136 research outputs found
Bisimple monogenic orthodox semigroups
We give a complete description of the structure of all bisimple orthodox
semigroups generated by two mutually inverse elements
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 left quotients; equivalently, we
look for the largest transition semigroup of an aperiodic finite automaton with
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 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 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
Lattice isomorphisms of bisimple monogenic orthodox semigroups
Using the classification and description of the structure of bisimple
monogenic orthodox semigroups obtained in \cite{key10}, we prove that every
bisimple orthodox semigroup generated by a pair of mutually inverse elements of
infinite order is strongly determined by the lattice of its subsemigroups in
the class of all semigroups. This theorem substantially extends an earlier
result of \cite{key25} stating that the bicyclic semigroup is strongly lattice
determined.Comment: Semigroup Forum (published online: 15 April 2011
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
Covers of acts over monoids II
In 1981 Edgar Enochs conjectured that every module has a flat cover and
finally proved this in 2001. Since then a great deal of effort has been spent
on studying different types of covers, for example injective and torsion free
covers. In 2008, Mahmoudi and Renshaw initiated the study of flat covers of
acts over monoids but their definition of cover was slightly different from
that of Enochs. Recently, Bailey and Renshaw produced some preliminary results
on the `other' type of cover and it is this work that is extended in this
paper. We consider free, divisible, torsion free and injective covers and
demonstrate that in some cases the results are quite different from the module
case
The algebra of adjacency patterns: Rees matrix semigroups with reversion
We establish a surprisingly close relationship between universal Horn classes
of directed graphs and varieties generated by so-called adjacency semigroups
which are Rees matrix semigroups over the trivial group with the unary
operation of reversion. In particular, the lattice of subvarieties of the
variety generated by adjacency semigroups that are regular unary semigroups is
essentially the same as the lattice of universal Horn classes of reflexive
directed graphs. A number of examples follow, including a limit variety of
regular unary semigroups and finite unary semigroups with NP-hard variety
membership problems.Comment: 30 pages, 9 figure
Electron energy loss and induced photon emission in photonic crystals
The interaction of a fast electron with a photonic crystal is investigated by
solving the Maxwell equations exactly for the external field provided by the
electron in the presence of the crystal. The energy loss is obtained from the
retarding force exerted on the electron by the induced electric field. The
features of the energy loss spectra are shown to be related to the photonic
band structure of the crystal. Two different regimes are discussed: for small
lattice constants relative to the wavelength of the associated electron
excitations , an effective medium theory can be used to describe the
material; however, for the photonic band structure plays an
important role. Special attention is paid to the frequency gap regions in the
latter case.Comment: 12 pages, 7 figure
A method for finding new sets of axioms for classes of semigroups
We introduce a general technique for finding sets of axioms for a given class of semigroups. To illustrate the technique, we provide new sets of defining axioms for groups of exponent n, bands, and semilattices
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