Finite basis problem for 2-testable monoids

Lee, Edmond W. H.

doi:10.2478/s11533-010-0080-x

Cited by 8 publications

(4 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Lemma 6. 8. Let u and v be any words in canonical form such that the identity u ≈ v is satisfied by the semigroup T m,n .…”

Section: Proofmentioning

confidence: 99%

“…(5) ℓ is finite meet irreducible (fmi) if, for any finite F ⊆ L , ℓ ≥ F =⇒ ℓ ≥ x for some x ∈ F ; (6) ℓ is strictly join irreducible (sji) if, for any set X ⊆ L , ℓ = X =⇒ ℓ ∈ X ; (7) ℓ is strictly finite join irreducible (sfji) if, for any finite F ⊆ L , ℓ = F =⇒ ℓ ∈ F ; (8) ℓ is strictly meet irreducible (smi) if, for any X ⊆ L , ℓ = X =⇒ ℓ ∈ X ;…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Join irreducible semigroups

Lee

Rhodes

Steinberg

2019

Int. J. Algebra Comput.

Self Cite

View full text Add to dashboard Cite

We begin a systematic study of those finite semigroups that generate join irreducible members of the lattice of pseudovarieties of finite semigroups, which are important for the spectral theory of this lattice. Finite semigroups S that generate join irreducible pseudovarieties are characterized as follows: whenever S divides a direct product A × B of finite semigroups, then S divides either A n or B n for some n ≥ 1. We present a new operator V → V bar that preserves the property of join irreducibility, as does the dual operator, and show that iteration of these operators on any nontrivial join irreducible pseudovariety leads to an infinite hierarchy of join irreducible pseudovarieties.We also describe all join irreducible pseudovarieties generated by a semigroup of order at most five. It turns out that there are 30 such pseudovarieties, and there is a relatively easy way to remember them. In addition, we survey most results known about join irreducible pseudovarieties to date and generalize a number of results in Chapter 7.3

show abstract

“…Lemma 6. 8. Let u and v be any words in canonical form such that the identity u ≈ v is satisfied by the semigroup T m,n .…”

Section: Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Join irreducible semigroups

Lee

Rhodes

Steinberg

2019

Int. J. Algebra Comput.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Now if i ě 3 and x i is nonlinear in w 1 , then equation ( 4) can be used to replace this power by 4. Similarly if the power of x 2 is 3, then equation (5) shows that it can be raised to 4. If the power of x 2 is 2 and the power of x 1 is not 1, then equation (6) shows that x 2 may be raised to the power 4.…”

Section: Compact Smi Pseudovarietiesmentioning

confidence: 99%

On the atoms of algebraic lattices arising in 𝔮-theory

Egri-Nagy

Jackson

Rhodes

et al. 2017

Int. J. Algebra Comput.

View full text Add to dashboard Cite

We determine many of the atoms of the algebraic lattices arising in q-theory of finite semigroups.Caution 3.4. In an algebraic lattice L, AtomspLq can be empty.Corollary 3.5. If L has no atoms, then rB, ℓ 2 s has no atoms.Remark 3.6. In an algebraic lattice L, the atoms of rℓ 1 , Ts are the covers of ℓ 1 in L, so in general, they are unrelated to AtomspLq.Principle 3.7 (No atoms for L, an algebraic lattice). If each compact element c ‰ B has a compact element other than B strictly below, then AtomspLq " ∅, and conversely, since the atoms are the compact covers.Principle 3.7 was used in [9, Proposition 7.1.24] to prove the following proposition.Proposition 3.8. The algebraic lattice CntpPVq has no atoms.As a consequence, we can prove that CC has no atoms.Proposition 3.9. The algebraic lattice CC has no atoms.Proof. By [9, Theorem 2.3.9], there is a surjective map q : CC Ñ CntpPVq preserving all sups and finite meets. The bottom of CntpPVq is the constant map to the trivial pseudovariety. In [9, Page 121] it is shown that each constant map has a unique preimage under q, hence if R is not the bottom of CC, then is does not map to the bottom of CntpPVq under q. Since CntpPVq has no atoms, we can find B ‰ α ă Rq. By surjectivity, there exists S with Sq " α. Since q preserves finite infs, we obtain pR X Sqq " α and so R X S ă R and R X S is not the bottom. Thus CC has no atoms.The reader is referred to [9, Proposition 2.1.11] for the definition of CC and [9, Page 75] for the definition of PVRM. Fact 3.10. If D denotes the class of all divisions, then( Proof. One way of calculating CC isCCpXq " tf | f Ď s d 1 pf 1ˆ¨¨¨ˆfn qd 2 , d 1 , d 2 P D, f i P Xu (see Proposition 2.1.14 in [9]) from which (1) follows. Also CCp1 V | V P PVq is closed under Axiom (co-re), as D is, so is a pseudovariety of relational morphisms in PVRM (see Proposition 2.1.8(c) in [9]) proving (2).Definition 3.11.CntpPVq´" tα P CntpPVq | α ď 1 PV u CC´" tβ P CC | β ď Du Fact 3.12. pCC´qq " CntpPVqṔ roof. Since Dq " 1 PV and q is order preserving pCC´qq Ď CntpPVq´. If α P CntpPVq´, then since α P CntpPVq, there exists R P CC, so Rq " α. Thus since q preserves finite intersections (intersection equals meet) pR X Dqq " α and R X D P CC´.Corollary 3.13. The lattices CC, CC´, CntpPVq, CntpPVq´have no atoms.

show abstract

“…Now the theorem follows immediately from Lemmas 3.4, 4.1, and 4.4-4.8. L 4.10 [7,Theorem 3.3]. Any variety that satisfies the identities x 2 ≈ x 3 , x 2 yx ≈ xyx, xyx 2 ≈ xyx, xyxzx ≈ xyzx, (4.13) is finitely based.…”

Section: The Maximal Subvarieties Of Var T T T 2 (F)mentioning

confidence: 99%

On the Variety Generated by the Monoid of Triangular 2×2 Matrices Over A two-Element field

Zhang

Luo³

2012

Bull. Aust. Math. Soc.

View full text Add to dashboard Cite

Let T n (F) denote the monoid of all upper triangular n × n matrices over a finite field F. It has been shown by Volkov and Goldberg that T n (F) is nonfinitely based if |F| > 2 and n ≥ 4, but the cases when |F| > 2 and n = 2, 3 or when |F| = 2 have remained open. In this paper, it is shown that the monoid T 2 (F) is finitely based when |F| = 2, and a finite identity basis for it is given. Moreover, all maximal subvarieties of the variety generated by T 2 (F) with |F| = 2 are determined.2010 Mathematics subject classification: primary 20M07.

show abstract

Finite basis problem for 2-testable monoids

Cited by 8 publications

References 21 publications

Join irreducible semigroups

Join irreducible semigroups

On the atoms of algebraic lattices arising in 𝔮-theory

On the Variety Generated by the Monoid of Triangular 2×2 Matrices Over A two-Element field

Contact Info

Product

Resources

About