# On the lattice of sub-pseudovarieties of DA

@article{Kufleitner2009OnTL, title={On the lattice of sub-pseudovarieties of DA}, author={Manfred Kufleitner and Pascal Weil}, journal={Semigroup Forum}, year={2009}, volume={81}, pages={243-254} }

The wealth of information that is available on the lattice of varieties of bands, is used to illuminate the structure of the lattice of sub-pseudovarieties of DA, a natural generalization of bands which plays an important role in language theory and in logic. The main result describes a hierarchy of decidable sub-pseudovarieties of DA in terms of iterated Mal’cev products with the pseudovarieties of definite and reverse definite semigroups.

## 13 Citations

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- 2012

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### The Word Problem for Omega-Terms over the Trotter-Weil Hierarchy

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It is shown that separability by the so-called corners of the Trotter-Weil Hierarchy is witnessed by ω-terms (this property is also known as ψ-reducibility).

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Polynomial closure is a standard operator which is applied to a class of regular languages. In this paper, we investigate three restrictions called left (LPol), right (RPol) and mixed polynomial…

### Quantifier Alternation in Two-Variable First-Order Logic with Successor Is Decidable

- Mathematics, Computer ScienceSTACS
- 2013

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### The Word Problem for Omega-Terms over the Trotter-Weil Hierarchy - (Extended Abstract)

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- 2016

The approach yields nondeterministic logarithmic space $$\text {NL}$$ and deterministic polynomial time algorithms, which are more efficient than straightforward translations of the NL-algorithms.

## References

SHOWING 1-10 OF 30 REFERENCES

### Complete endomorphisms of the lattice of pseudovarieties of finite semigroups

- MathematicsBulletin of the Australian Mathematical Society
- 1997

The main result established that the mapping V → V ∩ W (V ∈ ℒ(F)) is a complete endomorphism of the lattice ℒ(F) of pseudovarieties of finite semigroups for certain particular pseudovarieties W,…

### The lattice of pseudovarieties of idempotent semigroups and a non-regular analogue

- Mathematics
- 1997

Abstract. We use classical results on the lattice
$ \cal L (\cal B) $ of varieties of band (idempotent) semigroups to obtain information on the structure of the lattice Ps (DA) of subpseudovarieties…

### Profinite Methods in Semigroup Theory

- MathematicsInt. J. Algebra Comput.
- 2002

The contribution of profinite methods and the way they enriched and modified finite semigroup theory are surveyed.

### All varieties of bands

- Mathematics
- 1970

The object of this paper is to describe the lattice of varieties of bands, and to present some related results. For example, each variety of bands is defined by exactly one identity from a countably…

### A Characterization of a Dot-Depth Two Analogue of Generalized Definite Languages

- Computer Science, LinguisticsICALP
- 1979

This work examines noncounting languages and their connection to G-trivial languages, a relationship analogous to the one between generalized definite languages and the definite and reverse definite languages.

### Logic Meets Algebra: the Case of Regular Languages

- Computer ScienceLog. Methods Comput. Sci.
- 2007

This work surveys the existing results relating the expressibility of regular languages in logical fragments of MSO[S] with algebraic properties of their minimal automata and shows that many of the best known results share the same underlying mechanics and rely on a very strong relation between logical substitutions and block-products of pseudovarieties of monoid.

### Algebraic theory of machines, languages and semigroups

- Mathematics
- 1968

Abstract : The book is an integrated exposition of the algebraic, and especially semigroup-theoretic, approach to machines and languages. It is designed to carry the reader from the elementary theory…

### Profinite Semigroups, Mal'cev Products, and Identities☆

- Mathematics
- 1996

Abstract We compute a set of identities defining the Mal'cev product of pseudovarieties of finite semigroups or finite ordered semigroups. We also characterize the pointlike subsets of a finite…

### Finite Semigroups and Universal Algebra

- Mathematics
- 1995

Part 1 Finite universal algebra: elements of universal algebra order and topology finite algebras decidability. Part 2 Finite semigroups and monoids: permutativity operators relating semigroups and…