# Profinite Methods in Semigroup Theory

@article{Weil2002ProfiniteMI, title={Profinite Methods in Semigroup Theory}, author={Pascal Weil}, journal={Int. J. Algebra Comput.}, year={2002}, volume={12}, pages={137-177} }

Many recent results in finite semigroup theory make use of profinite methods, that is, they rely on the study of certain infinite, compact semigroups which arise as projective limits of finite semigroups. These ideas were introduced in semigroup theory in the 1980s, first to describe pseudovarieties in terms of so-called pseudo-identities: this is Reiterman's theorem, which can be viewed as the (much more complex) finite algebra analogue of Birkhoff's variety theorem. Soon, these methods were…

## 41 Citations

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## References

SHOWING 1-10 OF 159 REFERENCES

Syntactic and Global Semigroup Theory: A Synthesis Approach

- Mathematics
- 2000

This paper is the culmination of a series of work integrating syntactic and global semigroup theoretical approaches for the purpose of calculating semidirect products of pseudovarieties of…

GLOBALS OF PSEUDOVARIETIES OF COMMUTATIVE SEMIGROUPS: THE FINITE BASIS PROBLEM, DECIDABILITY AND GAPS

- MathematicsProceedings of the Edinburgh Mathematical Society
- 2001

Abstract Whereas pseudovarieties of commutative semigroups are known to be finitely based, the globals of monoidal pseudovarieties of commutative semigroups are shown to be finitely based (or of…

Free Profinite ℛ-Trivial Monoids

- MathematicsInt. J. Algebra Comput.
- 1997

The structure of semigroups of implicit operations on R, the pseudovariety of all ℛ-trivial semig groups, is described by means of labeled ordinals and of labeled infinite trees of finite depth.

PROFINITE IDENTITIES FOR FINITE SEMIGROUPS WHOSE SUBGROUPS BELONG TO A GIVEN PSEUDOVARIETY

- MathematicsJournal of Algebra and Its Applications
- 2003

We introduce a series of new polynomially computable implicit operations on the class of all finite semigroups. These new operations enable us to construct a finite pro-identity basis for the…

INTRODUCTION TO SEMIGROUP THEORY

- Mathematics
- 1993

Any short book on semigroup theory would inevitably be shallow were its scope not limited to certain parts of the theory. This volume concentrates primarily on regular semigroups, for which a well…

The Birkhoff theorem for finite algebras

- Mathematics
- 1982

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…

Inverse Semigroups and Extensions of Groups by Semilattices

- Mathematics
- 2003

This paper is the first part of a series of three papers devoted to the study of inverse semigroups. The subject of our second paper [7] is free inverse semigroups, the third one [S] is dedicated to…

Normal Forms for Free Aperiodic Semigroups

- MathematicsInt. J. Algebra Comput.
- 2001

This article introduces a specific and rather elementary list of pseudoidentitites, and shows that for each n, the n-generated free aperiodic semigroup is defined by this list of Pseudoidentities, and uses this identification to show that it has a decidable word problem.