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Combinatorial and decidability questions in semigroups of words

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Edmund Robertson

## Abstract

Let A be a finite alphabet and let F(A) be the set of all words in A.
F(A) becomes a semigroup by defining multiplication as concatenation
of words. F(A) is called the free semigroup on A. Given a finite
subset B of F(A), let T be the smallest subset of F(A) containing B
which is closed under multiplication. We show that T need not be a
free semigroup and examine questions relating to how close T is to
being free. A general T may be free or may not be finitely
presented. By way of contrast, if T is an ideal of S then T is never
free but is always finitely presented.