initial object called the free idempotent generated semigroup over E, defined by a presentation over alphabet E, and denoted by IG(E). Recently, much effort has been put into investigating the structure of semigroups of the form IG(E), especially regarding their maximal subgroups. In this paper we take these investigations in a new direction by considering the word problem for IG(E). We prove two principal results, one positive and one negative. We show that, for a finite biordered set E, it is decidable whether a given word w ā Eārepresents a regular element; if in addition one assumes that all maximal subgroups of IG(E) have decidable word problems, then the word problem in IG(E) restricted to regular words is decidable. On the other hand, we exhibit a biorder E arising from a finite idempotent semigroup S, such that the word problem for IG(E) is undecidable, even though all the maximal subgroups have decidable word problems. This is achieved by relating the word problem of IG(E) to the subgroup membership problem in finitely presented groups
Homotopy bases and finite derivation type for Schutzenberger groups of monoids
Given a finitely presented monoid and a homotopy base for the monoid, and
given an arbitrary Schutzenberger group of the monoid, the main result of this
paper gives a homotopy base, and presentation, for the Schutzenberger group. In
the case that the R-class R' of the Schutzenberger group G(H) has only finitely
many H-classes, and there is an element s of the multiplicative right pointwise
stabilizer of H, such that under the left action of the monoid on its R-classes
the intersection of the orbit of the R-class of s with the inverse orbit of R'
is finite, then finiteness of the presentation and of the homotopy base is
preserved.Comment: 24 page
- ā¦