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Example of monoid $M$ such that $\operatorname{RAT}(M) \not\subseteq \operatorname{REC}(M)$

Let $M$ be a monoid, the family of rational sets $\operatorname{RAT}(M)$ is defined as the smallest set containing the finite subsets, and closed under union, concatentaion and the star operation. The ...
StefanH's user avatar
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9 votes
1 answer
319 views

Generalisation of the statement that a monoid recognizes language iff syntactic monoid divides monoid

Let $A$ be a finite alphabet. For a given language $L \subseteq A^{\ast}$ the syntactic monoid $M(L)$ is a well-known notion in formal language theory. Furthermore, a monoid $M$ recognizes a language $...
StefanH's user avatar
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15 votes
3 answers
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On the realisation of monoids as syntactic monoids of languages

Let $L \subseteq X^{\ast}$ be some language, then we define the syntactic congruence as $$ u \sim v :\Leftrightarrow \forall x, y\in X^{\ast} : xuy \in L \leftrightarrow xvy \in L $$ and the quotient ...
StefanH's user avatar
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9 votes
2 answers
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Formal representation of an abstraction hierarchy

Introduction I'm writing my PhD thesis on Abstract Delta Modeling (ADM), an abstract algebraic description of modifications (known as deltas) able to act on products (as in 'software products'). This ...
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