219 research outputs found
Green's Relations in Finite Transformation Semigroups
We consider the complexity of Green's relations when the semigroup is given
by transformations on a finite set. Green's relations can be defined by
reachability in the (right/left/two-sided) Cayley graph. The equivalence
classes then correspond to the strongly connected components. It is not
difficult to show that, in the worst case, the number of equivalence classes is
in the same order of magnitude as the number of elements. Another important
parameter is the maximal length of a chain of components. Our main contribution
is an exponential lower bound for this parameter. There is a simple
construction for an arbitrary set of generators. However, the proof for
constant alphabet is rather involved. Our results also apply to automata and
their syntactic semigroups.Comment: Full version of a paper submitted to CSR 2017 on 2016-12-1
On certain combinatorial problems of the semigroup of partial and full contractions of a finite chain
Denote[n]=(1,2,...,n) to be finite n-chain, where is a natural number. Let Pn and Tn denote the semigroups of partial and full transformations of [n], respectively. Let CPn=(α â Pn:|xα-yα|â€|x-y|âx,y â Dom α) and CTn = (α â Tn:|xα-yα|â€|x-y| âx,y â [n], then CPn and CTn are known to be subsemigroup of Pn and Tn, respectively. The algebraic properties of these semigroup have been investigated, however the combinatorial properties are yet to be investigated. In this paper, combinatorial problems (or questions) of these subsemigroups where explored. Let DCPn =(α â DPn: |xα - yα|â€|x-y|âx,y â Dom α) and DCTn = (α â DTn: |xα - yα|â€|x-y|âx,y â [n]) (where DPn and DTn are the semigroup of order decreasing partial and full transformations, respectively. ) Then DCPn and DCTn are known to be the semigroup of order decreasing partial and full contractions, respectively. In this paper we give a necessary and sufficient conditions for an element to be regular for the semigroups DCPn and DCTnKeywords: Transformation semigroup, Contractions, Number of fixed point, equivalence
The structure of End()
The full transformation semigroups , where ,
consisting of all maps from a set of cardinality to itself, are arguably
the most important family of finite semigroups. This article investigates the
endomorphism monoid End() of . The determination
of the elements of End() is due Schein and Teclezghi.
Surprisingly, the algebraic structure of End() has not been
further explored. We describe Green's relations and extended Green's relations
on End(), and the generalised regularity properties of these
monoids. In particular, we prove that (with equality if and only if ); the
idempotents of End() form a band (which is equal to
End() if and only if ) and also the regular elements of
End() form a subsemigroup (which is equal to
End() if and only if ). Further, the regular elements
of End() are precisely the idempotents together with all
endomorphisms of rank greater than . We also provide a presentation for
End() with respect to a minimal generating set
Towards an Effective Theory of Reformulation
This paper describes an investigation into the structure of representations of sets of actions, utilizing semigroup theory. The goals of this project are twofold: to shed light on the relationship between tasks and representations, leading to a classification of tasks according to the representations they admit; and to develop techniques for automatically transforming representations so as to improve problem-solving performance. A method is demonstrated for automatically generating serial algorithms for representations whose actions form a finite group. This method is then extended to representations whose actions form a finite inverse semigroup
Permutations of a semigroup that map to inverses
We investigate the question as to when the members of a finite regular semigroup may be permuted in such a way that each member is mapped to one of its inverses. In general this is not possible. However we reformulate the problem in terms of a related graph and, using an application of Hallâs Marriage Lemma, we show in particular that the finite full transformation semigroup does enjoy this property
Computational techniques in finite semigroup theory
A semigroup is simply a set with an associative binary operation; computational semigroup theory is the branch of mathematics concerned with developing techniques for computing with semigroups, as well as investigating semigroups with the help of computers. This thesis explores both sides of computational semigroup theory, across several topics, especially in the finite case.
The central focus of this thesis is computing and describing maximal subsemigroups of finite semigroups. A maximal subsemigroup of a semigroup is a proper subsemigroup that is contained in no other proper subsemigroup. We present novel and useful algorithms for computing the maximal subsemigroups of an arbitrary finite semigroup, building on the paper of Graham, Graham, and Rhodes from 1968. In certain cases, the algorithms reduce to computing maximal subgroups of finite groups, and analysing graphs that capture information about the regular â-classes of a semigroup. We use the framework underpinning these algorithms to describe the maximal subsemigroups of many families of finite transformation and diagram monoids. This reproduces and greatly extends a large amount of existing work in the literature, and allows us to easily see the common features between these maximal subsemigroups.
This thesis is also concerned with direct products of semigroups, and with a special class of semigroups known as Rees 0-matrix semigroups. We extend known results concerning the generating sets of direct products of semigroups; in doing so, we propose techniques for computing relatively small generating sets for certain kinds of direct products. Additionally, we characterise several features of Rees 0-matrix semigroups in terms of their underlying semigroups and matrices, such as their Green's relations and generating sets, and whether they are inverse. In doing so, we suggest new methods for computing Rees 0-matrix semigroups
Algebraic hierarchical decomposition of finite state automata : a computational approach
The theory of algebraic hierarchical decomposition of finite state automata
is an important and well developed branch of theoretical computer science
(Krohn-Rhodes Theory). Beyond this it gives a general model for some
important aspects of our cognitive capabilities and also provides possible
means for constructing artificial cognitive systems: a Krohn-Rhodes decomposition
may serve as a formal model of understanding since we comprehend
the world around us in terms of hierarchical representations. In order to
investigate formal models of understanding using this approach, we need
efficient tools but despite the significance of the theory there has been no
computational implementation until this work.
Here the main aim was to open up the vast space of these decompositions
by developing a computational toolkit and to make the initial steps of the
exploration. Two different decomposition methods were implemented: the
VuT and the holonomy decomposition. Since the holonomy method, unlike
the VUT method, gives decompositions of reasonable lengths, it was chosen
for a more detailed study.
In studying the holonomy decomposition our main focus is to develop
techniques which enable us to calculate the decompositions efficiently, since
eventually we would like to apply the decompositions for real-world problems.
As the most crucial part is finding the the group components we
present several different ways for solving this problem. Then we investigate
actual decompositions generated by the holonomy method: automata with
some spatial structure illustrating the core structure of the holonomy decomposition,
cases for showing interesting properties of the decomposition
(length of the decomposition, number of states of a component), and the
decomposition of finite residue class rings of integers modulo n.
Finally we analyse the applicability of the holonomy decompositions as
formal theories of understanding, and delineate the directions for further
research
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