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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Congruences associated with DOL-schemes


Authors: Mario Petrich and Gabriel Thierrin
Journal: Proc. Amer. Math. Soc. 102 (1988), 787-793
MSC: Primary 68Q45; Secondary 20M05, 20M35
MathSciNet review: 934843
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Abstract: For a DOL-scheme $ (X,\varphi )$, where $ X$ is a finite alphabet and $ \varphi $ is an endomorphism of $ {X^*}$, we study the properties of the congruence $ \bar \varphi $ induced by $ \varphi $ in terms of the properties of $ {X^*}\varphi $. We prove that every submonoid of $ {X^*}$ has a disjunctive subset (for any $ X$) and deduce that $ \bar \varphi $ is a syntactic congruence. As special cases, we consider the conditions on $ \varphi $ which are equivalent to $ \bar \varphi $ being perfect or uniquely perfect or linear. The latter is introduced in the paper together with a ramification.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0934843-7
PII: S 0002-9939(1988)0934843-7
Keywords: Free monoid, endomorphism, DOL-scheme, syntactic congruence, perfect congruence, linear congruence, code, hypercode
Article copyright: © Copyright 1988 American Mathematical Society