Journal ArticleDOI
The Birkhoff theorem for finite algebras
TLDR
In this article, a finite analogue of the Birkhoff variety theorem is proved: a non-void class of finite algebras of a finite type τ is closed under the formation of finite products, subalgesas and homomorphic images if and only if it is definable by equations for implicit operations, that is, roughly speaking, operations which are not necessarily induced by τ-terms but which are compatible with all homomorphisms.Abstract:
A finite analogue of the Birkhoff variety theorem is proved: a non-void class of finite algebras of a finite type τ is closed under the formation of finite products, subalgebras and homomorphic images if and only if it is definable by equations for implicit operations, that is, roughly speaking, operations which are not necessarily induced by τ-terms but which are compatible with all homomorphisms. It is well-known that explicit operations (those induced by τ-terms) do not suffice for such an equational description. Topological aspects of implicit operations are considered. Various examples are given.read more
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Book
The q-theory of Finite Semigroups
John Rhodes,Benjamin Steinberg +1 more
TL;DR: The q-theory of finite semigroups as mentioned in this paper is a theory that provides a unifying approach to finite semigroup theory via quantization, and it is the only contemporary exposition of the complete theory of the complexity of finite semiigroups.
Book
Syntactic semigroups
TL;DR: HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not, for teaching and research institutions in France or abroad, or from public or private research centers.
Journal ArticleDOI
Polynomial closure and unambiguous product
Jean-Eric Pin,Pascal Weil +1 more
TL;DR: It is shown that if a language and its complement are in the polynomial closure of a variety of languages, then this language can be written as a disjoint union of marked unambiguous products of languages of the variety.
Book ChapterDOI
Duality and Equational Theory of Regular Languages
TL;DR: This paper presents a new result in the equational theory of regular languages, which emerged from lively discussions between the authors about Stone and Priestley duality, and shows for instance that any class ofregular languages defined by a fragment of logic closed under conjunctions and disjunctions admits an equational description.
Journal ArticleDOI
Ash's type ii theorem, profinite topology and malcev products: part i
TL;DR: In this paper, it was shown that the type II conjecture is equivalent with two other conjectures on the structure of closed sets (one conjecture for the free monoid and another one for free groups).
References
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Automata, Languages, and Machines
TL;DR: This book attempts to provide a comprehensive textbook for undergraduate and postgraduate mathematicians with an interest in formal languages and automata, written by Professor Ian Chiswell.
Journal ArticleDOI
On the Structure of Abstract Algebras
Garrett Birkhoff,P. Hall +1 more
TL;DR: In this paper, a study of abstract algebra qua abstract algebras is presented, where no vocabulary suitable for this purpose is current, and they have been forced to use a number of new terms, and extend the meaning of some accepted ones.