Finite bands are finitely related

Dolinka, Igor

doi:10.1007/s00233-017-9897-y

Cited by 4 publications

(5 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The classification of finitely related algebras generating congruence distributive varieties [3] and eventually, congruence modular varieties [4], were landmark contributions in this regard. For semigroups, extensive characterizations of finite relatedness were presented in [8,13,9]. We build upon these foundational contributions in the present paper.…”

Section: Introductionmentioning

confidence: 94%

See 1 more Smart Citation

Not all nilpotent monoids are finitely related

Steindl

2024

Algebra Univers.

View full text Add to dashboard Cite

A finite semigroup is finitely related (has finite degree) if its term functions are determined by a finite set of finitary relations. For example, it is known that all nilpotent semigroups are finitely related. A nilpotent monoid is a nilpotent semigroup with adjoined identity. We show that every 4-nilpotent monoid is finitely related. We also give an example of a 5-nilpotent monoid that is not finitely related. To our knowledge, this is the first example of a finitely related semigroup where adjoining an identity yields a semigroup which is not finitely related. We also provide examples of finitely related semigroups which have subsemigroups, homomorphic images, and in particular Rees quotients, that are not finitely related.

show abstract

Section: Introductionmentioning

confidence: 94%

“…He also gave the first known example of a semigroup that is not finitely related, the 6-element Brandt monoid. In [9] Dolinka showed that all finite bands are finitely related.…”

Section: Introductionmentioning

confidence: 99%

Not all nilpotent monoids are finitely related

Steindl

2024

Algebra Univers.

View full text Add to dashboard Cite

show abstract

“…In this paper we investigate term functions of semigroups and the relations that determine them. We continue the work started in [7], [12], and [8].…”

Section: Introductionmentioning

confidence: 96%

“…He also gave the first known example of a semigroup that is not finitely related, the 6-element Brandt Monoid. Dolinka [8] showed that all finite bands are finitely related.…”

Section: Introductionmentioning

confidence: 99%

Not all nilpotent monoids are finitely related

Steindl¹

2022

Preprint

View full text Add to dashboard Cite

A finite semigroup is finitely related (has finite degree) if its term functions are determined by a finite set of finitary relations. For example, it is known that all nilpotent semigroups are finitely related. A nilpotent monoid is a nilpotent semigroup with adjoined identity. We show that every 4-nilpotent monoid is finitely related. We also give an example of a 5-nilpotent monoid that is not finitely related. This is the first known example where adjoining an identity to a finitely related semigroup yields a semigroup which is not finitely related. We also provide examples of finitely related semigroups which have subsemigroups, homomorphic images, and in particular Rees quotients, that are not finitely related.

show abstract

“…, e n ∈ A(M) m−1 (on coordinates L ∪ {p}) as in Lemma 8.15 so that e i (L) = g i (L) for all i, but the e i have an "extra" row p ∈ [m]. Since g j (ℓ 2 ) ∈ X ∪D and g j (ℓ 1 ) ∈ X ∪D, from the description of e j in Lemma 8 15. we have e j (p) ∈ X.Let I = {e 1 , .…”

mentioning

confidence: 99%

Finite degree clones are undecidable

Moore

2019

Theoretical Computer Science

View full text Add to dashboard Cite

A clone of functions on a finite domain determines and is determined by its system of invariant relations (=predicates). When a clone is determined by a finite number of relations, we say that the clone is of finite degree. For each Minsky machine M we associate a finitely generated clone C such that C has finite degree if and only if M halts, thus proving that deciding whether a given clone has finite degree is impossible.

show abstract

Finite bands are finitely related

Abstract: Abstract. We prove that every finite idempotent semigroup (band) is finitely related, which means that the clone of its term operations (i.e. operations induced by words) is determined by finitely many relations. This solves an open problem posed by Peter Mayr in 2013.

Cited by 4 publications

References 18 publications

Not all nilpotent monoids are finitely related

Not all nilpotent monoids are finitely related

Not all nilpotent monoids are finitely related

Finite degree clones are undecidable

Contact Info

Product

Resources

About