Periodic elements of the free idempotent generated semigroup on a biordered set

We show that every periodic element of the free idempotent generated semigroup on an arbitrary biordered set belongs to a subgroup of the semigroup

Largeness and SQ-universality of cyclically presented groups

Largeness, SQ-universality, and the existence of free subgroups of rank 2 are measures of the complexity of a finitely presented group. We obtain conditions under which a cyclically presented group possesses one or more of these properties. We apply our results to a class of groups introduced by Prishchepov which contains, amongst others, the various generalizations of Fibonacci groups introduced by Campbell and Robertson

Possible detection of singly-ionized oxygen in the Type Ia SN 2010kg

We present direct spectroscopic modeling of 11 high-S/N observed spectra of the Type Ia SN 2010kg, taken between -10 and +5 days with respect to B-maximum. The synthetic spectra, calculated with the SYN++ code, span the range between 4100 and 8500 \r{A}. Our results are in good agreement with previous findings for other Type Ia SNe. Most of the spectral features are formed at or close to the photosphere, but some ions, like Fe II and Mg II, also form features at ~2000 - 5000 km s$^{-1}$ above the photosphere. The well-known high-velocity features of the Ca II IR-triplet as well as Si II $\lambda$6355 are also detected. The single absorption feature at ~4400 \r{A}, which usually has been identified as due to Si III, is poorly fit with Si III in SN 2010kg. We find that the fit can be improved by assuming that this feature is due to either C III or O II, located in the outermost part of the ejecta, ~4000 - 5000 km s$^{-1}$ above the photosphere. Since the presence of C III is unlikely, because of the lack of the necessary excitation/ionization conditions in the outer ejecta, we identify this feature as due to O II. The simultaneous presence of O I and O II is in good agreement with the optical depth calculations and the temperature distribution in the ejecta of SN 2010kg. This could be the first identification of singly ionized oxygen in a Type Ia SN atmosphere.Comment: Submitted to MNRA

Groups of Fibonacci type revisited

This article concerns a class of groups of Fibonacci type introduced by Johnson and Mawdesley that includes Conway?s Fibonacci groups, the Sieradski groups, and the Gilbert-Howie groups. This class of groups provides an interesting focus for developing the theory of cyclically presented groups and, following questions by Bardakov and Vesnin and by Cavicchioli, Hegenbarth, and Repov?s, they have enjoyed renewed interest in recent years. We survey results concerning their algebraic properties, such as isomorphisms within the class, the classification of the finite groups, small cancellation properties, abelianizations, asphericity, connections with Labelled Oriented Graph groups, and the semigroups of Fibonacci type. Further, we present a new method of proving the classification of the finite groups that deals with all but three groups

Generalized Green'S Equivalences on the Subsemigroups of the Bicyclic Monoid

We study generalized Green's equivalences on all subsemigroups of the bicyclic monoid B and determine the abundant (and adequate) subsemigroups of B. © 2010 Copyright Taylor and Francis Group, LLC

The structure of one-relator relative presentations and their centres

Suppose that G is a nontrivial torsion-free group and w is a word in the alphabet G\cup\{x_1^{\pm1},...,x_n^{\pm1}\} such that the word w' obtained from w by erasing all letters belonging to G is not a proper power in the free group F(x_1,...,x_n). We show how to reduce the study of the relative presentation \^G= to the case n=1. It turns out that an "n-variable" group \^G can be constructed from similar "one-variable" groups using an explicit construction similar to wreath product. As an illustration, we prove that, for n>1, the centre of \^G is always trivial. For n=1, the centre of \^G is also almost always trivial; there are several exceptions, and all of them are known.Comment: 15 pages. A Russian version of this paper is at http://mech.math.msu.su/department/algebra/staff/klyachko/papers.htm . V4: the intoduction is rewritten; Section 1 is extended; a short introduction to Secton 5 is added; some misprints are corrected and some cosmetic improvements are mad

Large Aperiodic Semigroups

The syntactic complexity of a regular language is the size of its syntactic semigroup. This semigroup is isomorphic to the transition semigroup of the minimal deterministic finite automaton accepting the language, that is, to the semigroup generated by transformations induced by non-empty words on the set of states of the automaton. In this paper we search for the largest syntactic semigroup of a star-free language having $n$ left quotients; equivalently, we look for the largest transition semigroup of an aperiodic finite automaton with $n$ states. We introduce two new aperiodic transition semigroups. The first is generated by transformations that change only one state; we call such transformations and resulting semigroups unitary. In particular, we study complete unitary semigroups which have a special structure, and we show that each maximal unitary semigroup is complete. For $n \ge 4$ there exists a complete unitary semigroup that is larger than any aperiodic semigroup known to date. We then present even larger aperiodic semigroups, generated by transformations that map a non-empty subset of states to a single state; we call such transformations and semigroups semiconstant. In particular, we examine semiconstant tree semigroups which have a structure based on full binary trees. The semiconstant tree semigroups are at present the best candidates for largest aperiodic semigroups. We also prove that $2^n-1$ is an upper bound on the state complexity of reversal of star-free languages, and resolve an open problem about a special case of state complexity of concatenation of star-free languages.Comment: 22 pages, 1 figure, 2 table

On covers of cyclic acts over monoids

In (Bull. Lond. Math. Soc. 33:385–390, 2001) Bican, Bashir and Enochs finally solved a long standing conjecture in module theory that all modules over a unitary ring have a flat cover. The only substantial work on covers of acts over monoids seems to be that of Isbell (Semigroup Forum 2:95–118, 1971), Fountain (Proc. Edinb. Math. Soc. (2) 20:87–93, 1976) and Kilp (Semigroup Forum 53:225–229, 1996) who only consider projective covers. To our knowledge the situation for flat covers of acts has not been addressed and this paper is an attempt to initiate such a study. We consider almost exclusively covers of cyclic acts and restrict our attention to strongly flat and condition (P) covers. We give a necessary and sufficient condition for the existence of such covers and for a monoid to have the property that all its cyclic right acts have a strongly flat cover (resp. (P)-cover). We give numerous classes of monoids that satisfy these conditions and we also show that there are monoids that do not satisfy this condition in the strongly flat case. We give a new necessary and sufficient condition for a cyclic act to have a projective cover and provide a new proof of one of Isbell’s classic results concerning projective covers. We show also that condition (P) covers are not unique, unlike the situation for projective covers