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Representation in mathematics
"A semigroup is defined to be set equipped with an associative binary
operation. A good example of a semigroup is provided by the set of
all binary strings; any two such strings can be composed by
concatenation to form a third binary string, an operation which is
clearly associative. This example also illustrates an important
feature of modern semigroup theory. Binary strings form the input to
computers: any program can be regarded as an, albeit extremely
lengthy, binary string. The collection of all syntactically correct
programs is then a subset of the set of all binary strings. In
mathematical terms, the semigroup of all binary strings is the free
semigroup on two generators, subsets of the free monoid are
languages, and the computer is an example of a finite state machine.
The relationship between semigroups, languages and automata is one
of the most important aspects of contemporary semigroup theory."
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