Semigroups are very natural and general structures and enter our mathematical life from the very beginning (N with respect to addition, multiplication, maximum or minimum, sets with respect to union or intersection). Due to their simple axioms, they are very often and easily found. If ???? is a nonempty set and ???? = ????M is the set of all mappings from ???? ???????? ????, then ???? is a semigroup with respect to composition, and in fact every semigroup can be realized as a subsemigroup of ????M from ????. The class of semigroups being so vast it might seem reasonable to limit somehow this generality. We shall do this in a very specific way by looking primarily to abelian semigroups. On the other hand, in many situations the semigroups we meet carry a natural involution , i.e. a mapping -: ???? → ???? such that (????????) -= ???? -????- and (????-)-= ????. On a group we always have the group involution ????-=????-1. As another example (used later on) consider the upper half –plane with usual addition and
(????, ????)-≔ (−????, ????). Contrary to the group case, the existence of a neutral element is not automatic in semigroups. Although there are important instances where this nonexistence is natural, we will – if not explicity told otherwise – tacitly assume
that the semigroups considered do have a neutral element. So a semigroup for us will be a triple ???? = (????, +,-
) where S is a (nonempty) set, ‘’+’’ is a commutative semigroup operation , and ‘’−‘’ is an involution on S (very often -= id in which case we simply write ???? = (????, +)). The neutral element will be denoted by ‘’0’’. The simplest functions on a semigroup ‘’respecting’’ the semigroup structure
are certainly the scalar – valued homomorphisms, i.e. the characters. They are
the building stones for many other important function families , such as
positive or negative definite functions, and some related classes as well. An
excellent survey on this topic from an analytical point of view has recently been
given by C. Berg [1] and is strongly recommended as a supplement to the
present paper.
When mentioning semigroups in connection with probability theory,the first
thought perhaps goes to one –parameter semigroups and infinitely divisible
distributions. This is certainly an important area of research and of beautiful
results, but it is by no means the only nontrivial occurrence of semigroups, as
will be demonstrated in the sequel. Notions from Harmonic Analysis on
semigroups play a dominant (and sometimes perhaps unexpected) role here,
too.
Semigroups can be used in biology to describe certain aspects in the crossing of
organisms , in genetics and in consideration of metabolisms. The growth of
plants can be described algebraically in Hermann and Rosenberg (1975).
Further material on this subject is contained in Holcombe (1982). Details on the
use of semiautomata in metabolic pathways and the aid of a computer therein,
including a theory of scientific experiments, can be found in Krohn , Langer and
Rhodes (1976). Rosen (1973) studies ways in which environmental changes can
affect the repair capacity of biological systems and considers carcinogenesis
and reversibility problems. Language theory is used in cell- development
problems, as introduced by Lindenmayes (1968), Hermann and Roscendalg
(1975). Suppose (1969) Kiesras (1976) develop a theory of learning in which a
subject is instructed to behave like a semiautomaton.
II. Semigroups and its important Applications
Now a days the theory of semigroups has been expanded greatly due to its
applications to computer science and we also finds its usage in biology science
and Partial Differential Equation. In this section we discuss some important
applications of semigroups in different areas.
Automaton-Semigroups.
A finite-state machine (????????????) or finite-state automaton (????????????,
plural:automata), finite automaton, or simply a state machine, is a
mathematical model of computation. It is an abstract machine that can be in
exactly one of a finite number of states at any given time. The FSM can change
from one state to another in response to some inputs; the change from one
state to another is called transition. An FSM is defined by a list of its states, its
initial state , and the inputs that trigger each transition. Finite – state machines
are of two types deterministic finite-state machines and non-deterministic
finite-state machines. A deterministic finite-state machine can be constructed
equivalent to any non-deterministic one.
The behavior of state machines can be observed in many devices in modern
society that perform a predetermined sequence of actions depending on a
sequence of events with ehich they are presented. Simple examples are vending
machines, which dispense products when the proper combination of coins is
deposited elevators, whose sequence of stops is determined by the floors
requested by riders, traffic lights. Which change sequence when cars are
waiting , and combination locks. Which require the input of a sequence of
numbers in the proper order.
The finite-state machine has less computational power than some other models
of computation such as the Turing machine.The computational power
distinction means there are aomputational tasks that a Turing machine can do
but an FSM cannot . This is because an FSM’s memory is limited by the number
of states it has FSM’s are studied in the more general field of automata theory.
Automata:
Automata have been defined as models of systems reacting upon sequences of
stimuli. Computers are among the most common examples of automata. Living
organisms, many types of machines, information transmission channels,
complex systems as e.g. traffic are further typical examples.
An automaton A consists of
(????) ????h???? set ???? ≠???? of elementary stimuli, called input symbols,
(????) ????h???? memory S, considered as a nonempty set, the set of states, which may
have an algebraical or topological structure,
(????) ????h???? set ???? ≠???? of elementary reactions, called output symbols,
(????) ????h???? function ????: ???? × ???? → ???? × ????, which to each input symbol ???? ∈ ???? and each
state ???? ∈ ???? associaties by ????(????, ????) = (????
′
, ????) the next state ????
’ and the
corresponding output symbol y.
Semigroups – Formal Languages.
Besides the ‘’natural’’ languages such as English, German , French or Chinese,
the so-called formal languages are of importance in the formal sciences,
especially in computing or information sciences. The underlying principle is
always the same: Given an ‘’alphabet’’ or ‘’vocabulary’’ ( consisting of letters,
words , punctuation symbols, number symbols, programming instructions ,etc.)
one has a method (‘’grammar’’) for constructing ‘’meaningful’’ words or
sentences (i.e. the ‘’language’’) from this alphabet. This immediately reminds
us of the term
‘’word semigroup’’ and , indeed, these free semigroups will play a major role,
the language ???? constructed will be a subset of the free semigroup ????A
on the alphabet ????.
There are essentially three ways to construct a language ????.
(????) Approach via grammar: given a collection rules (‘’grammar’’) , generate ????
from .
(????) Approach via automata: consider an initial semiautomaton which processes
the elements of ???? in a suitable way.
(????) Algebraic approach: ???? is constructed by the algerbraic combination of
certain subsets of ????A .
In all three approaches we shall use algebraic methods. First we study the
different approaches, then investigate connections between them. The whole
theory is part of mathematical linguistics, which also uses probabilistic
methods. We use the following notation : ????* is the free semigroup over a set ????,
????
* is the free monoid over ????, so ????
*= ????*∪ {∧} , where ∧ denotes the empty word.
Semigroups in Biology.
The general approach that allows to construct the Markov processes describing
various processes in mathematical biology (or in other applied sciences) is
presented. The Markov processes are of a jump type and the starting point is
the related linear equations. They describe at the micro-scale level the behavior
of a large number ???? of interacting individuals (entities). The large individual
limit
(′′???? → ∞′′) is studied and the intermediate level (the meso-scale level) is given
in terms of nonlinear kinetic-type equations. Finally the corresponding systems
of nonlinear ODEs (or PDEs) at the macroscopic level (in terms of densities of
the interacting subpopulations) are obtained. Mathematical relationships
between these three possible descriptions are presented and explicit error
estimates are given. The general framework is applied to propose the
microscopic and mesoscopic models that correspond to well known systems of
nonlinear equations in biomathematics.
Semigroups in Probability Theory.
One of the most fundamental operations im probability theory is convolution,
and a large part of classical probability theory deals with ????+
1
(R), the
probability distributions on the real line, being a (commutative) semigroup with
respect to convolution and carrying the natural involution ????
–
(????) ≔ ????(−????). Two
basic procedures of obtaining new distributions from given ones are finite
convolution products and limits of such finite convolutions (in the weak
topology). Both these operations are in general difficult to perform by direct
calculation. As an aid of great importance we have the classical Fourier
transformation
????: ????+
1
(R) → ????
R
???? ↦ (???? ↦ ∫ ????
itx????????(????))
being a semigroup isomorphism and also a homeomorphism onto its images, if
we consider ????
R with pointwise operations.
Semigroups in Partial differential equations.
Semigroup theory can be used to study some problems in the field of partial
differential equations. Roughly speaking , the semigroup approach is to regard
a time-dependent partial differential equation as an ordinary differential
equation on a function space. For example , consider the following initial/
boundary value problem for the heat equation on the spatial interval (0,1) ⊂ ????
and times ???? ≥ 0.
????????????(????,????) = ????????????(????,????) ,???? ∈ (0,1),???? > 0 {????(????, ????) = 0 , ???? ∈ {0,1}, ???? > 0
????(????.????)=????(????), ????∈(0,1),????=0
Let ???? = ????
2
((0,1)????) be the ????
p space of square –integrable real-valued functions
with domain the interval (0,1) and let A be the second –derivative operator
with domain ????(????) = {???? ∈ ????2
((0,1); ????)|????(0) = ????(1) = 0},
where ????2 is a Sobolev space . Then the above initial/boundary value problem
can be interpreted as an initial value problem for an ordinary differential
equation on the space X.
{
????(????) = ̇????????(????); ????(0) = ????
On an heuristic level, the solution to this problem ‘’ought’’ to be ????(????) =
exp(????????) ????0 . However , for a rigorous treatment, a meaning must be given to
the exponential of tA. As a function of t, exp(tA) is a semigroup of operators
from X to itself , taking the initial state u0 at time t=0 to the state u(t)=exp(tA)u0
at time t. The operator A is said to be the infinitesimal.
